Capital Market Theory Rsm 332 – Week 2

CAPITAL MARKET THEORY RSM 332 – Week 2 Week 1 – Introduction – Financial Accounting (Review) Week 2 – Financial Markets and Net Present Value Week 3 – Present Value Concepts Week 4 – Bond Valuation and Term Structure Theory Week 5 – Valuation of Stocks Week 6 – Risk and Return – Problem Set #1 Due Week 7* – Midterm (Tuesday*) Week 8 – Portfolio Theory Week 9 – Capital Asset Pricing Model Week 10 – Arbitrage Pricing Theory Week 11 – Operation and Efficiency of Capital Markets Week 12 – Course Review – Problem Set #2 Due Contact: otto. [email  protected] utoronto. ca CAPITAL MARKET THEORY RSM 332 – Week 2AGENDA 1. 2. 3. 4. 5. Announcements Financial Markets and Net Present Value Survey Results Optional Material (e. g. Cases, Practical Knowledge, News, etc. ) Suggestions/Practice for Exam(s) Contact: otto. [email  protected] utoronto. ca Extended Office Hours Friday , October 19th (11:00am-3:00pm) †¢ Room 6 – TZ6 (Tanz Neuroscience Bldg – 6 Queen’s Park Crescent West) TBD – Saturday, October 20th †¢ Depends if there is enough demand Thursday, October 25th (5:00pm-7:00pm and 7:00pm-9:00pm) †¢ During regular timeslot †¢ Cover optional material (e. g. cases, practical knowledge, etc. ) Contact: otto. [email  protected] utoronto. ca Exams Midterm (Tuesday, October 23rd – 8:00pm-10:00pm): †¢ EX 100 (Examination Facility – 255 McCaul Street) †¢ 2 Hours Final (TBA): †¢ 2 Hours Preparation: †¢ Problem Sets 1 & 2 †¢ Crib Sheet (Start Early and 1-Sided) †¢ Calculator (Silent) Contact: otto. [email  protected] utoronto. ca Tutorials †¢ Starting – September 19/20/21 †¢ Wednesday (6:00pm-8:00pm) †¢ TZ6 (Tanz Neuroscience Bldg – 6 Queen’s Park Crescent West) †¢ Thursday (11:00am-1:00pm) †¢ RW 110 (Ramsay Wright Laborat ories – 25 Harbord Street) †¢ Friday (5:00pm-7:00pm) †¢ RW 110 (Ramsay Wright Laboratories – 25 Harbord Street) Review: †¢ Midterms and Finals (2008-2011) Xiaofei Zhao (xiaofei. [email  protected] utoronto. ca) †¢ http://332ta. raykan. com †¢ Contact: otto. [email  protected] utoronto. ca Outside of Lecture Office Hours (Drop-In): †¢ Wednesdays: 4:00pm-6:00pm †¢ 105 St. George Street – Rotman (North Building) Room 413 or 417 Office Hours (Other Days/Times): †¢ Extended Hours †¢ By Appointment Contact: otto. [email  protected] utoronto. ca Corporate Finance: What is Going On? 3) Firm’s Financial (5) Investors (4) (Financial Institutions, (1) Individuals, Other Firms) (1) (2) (3) (4) (5) Cash raised from investors by selling financial assets Cash invested in real assets (some are intangible) Cash generated by operations Cash reinvested in the firm (retained earnings) Cash repaid to investors (interest, divi dends, etc. ) Operations (2) Decision Maker Reference: Alex MacKay Financial Markets: What is Going On? Firms (Users of Capital) Initial Public Offering (IPO) Secondary Offerings (SEO) Borrowing (Loans, Bonds)Dividends, $ Repurchases, Interest Payments $ Market Mechanisms or Market Makers (Stock Exchanges, Banks, Investment Funds, †¦) $ $ Firms Issue Stock Certificates and Bonds $ $$$ Invested in Stocks and Bonds Investors (Providers of Capital) Investment Banks help firms make transactions Brokers/Dealers help investors make transactions Reference: Alex MacKay Financial Theory and Corporate Policy Chapter 1 (Copeland, Weston and Shastri) Course Reserve FINANCIAL MARKETS AND NET PRESENT VALUE Consumption Plan and Investment RuleConsider 1 period problems Assumptions: †¢ No uncertainty †¢ One period (two dates), consumptions occur on date 0 and date 1 †¢ A consumer is endowed with initial wealth (Y0) on date 0, and will receive income (Y1) on date 1 †¢ Simple interest rate (r) Date 0 Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Date 1 Consumption Plan and Investment Rule 4 CASES †¢ Case I: †¢ Case II: No Capital Market, No Production Opportunities With Capital Market, No Production Opportunities †¢ Case III: No Capital Market, With Production Opportunities †¢ Case IV: With Capital Market, With Production OpportunitiesReference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Consumption and Investment without Capital Markets C1 U2 U1 U0 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment without Capital Markets C1 Slope of the Tangent (-ve) = (Marginal Rate of Substitution) (MRS) MRS = ? C1 ? C0 U1 U(C0, C1) MRS = ? U / ? C0 ?U / ? C1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Poli cy) 4 th Edition 2004 Consumption and Investment without Capital MarketsC1 Production/Investment Opportunity Set C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment without Capital Markets C1 Rate at which a dollar of consumption today (C0) is transformed by productive investment into a dollar of consumption (C1) tomorrow. C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment without Capital Markets C1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment without Capital Markets C1 Marginal Rate of Transformation (MRT) MRT = ? C1 ? C0 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment without Capital Markets C1 U1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment without Capital Markets C1 Y1 U1 Resource Bundle: (Y0, Y1) Y0Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0 Edition 2004 Consumption and Investment without Capital Markets C1 Increase investment until MRT = MRS U1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment without Capital Markets C1 U2 U1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment without Capital MarketsC1 MRT = MRS U2 U1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment without Capital Markets C1 U2 U1 (Increase Investment) Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0 Edition 2004 Consumption and Investment with Capital Markets C1 Slope = -(1+r) Borrowing and Lending opportunities (Capital Market Line) (at market interest rate r) C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets C1 Interest plus Principal (Invest/Lending) Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0 Edition 2004 Consumption and Investment with Capital Markets C1 Interest plus Princip al (Borrowed Amount – Principal) Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0 Edition 2004 Consumption and Investment with Capital Markets C1 U1 C0Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets C1 Y1 U1 Endowment: (Y0, Y1) Y0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0 Edition 2004 Consumption and Investment with Capital Markets C1 (Invest) Y1 U1 Y0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0 Edition 2004 Consumption and Investment with Capital Markets C1Market rate of return > Subjective Time Preference (1+r) > (1+rtime preference) Y1 U1 Y0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0 Edition 2004 Consumption and Investment with Capital Markets C1 U1 (Consume Less) Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0 Edition 2004 Consumption and Investment with Capital Markets C1 U1 (Invest) Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0Edition 2004 Consumption and Investment with Capital Markets C1 U2 Y1 U1 Y0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0 Edition 2004 Consumption and Investment with Capital Markets C1 Market Interest Rate = Subjective Time Preference U2 Y1 U1 Y0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th C0 Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 U1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] toronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 C0 Reference: Copeland, W eston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 U3 = (production and capital market) U2 = (with production alone) U1 = (initial endowment) C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption Plan and Investment RuleConsider 1 period problems Assumptions: †¢ No uncertainty †¢ One period (two dates), consumptions occur on date 0 and date 1 †¢ A consumer is endowed with initial wealth (Y0) on date 0, and will receive income (Y1) on date 1 †¢ Simple interest rate (r) Date 0 Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Date 1 Consumption Plan and Investment Rule 4 CASES †¢ Case I: †¢ Case II: No Capital Market, No Production Opportunities With Capital Market, No Production Opportuni ties †¢ Case III: No Capital Market, With Production Opportunities †¢ Case IV: With Capital Market, With Production Opportunities Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Consumption Plan and Investment Rule CASE I – No Capital Market, No Production Opportunities †¢ Consumer can consume Y0 on date 0, and Y1 on date 1 Date 0 Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Date 1 Consumption Plan and Investment Rule CASE II – With Capital Market, No Production Opportunities The set of consumption plans is broadened 1. 2. Consumer can save from Y0, invests in financial assets, and consumes more on date 1 Borrow against Y1, consume more on date 0, pay back loan with interest on date 1 from Y1, and consume less on date 1 Date 0 Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Date 1 Consumption Plan and Investment Rule CASE II – With Capital Market, No Production Opportunitie s †¢ Denote C0 and C1 as date 0 and date 1 consumption respectively †¢ Constraint on them is: C1 = (Y0 – C0) (1+r) + Y1 Consumption Budget Line (Constraint) C0 + C1 = Y0 + Y1 1+ r 1+ r Y Date 0 Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Date 1 In general, the consumer will be better off with capital markets Consumption Plan and Investment Rule CASE II – With Capital Market, No Production Opportunities Present Value †¢ For any cash flow, C0, C1, define its present value as: PV = C0 + C1 + r †¢ Budget constraint can be restated as: †¢ The present value of consumption equals the present value of income Date 0 Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Date 1 Consumption Plan and Investment Rule CASE II – With Capital Market, No Production Opportunities Example: †¢ Assume an investor has a wealth of $1. 5M on date 0, and will have an income of $0. 55M on date 1 †¢ The intere st rate is 10%. †¢ The present value of total income is: $2M = $1. 5M + $0. 55M (1+ 0. 10) Date 0 Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Date 1 Consumption Plan and Investment RuleCASE III – No Capital Market, With Production Opportunities Physical Investment †¢ Suppose the consumer is also an entrepreneur who identifies a physical investment opportunity †¢ Initial investment requires $0. 5M on date 0 †¢ Return of $0. 85M on date 1 †¢ Should this consumer/investor take this project? †¢ Without a capital market, it depends on her/his utility function Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Consumption Plan and Investment Rule CASE IV – With Capital Market, With Production Opportunities †¢ By investing $0. 5M in a financial asset, receive $0. 55M in return (i. . 10% return) †¢ By investing $0. 5M in a physical asset, receive $0. 85M in return (i. e. 70% return) †¢ Consumer/Investor should take this project †¢ Interest rate is also called the opportunity cost of capital †¢ i. e. Return foregone by investing in a project rather than in comparable investment alternatives Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Consumption Plan and Investment Rule CASE IV – With Capital Market, With Production Opportunities Net Present Value (NPV) †¢ Is the project’s net contribution to wealth (i. e. present value minus initial investment) NPV = C0 + C1 1+ r In the above example, the NPV of the project is: NPV = -$0. 5M + $0. 85M = $0. 2727M (1 + 0. 10) Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca Consumption Plan and Investment Rule CASE IV – With Capital Market, With Production Opportunities NPV Rule †¢ States that: †¢ If a project has a positive NPV, we should accept it †¢ If a project has a negative NPV, we should reject it Equivalent Rules †¢ NPV Rule – Accept positive NPV projects †¢ Rate-of-Return Rule – Invest in projects which offer a rate higher than the cost of capital Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca A Separation Theorem You are at a Honda (HMC) shareholders’ meeting †¢ Three shareholders are quite vocal about what the company should do Shareholder #1 – Old Lady †¢ Wants money right now †¢ Wants HMC to invest in sports cars which will yield a quick profit Shareholder #2 – Representative of a Little Boy’s Trust Fund †¢ Wants money a long way in the future †¢ Wants HMC to invest in building electric cars Shareholder #3 – Young Professional †¢ Wants money at some specified time in future (i. e. 10 years) †¢ Wants HMC to build smaller cars because of an expected oil crisis Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca A Separation TheoremWhat do you think Honda manag ers should do? Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca A Separation Theorem What do you think Honda managers should do? MAXIMIZE VALUE Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca A Separation Theorem In general, each shareholder may want: †¢ Maximum wealth †¢ Ability to transfer wealth across time into consumption †¢ Choose risk characteristics of consumption plan Each shareholder, however, can: †¢ Achieve own consumption plan through investments in financial assets †¢ Achieve risk characteristics of plan by investing in more or less risky securitiesEQUITY (CAPITAL GAINS, DIVIDENDS) Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca A Separation Theorem In general, each shareholder may want: †¢ Maximum wealth †¢ Ability to transfer wealth across time into consumption †¢ Choose risk characteristics of consumption plan Each shareholder, however, can: †¢ Achieve own consumption plan through investments in financial assets †¢ Achieve risk characteristics of plan by investing in more or less risky securities EQUITY (CAPITAL GAINS, DIVIDENDS) DEBT (INTEREST) TAX AGENCY COSTS Reference: Raymond Kan Contact: otto. [email  protected] utoronto. ca A Separation Theorem In general, each shareholder may want: †¢ Maximum wealth †¢ Ability to transfer wealth across time into consumption †¢ Choose risk characteristics of consumption plan Each shareholder, however, can: WHAT TYPE OF INCOME DO YOU PREFER? †¢ Achieve own consumption plan through investments in financial assets †¢ Achieve risk characteristics of plan by investing in more or less risky securities EQUITY (CAPITAL GAINS, DIVIDENDS) DEBT (INTEREST) TAX AGENCY COSTS Reference: Raymond Kan Contact: otto. [email  protected] utoronto. caConsumption and Investment with Capital Markets (With Production Set) C1 U3 = (production and capital market) U2 = (with producti on alone) U1 = (initial endowment) C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 Choose the optimal production decision by taking on projects until the marginal rate of return on investment equals the objective market rate) C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 (Choose the optimal consumption pattern by borrowing or lending along the capital market line to equate your subjective time preference with the market rate of return) C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] toronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 (Production/Investment Decision) (Consumption Decision) C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 (Production/Investment Decision) (Consumption Decision) (Fisher Separation Theorem) C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004Consumption and Investment with Capital Markets (With Production Set) C1 (Fisher Separation Theorem) Given perfect and complete capital markets, the production decision is governed solely by an objective market criterion (represented by maximizing attained wealth ) without regard to individuals’ subjective preferences that enter into consumption decisions C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 (Production/Investment Decision) (Consumption Decision) Fisher Separation Theorem) C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 (Production/Investment Decision) (Consumption Decision) (Fisher Separation Theorem) MRS = MRT = 1+r C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Consumption and Investment with Capital Markets (With Production Set) C1 ALL INDIVIDUALS USE THE SAME TIME VALUE OF MONEY (i. e. ame market interest rate) IN MAKING THEIR PRODUCTION/INVESTMENT DECISIONS (Fisher Separation Theorem) MRS = MRT = 1+r C0 Reference: Copeland, Weston, Shastri (Financial Theory Contact: otto. [email  protected] utoronto. ca and Corporate Policy) 4 th Edition 2004 Example Ronald, a finance student, has $400 cash-on-hand and has $1,100 in trust from his grandmother. Ronald will receive the trust funds next year. (Market interest rate is 10% or trust funds worth $1,000 today). Ronald has 2 mutually exclusive investment opportunities (i. e. AAA and BBB rated investments). Contact: otto. [email  protected] utoronto. caReference: Don Brean Example Ronald, a finance student, has $400 cash-on-hand and has $1,100 in trust from his grandmother. Ronald will receive the trust funds next year. (Market interest rate is 10% or trust funds worth $1,000 today). Ronald has 2 mutually exclusive investment opportunities (i. e. AAA and BBB rated investments). 1. 2. 3. Which investment should Ronal d invest in, AAA or BBB? How much should he invest? If Ronald makes investment describe his cash flows? (i. e. Consumption spending divided equally in present value terms) Contact: otto. [email  protected] utoronto. ca Reference: Don Brean ExampleRonald, a finance student, has $400 cash-on-hand and has $1,100 in trust from his grandmother. Ronald will receive the trust funds next year. (Market interest rate is 10% or trust funds worth $1,000 today). Ronald has 2 mutually exclusive investment opportunities (i. e. AAA and BBB rated investments). 1. Which investment should Ronald invest in, AAA or BBB? Contact: otto. [email  protected] utoronto. ca Reference: Don Brean Example Ronald, a finance student, has $400 cash-on-hand and has $1,100 in trust from his grandmother. Ronald will receive the trust funds next year. (Market interest rate is 10% or trust funds worth $1,000 today).Ronald has 2 mutually exclusive investment opportunities (i. e. AAA and BBB rated investments). 1. Which investment should Ronald invest in, AAA or BBB? Contact: otto. [email  protected] utoronto. ca Reference: Don Brean Example Ronald, a finance student, has $400 cash-on-hand and has $1,100 in trust from his grandmother. Ronald will receive the trust funds next year. (Market interest rate is 10% or trust funds worth $1,000 today). Ronald has 2 mutually exclusive investment opportunities (i. e. AAA and BBB rated investments). 1. Which investment should Ronald invest in, AAA or BBB? 2. How much should he invest? Contact: otto. [email  protected] toronto. ca Reference: Don Brean Example Ronald, a finance student, has $400 cash-on-hand and has $1,100 in trust from his grandmother. Ronald will receive the trust funds next year. (Market interest rate is 10% or trust funds worth $1,000 today). Ronald has 2 mutually exclusive investment opportunities (i. e. AAA and BBB rated investments). 3. If Ronald makes investment describe his cash flows? (i. e. Consumption spending divided equally i n present value terms) Contact: otto. [email  protected] utoronto. ca Reference: Don Brean Example Ronald, a finance student, has $400 cash-on-hand and has $1,100 in trust from his grandmother.Ronald will receive the trust funds next year. (Market interest rate is 10% or trust funds worth $1,000 today). Ronald has 2 mutually exclusive investment opportunities (i. e. AAA and BBB rated investments). 3. If Ronald makes investment describe his cash flows? (i. e. Consumption spending divided equally in present value terms) PV of Wealth = PV of Consumption PV (C0) = PV (C1) (i. e. C0 = C1 / (1+r) ) NPVBBB Ronald’s PV of Wealth = $400 + $1,000 + $87. 27 = $1,487. 27 $1,487. 27 = C0 + C1 / (1+r) = C0 + [C0 (1+r)] / (1+r) C0 = $743. 64 and C1 = $818 Contact: otto. [email  protected] utoronto. ca Reference: Don Brean ExampleRonald, a finance student, has $400 cash-on-hand and has $1,100 in trust from his grandmother. Ronald will receive the trust funds next year. (Market interest r ate is 10% or trust funds worth $1,000 today). Ronald has 2 mutually exclusive investment opportunities (i. e. AAA and BBB rated investments). 3. If Ronald makes investment describe his cash flows? (i. e. Consumption spending divided equally in present value terms) C0 = $743. 64 Investment in BBB Cash Flow Requirement (CF0) = ($743. 64 + $300) = $1,043. 64 Borrowing Requirement = CF0 – $400 = $643. 64 Contact: otto. [email  protected] utoronto. ca Reference: Don Brean ExampleRonald, a finance student, has $400 cash-on-hand and has $1,100 in trust from his grandmother. Ronald will receive the trust funds next year. (Market interest rate is 10% or trust funds worth $1,000 today). Ronald has 2 mutually exclusive investment opportunities (i. e. AAA and BBB rated investments). 3. If Ronald makes investment describe his cash flows? (i. e. Consumption spending divided equally in present value terms) C1 = $818 Return from BBB Cash Inflow (next year) = $1,100 + $426 = $1,526 Cash Ou tflow (next year) = $818 + $643. 64 + $64. 36 = $1,526 Loan Repayment Interest on Loan @ 10% Contact: otto. [email  protected] utoronto. a Reference: Don Brean Example Ronald, a finance student, has $400 cash-on-hand and has $1,100 in trust from his grandmother. Ronald will receive the trust funds next year. (Market interest rate is 10% or trust funds worth $1,000 today). Ronald has 2 mutually exclusive investment opportunities (i. e. AAA and BBB rated investments). CONCLUDING THOUGHT Ronald’s optimal investment decision (i. e. $300 in BBB) is independent or separate from his decision as to how he inter-temporally allocates his consumption (i. e. C0 and C1) The independence of those two decisions is referred to as the Fisher Separation Theorem. Contact: otto. [email  protected] utoronto. ca Reference: Don Brean â€Å"GET TO KNOW YOU† SURVEY (Name: Optional) Question #1: †¢ What has occurred in your other courses that you were happy about and would like to be i ncorporated into this course ? †¢ What has occurred in your other courses that you were NOT happy about? Question #2: †¢ Anything specific you would like to learn? What are your learning goals in this course? †¢ Any specific requests from the instructor, TAs, program, other support staff, etc? Question #3: †¢ Are you thinking of pursuing further education in Finance, if not then what do you have in mind? And/or†¦ What job(s) are you interested in?Question #4: †¢ Tell me more about yourself (e. g. goals, program concentration, 2nd or 3rd year, etc†¦ ) Question #5: †¢ Any other comments, requests, suggestions, etc? TAKE ~3 MINUTES INDIVIDUALLY TO FILL OUT SURVEY TAKE ~ 5 MINUTES TO TALK TO 5 CLASSMATES WHOM YOU HAVEN’T MET YET (write down initials) SURVEY RESULTS (SUMMARY) †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ Real world experiences, practica l (real-world) examples, cases†¦ Relevant news (where to find news), Current issues in the market Relate course material to real world Exam tips/techniques Applications and excel models used in the real world Interactive class, games, videos†¦Extended office hours (availability) to address questions Humour Practice questions and solutions; Past exams and solutions Capital markets (high-level overview) Typical jobs in finance, Leading finance organizations Additional tutorial time Stock picking, portfolio allocation/analysis, investment tools/strategies, trading tips Learning topics that can be applied in real life Relate designations/roles to course material and applications Better understanding of financial instruments (e. g. Mortgages, bonds, etc†¦ ) View of finance from other functional areas (e. g. Marketing) 13 Popular Case Studies (Failures) 1. 2. 3. 4. 5. . Barings (Bank) – Operational Risk (Trading Activities – From arbitrageur to speculator) Nat ional Australia Bank – Operational and Market Risks (Currency Trading) Bankgesellschaft Berlin (Bank) – Credit and Operational Risks (Loans to Property Developers) Taisei Fire and Marine Insurance Co – Insurance & Governance Risks (Uninsured exposure – Lack of understanding) Washington Mutual (Bank) – Credit, Regulatory and Governance Risks; Stress and Scenario Testing (Low lending standards and bad quality acquisitions) Fannie Mae and Freddie Mac – Credit, Market, Operational, Regulatory Governance and Moral Risk; Politicians vs.Financial Risk Management (Sub-prime loans) Long-Term Capital Management – LTCM (Hedge Fund) – Market & Model Risks (Short liquid vs. Long Illiquid Investments (e. g.Bonds) – Russia Defaulted) Bankers Trust (Bank) – Operational Risk (Misled clients on derivatives sold to them) Orange County – Market and Interest Rate Risks (Wrong way bet on interest rates – Borrowing Short a nd Investing Long – Interest Rates Increased) Northern Rock (UK Bank) – Portfolio, Capital Funding, Operational and Reputational Risks; Stress and Scenario Testing (Sub-prime mortgages – Bank Run) Metallgesellschaft AG (Energy Group) – Market Risks (Cash Flow Issues from Written Forwards) Worldcom (Telecom) – Operational Risks (Accounting Fraud – Massive cquisitions & Debt) China Aviation Oil (Singapore) – Market and Governance Risks (Misreported oil futures trading losses, Un-hedged open short positions, Oil Prices Increased) Source: PRMIA 7. 8. 9. 10. 11. 12. 13. SURVEY AND BREAK 13 POPULAR CASE STUDIES Midterm 2011 – Q3 Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part A – Assume that there is no capital market, which investment, A or B, will Jack choose? Justify your answer with calculations (6 marks) Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part A  œ Assume that there is no capital market, which investment, A or B, will Jack choose?Justify your answer with calculations (6 marks) †¢ If Jack does not invest, his utility is zero †¢ If Jack makes investment A (Utility is ? ) †¢ If Jack makes investment B (Utility is ? ) †¢ Y0 = $500 and Y1 = $0 †¢ Savings = Investment = Y – C Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part A – Assume that there is no capital market, which investment, A or B, will Jack choose? Justify your answer with calculations (6 marks) †¢ Investment A †¢ UA = (500-244)1/4 (400)1/2 = 80 Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Contact: otto. [email  protected] utoronto. caPart A – Assume that there is no capital market, which investment, A or B, will Jack choose? Justify your answer with calculations (6 marks) †¢ Investment B †¢ UB (I) = (500-I)1/4 (50(I)1/2)1/2 †¢ UB (I) = (5 0)1/2 [(500-I)I]1/4 †¢ Find I* by differentiating UB (I) wrt I (set to zero) †¢ dUB(I) = (50)1/2 (1/4) [(500-I)I]-3/4 (500-2I) dI I* = 250 Derivatives (Review) Reference: Martin J. Osborne http://www. economics. utoronto. ca/osborne/MathTutorial/CLCF. HTM Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part A – Assume that there is no capital market, which investment, A or B, will Jack choose?Justify your answer with calculations (6 marks) †¢ Investment B †¢ UB (250) = (50)1/2 [(500-I)I]1/4 †¢ UB (250) = 111. 80 †¢ UB > UA Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part A – Assume that there is no capital market, which investment, A or B, will Jack choose? Justify your answer with calculations (6 marks) †¢ Note: Two methods to calculate I* †¢ 1st method (take derivative of Utility Function) †¢ What’s the 2nd method? Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Alternatively – Investment B Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3Part B – Which investment, A or B will Jack choose? What is his utilitymaximizing investment I* and the optimal consumption plan? (6 marks) (Assume a perfect capital market for borrowing and lending exists and the market interest rate is 20%) Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part B – Which investment, A or B will Jack choose? What is his utilitymaximizing investment I* and the optimal consumption plan? (6 marks) †¢ Jack will choose the investment with the highest NPV †¢ Calculate NPVA and NPVB Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3Part B – Which investment, A or B will Jack choose? What is his utilitymaximizing investment I* and the optimal consumption plan? (6 marks) †¢ NPVA = -$244 + ($400)/(1+0. 20) = $89. 33 Contact: otto. [email  p rotected] utoronto. ca Midterm 2011 – Q3 Contact: otto. [email  protected] utoronto. ca Part B – Which investment, A or B will Jack choose? What is his utilitymaximizing investment I* and the optimal consumption plan? (6 marks) †¢ To solve for NPVB †¢ Need to find optimal investment (I*) †¢ set MRT = -(1+r) = -1. 20 I* = $434. 03 †¢ MRT = – dF/dI = -25/(I1/2) = -1. 20 †¢ F = 50 ($434. 31/2) = $1041. 67 †¢ NPVB = -$434. 03 + ($1041. 67/1. 20) = $434. 03 Midterm 2011 – Q3 Part B – Which investment, A or B will Jack choose? What is his utilitymaximizing investment I* and the optimal consumption plan? (6 marks) †¢ To solve for optimal consumption plan (i. e. C0*and C1*) Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part B – Which investment, A or B will Jack choose? What is his utilitymaximizing investment I* and the optimal consumption plan? (6 marks) †¢ To solve for optimal consumption plan (i. e. C0*and C1*) †¢ Total Wealth = $500 + $434. 03 = $934. 3 (set equal to C0 + C1/(1+r)) †¢ PV Wealth = PV Consumption †¢ C1 = 1120. 84 – 1. 2C0 Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Contact: otto. [email  protected] utoronto. ca Part B – Which investment, A or B will Jack choose? What is his utilitymaximizing investment I* and the optimal consumption plan? (6 marks) †¢ To solve for optimal consumption plan (i. e. C0*and C1*) †¢ Total Wealth = $500 + $434. 03 = $934. 03 (set equal to C0 + C1/(1+r)) †¢ U(C0, C1) = C01/4 (1120. 84 – 1. 2C0 )1/2 †¢ dU/dC0 = (1/4)C0-3/4 (1120. 84 – 1. 2C0)1/2 – 1. 2 x (1/2)C01/4(1120. 4 – 1. 2C0)-1/2 †¢ Setting it equal to zero: 1120. 84 – 1. 2C0 = 2. 4C0 C0* = $311. 34 †¢ C1* = 1120. 84 – 1. 2C0 = $747. 22 Midterm 2011 – Q3 Part B – Which investment, A or B will Jack choose? What is his utilitymaximizing investment I* and the optimal consumption plan? (6 marks) (Assume a perfect capital market for borrowing and lending exists and the market interest rate is 20%) †¢ Alternatively: To solve for optimal consumption plan (i. e. C0*and C1*) Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part B – Which investment, A or B will Jack choose?What is his utilitymaximizing investment I* and the optimal consumption plan? (6 marks) †¢ Alternatively: To solve for optimal consumption plan (i. e. C0*and C1*) †¢ MRS = – (1+r), which leads to †¢ – (C1/2C0) = 1. 2 C1 = 2. 4 C0 †¢ Budget constraint: C0 + C1 / (1+r) = Total Wealth = $934. 03 C1 = 1120. 84 – 1. 2C0 C0* = $311. 34 C1* = $747. 22 Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part C – Jack can hire a worker to supervise one investment for him. As a result, he can now invest in both production opportunities if he wants.If he hires a worker, he has to pay wages in equal instalments (i. e. Same wage today and next period). What maximum wage per period would Jack be willing to pay? (4 marks) Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part C – Jack can hire a worker to supervise one investment for him. As a result, he can now invest in both production opportunities if he wants. If he hires a worker, he has to pay wages in equal instalments (i. e. Same wage today and next period). What maximum wage per period would Jack be willing to pay? (4 marks) †¢ NPVA = $89. 33 = W + (W/1. 20) †¢ W = $48. 3 (i. e. Maximum wage per period) Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part D – Jill earns an income of $250 today and $250 next period but has no access to any production opportunities. She can, however spend some money today to purchase investment opportunity B. Her utility function is: U(C0, C1) = C0 + 2C1 + min(C0, C1) What is the highest price that Jill is willing to pay? (4 marks) Contact: otto. [email  protected] utoronto. ca Midterm 2011 – Q3 Part D – Jill earns an income of $250 today and $250 next period but has no access to any production opportunities.She can, however spend some money today to purchase investment opportunity B. Her utility function is: U(C0, C1) = C0 + 2C1 + min(C0, C1) What is the highest price that Jill is willing to pay? (4 marks) †¢ With a perfect capital market, the Fisher Separation Theorem applies †¢ So the maximum amount she will pay is $434. 03 (i. e. NPVB) Contact: otto. [email  protected] utoronto. ca FINANCIAL MARKETS AND NET PRESENT VALUE (TO SUCCEED – PRACTICE, PRACTICE, PRACTICE) Week 3 – Quick Review (Self-Evaluation) of Week 2 â€Å"GET TO KNOW YOU† SURVEY (Name: Optional)Question #1: †¢ What has occurred in your other courses that you were happy about and would like to be incorporated int o this course ? †¢ What has occurred in your other courses that you were NOT happy about? Question #2: †¢ Anything specific you would like to learn? What are your learning goals in this course? †¢ Any specific requests from the instructor, TAs, program, other support staff, etc? Question #3: †¢ Are you thinking of pursuing further education in Finance, if not then what do you have in mind? And/or†¦ What job(s) are you interested in? Question #4: †¢ Tell me more about yourself (e. . goals, program concentration, 2nd or 3rd year, etc†¦ ) Question #5: †¢ Any other comments, requests, suggestions, etc? TAKE ~3 MINUTES INDIVIDUALLY TO FILL OUT SURVEY TAKE ~ 5 MINUTES TO TALK TO 5 CLASSMATES WHOM YOU HAVEN’T MET YET (write down initials) SURVEY RESULTS (SUMMARY) †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ †¢ Real world experiences, practical (real-world) examples, cases†¦ Relevant news (where to find news), Current issues in the market Relate course material to real world Exam tips/techniques Applications and excel models used in the real world Interactive class, games, videos†¦Extended office hours (availability) to address questions Humour Practice questions and solutions; Past exams and solutions Capital markets (high-level overview) Typical jobs in finance, Leading finance organizations Additional tutorial time Stock picking, portfolio allocation/analysis, investment tools/strategies, trading tips Learning topics that can be applied in real life Relate designations/roles to course material and applications Better understanding of financial instruments (e. g. Mortgages, bonds, etc†¦ ) View of finance from other functional areas (e. g. Marketing) http://www. explorefinancialservices. om/Options http://www. explorefinancialservices. com/ Financial Markets: What is Going On? Firms (Users of Capital) Initial Public Off ering (IPO) Secondary Offerings (SEO) Borrowing (Loans, Bonds) Dividends, $ Repurchases, Interest Payments $ Market Mechanisms or Market Makers (Stock Exchanges, Banks, Investment Funds, †¦) $ $ Firms Issue Stock Certificates and Bonds $ $$$ Invested in Stocks and Bonds Investors (Providers of Capital) Investment Banks help firms make transactions Brokers/Dealers help investors make transactions Reference: Alex MacKay 113 Hedge Fund Strategies Dedicated ShortSource: AIMA Canada Further Reading Hedge Funds – Emerging Market Strategy †¢ Emerging Markets (American Depository Receipts – ADRs vs. Foreign Securities) http://www. sec. gov/pdf/ininvest. pdf (Page 12) (SAP) Hedge Fund – Quants †¢ Jim Simons (Renaissance Technologies) – Commodities/Futures – (Rapid Fire Trading) – (computer and system specialists, researchers and traders) (computational linguists–speech recognition/investing) †¢ http://chinese-school. netfir ms. com/abacus-hedge-funds-Jim-Simons. html †¢ †¢ Kenneth Griffin (Citadel Investment Group) – Convertible Bonds –> Long-Short †¢ http://money. cnn. om/2008/12/08/news/companies/citadel_vickers. boyd. fortune/index. htm The Quants (Scott Patterson – Wall Street Journal Reporter) †¢ †¢ http://www. businessweek. com/magazine/content/10_09/b4168070829612. htm http://online. wsj. com/article/SB10001424052748704509704575019032416477138. html Steven Palmer (AlphaNorth Asset Management Inc) (Microcap – Tech) †¢ http://www. theglobeandmail. com/globe-investor/funds-and-etfs/funds/top-hedge-fund-manager-turns-to-techmicro-caps/article1884049/ House Dems propose taxing equity trades to fund new federal programs †¢ †¢ †¢ Financial transaction tax on all stock (0. 5%), bond (0. %) and derivatives (0. 005%) trades Protects financial markets from speculation Make high-frequency trading â€Å"unprofitable† http://thehi ll. com/blogs/floor-action/house/249893-house-dems-propose-taxing-equity-trades-to-fund-new-federal-programs Harsh HFT curbs could sneak into MiFID †¢ †¢ †¢ †¢ Introduction of minimum resting times between trades Could force HFT firms out of the market, widening spreads and making trading more costly Meetings held with the European Parliament’s Economic and Monetary Affairs Committee (ECON) MiFID (Markets in Financial Instruments Directive) – European Union Law http://www. hetradenews. com/news/Regions/Europe/Harsh_HFT_curbs_could_sneak_into_MiFID_II. aspx CAPITAL MARKET THEORY RSM 332 – Week 2 Week 1 – Introduction – Financial Accounting (Review) Week 2 – Financial Markets and Net Present Value Week 3 – Present Value Concepts Week 4 – Bond Valuation and Term Structure Theory Week 5 – Valuation of Stocks Week 6 – Risk and Return – Problem Set #1 Due Week 7* – Midterm (Tuesday*) We ek 8 – Portfolio Theory Week 9 – Capital Asset Pricing Model Week 10 – Arbitrage Pricing Theory Week 11 – Operation and Efficiency of Capital Markets Week 12 – Course Review – Problem Set #2 DueContact: otto. [email  protected] utoronto. ca THANK YOU SEE YOU NEXT WEEK! OFFICE HOURS WEDNESDAYS – 4:00PM-6:00PM ROOM 413 OR 417 105 ST. GEORGE STREET ROTMAN (NORTH BUILDING)

Why Was The Constitution A Controversial Document Even As It Was Being Written?

The United States Constitution was written more than 200 years ago and it has been used as the foundation for the government. The constitution has been and still remains the most durable political agreement in the history of the world. Even though an inspiring document, its creation was not that easy but controversial. Its framers were divided over a number of major issues which led to lengthy arguments when it was being written.After its creation the Constitution still remains controversial. It has turned out to be the epicenter of Civil Rights activism, conflict between state sovereignty and national supremacy, questionable over universal healthcare as well as marriage controversy1. The compromises on the key issues that characterized its genesis are responsible for the controversy of the imperative document. From the beginning of its creation the constitution ideas were divided between the advocates.The advocates understood that that the constitution was an idea of the republican thinking, providing for a new level to the people’s elected government. On the other hand, the critics were convinced that the republicans aligned themselves in small political units, the states. The separation of powers between the central government and the state governments was thus a major controversy as the constitution document was being written and stands out as a crucial issue up to date. The delegates drafting the constitution encountered issues over the state representation.The delegates that were representing bigger states advocated for a population based representation while the delegates from smaller states thought an equal number of representatives for every state was fair leading to the Electoral College2. Despite the agreement, the not all parties were satisfied making it a controversy. The creation of the constitution was strongly debated as some prolific and influential leaders were opposed it. The issue of a two party system was another issue that sparked c ontroversy as the American constitution was being drafted. The delegates failed to accept that it was a two party system.This continued to create a growing rift among the advocates. It led to the separation between advocates for a large and stronger versus those for a small and a weak federal government. Slavery was another controversial issue in the drafting of the constitution. It was not about its abolition, but rather division on to which state the slaves would be counted to in working out a state’s population as well as the amount of property tax to be implemented over the owners3. The government decided to it would not interfere with the slave trade for no less than 20 years, but this move received opposition from the southern.Eventually, another controversy of the constitution was the issue of the citizenship. The written constitution did not clarify what composed citizenship. It did not also deliver the particular rights for a United States citizen4. These issues led to the controversy in the Bill of Rights. The Bill of rights still remains controversial due to the changing times. Conclusion As the constitution was being written ideas were divided between the advocates. The advocates understood that that the constitution was an idea of the republican thinking, providing for a new level to the people’s elected government.The delegates drafting the constitution encountered issues over the state representation. A two party system was another issue that sparked controversy, Slavery as well as issues concerning citizenship formed the controversy of the constitution was being drafted. The U. S. was controversial even as it was being written and will always remain controversial since compromises rise up with time. There are also several interests of various people that cannot be reconciled all of them without compromise.

A Game of Thrones Chapter Forty

Pale white mists rose off Alyssa’s Tears, where the ghost waters plunged over the shoulder of the mountain to begin their long tumble down the face of the Giant’s Lance. Catelyn could feel the faint touch of spray on her face. Alyssa Arryn had seen her husband, her brothers, and all her children slain, and yet in life she had never shed a tear. So in death, the gods had decreed that she would know no rest until her weeping watered the black earth of the Vale, where the men she had loved were buried. Alyssa had been dead six thousand years now, and still no drop of the torrent had ever reached the valley floor far below. Catelyn wondered how large a waterfall her own tears would make when she died. â€Å"Tell me the rest of it,† she said. â€Å"The Kingslayer is massing a host at Casterly Rock,† Ser Rodrik Cassel answered from the room behind her. â€Å"Your brother writes that he has sent riders to the Rock, demanding that Lord Tywin proclaim his intent, but he has had no answer. Edmure has commanded Lord Vance and Lord Piper to guard the pass below the Golden Tooth. He vows to you that he will yield no foot of Tully land without first watering it with Lannister blood.† Catelyn turned away from the sunrise. Its beauty did little to lighten her mood; it seemed cruel for a day to dawn so fair and end so foul as this one promised to. â€Å"Edmure has sent riders and made vows,† she said, â€Å"but Edmure is not the Lord of Riverrun. What of my lord father?† â€Å"The message made no mention of Lord Hoster, my lady.† Ser Rodrik tugged at his whiskers. They had grown in white as snow and bristly as a thornbush while he was recovering from his wounds; he looked almost himself again. â€Å"My father would not have given the defense of Riverrun over to Edmure unless he was very sick,† she said, worried. â€Å"I should have been woken as soon as this bird arrived.† â€Å"Your lady sister thought it better to let you sleep, Maester Colemon told me.† â€Å"I should have been woken,† she insisted. â€Å"The maester tells me your sister planned to speak with you after the combat,† Ser Rodrik said. â€Å"Then she still plans to go through with this mummer’s farce?† Catelyn grimaced. â€Å"The dwarf has played her like a set of pipes, and she is too deaf to hear the tune. Whatever happens this morning, Ser Rodrik, it is past time we took our leave. My place is at Winterfell with my sons. If you are strong enough to travel, I shall ask Lysa for an escort to see us to Gulltown. We can take ship from there.† â€Å"Another ship?† Ser Rodrik looked a shade green, yet he managed not to shudder. â€Å"As you say, my lady.† The old knight waited outside her door as Catelyn summoned the servants Lysa had given her. If she spoke to her sister before the duel, perhaps she could change her mind, she thought as they dressed her. Lysa’s policies varied with her moods, and her moods changed hourly. The shy girl she had known at Riverrun had grown into a woman who was by turns proud, fearful, cruel, dreamy, reckless, timid, stubborn, vain, and, above all, inconstant. When that vile turnkey of hers had come crawling to tell them that Tyrion Lannister wished to confess, Catelyn had urged Lysa to have the dwarf brought to them privately, but no, nothing would do but that her sister must make a show of him before half the Vale. And now this . . . â€Å"Lannister is my prisoner,† she told Ser Rodrik as they descended the tower stairs and made their way through the Eyrie’s cold white halls. Catelyn wore plain grey wool with a silvered belt. â€Å"My sister must be reminded of that.† At the doors to Lysa’s apartments, they met her uncle storming out. â€Å"Going to join the fool’s festival?† Ser Brynden snapped. â€Å"I’d tell you to slap some sense into your sister, if I thought it would do any good, but you’d only bruise your hand.† â€Å"There was a bird from Riverrun,† Catelyn began, â€Å"a letter from Edmure . . . â€Å" â€Å"I know, child.† The black fish that fastened his cloak was Brynden’s only concession to ornament. â€Å"I had to hear it from Maester Colemon. I asked your sister for leave to take a thousand seasoned men and ride for Riverrun with all haste. Do you know what she told me? The Vale cannot spare a thousand swords, nor even one, Uncle, she said. You are the Knight of the Gate. Your place is here.† A gust of childish laughter drifted through the open doors behind him, and her uncle glanced darkly over his shoulder. â€Å"Well, I told her she could bloody well find herself a new Knight of the Gate. Black fish or no, I am still a Tully. I shall leave for Riverrun by evenfall.† Catelyn could not pretend to surprise. â€Å"Alone? You know as well as I that you will never survive the high road. Ser Rodrik and I are returning to Winterfell. Come with us, Uncle. I will give you your thousand men. Riverrun will not fight alone.† Brynden thought a moment, then nodded a brusque agreement. â€Å"As you say. It’s the long way home, but I’m more like to get there. I’ll wait for you below.† He went striding off, his cloak swirling behind him. Catelyn exchanged a look with Ser Rodrik. They went through the doors to the high, nervous sound of a child’s giggles. Lysa’s apartments opened over a small garden, a circle of dirt and grass planted with blue flowers and ringed on all sides by tall white towers. The builders had intended it as a godswood, but the Eyrie rested on the hard stone of the mountain, and no matter how much soil was hauled up from the Vale, they could not get a weirwood to take root here. So the Lords of the Eyrie planted grass and scattered statuary amidst low, flowering shrubs. It was there the two champions would meet to place their lives, and that of Tyrion Lannister, into the hands of the gods. Lysa, freshly scrubbed and garbed in cream velvet with a rope of sapphires and moonstones around her milk-white neck, was holding court on the terrace overlooking the scene of the combat, surrounded by her knights, retainers, and lords high and low. Most of them still hoped to wed her, bed her, and rule the Vale of Arryn by her side. From what Catelyn had seen during her stay at the Eyrie, it was a vain hope. A wooden platform had been built to elevate Robert’s chair; there the Lord of the Eyrie sat, giggling and clapping his hands as a humpbacked puppeteer in blue-and-white motley made two wooden knights hack and slash at each other. Pitchers of thick cream and baskets of blackberries had been set out, and the guests were sipping a sweet orange-scented wine from engraved silver cups. A fool’s festival, Brynden had called it, and small wonder. Across the terrace, Lysa laughed gaily at some jest of Lord Hunter’s, and nibbled a blackberry from the point of Ser Lyn Corbray’s dagger. They were the suitors who stood highest in Lysa’s favor . . . today, at least. Catelyn would have been hard-pressed to say which man was more unsuitable. Eon Hunter was even older than Jon Arryn had been, half-crippled by gout, and cursed with three quarrelsome sons, each more grasping than the last. Ser Lyn was a different sort of folly; lean and handsome, heir to an ancient but impoverished house, but vain, reckless, hot-tempered . . . and, it was whispered, notoriously uninterested in the intimate charms of women. When Lysa espied Catelyn, she welcomed her with a sisterly embrace and a moist kiss on the cheek. â€Å"Isn’t it a lovely morning? The gods are smiling on us. Do try a cup of the wine, sweet sister. Lord Hunter was kind enough to send for it, from his own cellars.† â€Å"Thank you, no. Lysa, we must talk.† â€Å"After,† her sister promised, already beginning to turn away from her. â€Å"Now.† Catelyn spoke more loudly than she’d intended. Men were turning to look. â€Å"Lysa, you cannot mean to go ahead with this folly. Alive, the Imp has value. Dead, he is only food for crows. And if his champion should prevail here—† â€Å"Small chance of that, my lady,† Lord Hunter assured her, patting her shoulder with a liver-spotted hand. â€Å"Ser Vardis is a doughty fighter. He will make short work of the sellsword.† â€Å"Will he, my lord?† Catelyn said coolly. â€Å"I wonder.† She had seen Bronn fight on the high road; it was no accident that he had survived the journey while other men had died. He moved like a panther, and that ugly sword of his seemed a part of his arm. Lysa’s suitors were gathering around them like bees round a blossom. â€Å"Women understand little of these things,† Ser Morton Waynwood said. â€Å"Ser Vardis is a knight, sweet lady. This other fellow, well, his sort are all cowards at heart. Useful enough in a battle, with thousands of their fellows around them, but stand them up alone and the manhood leaks right out of them.† â€Å"Say you have the truth of it, then,† Catelyn said with a courtesy that made her mouth ache. â€Å"What will we gain by the dwarf’s death? Do you imagine that Jaime will care a fig that we gave his brother a trial before we flung him off a mountain?† â€Å"Behead the man,† Ser Lyn Corbray suggested. â€Å"When the Kingslayer receives the Imp’s head, it will be a warning to him,† Lysa gave an impatient shake of her waist-long auburn hair. â€Å"Lord Robert wants to see him fly,† she said, as if that settled the matter. â€Å"And the Imp has only himself to blame. It was he who demanded a trial by combat.† â€Å"Lady Lysa had no honorable way to deny him, even if she’d wished to,† Lord Hunter intoned ponderously. Ignoring them all, Catelyn turned all her force on her sister. â€Å"I remind you, Tyrion Lannister is my prisoner.† â€Å"And I remind you, the dwarf murdered my lord husband!† Her voice rose. â€Å"He poisoned the Hand of the King and left my sweet baby fatherless, and now I mean to see him pay!† Whirling, her skirts swinging around her, Lysa stalked across the terrace. Ser Lyn and Ser Morton and the other suitors excused themselves with cool nods and trailed after her. â€Å"Do you think he did?† Ser Rodrik asked her quietly when they were alone again. â€Å"Murder Lord Jon, that is? The Imp still denies it, and most fiercely . . . â€Å" â€Å"I believe the Lannisters murdered Lord Arryn,† Catelyn replied, â€Å"but whether it was Tyrion, or Ser Jaime, or the queen, or all of them together, I could not begin to say.† Lysa had named Cersei in the letter she had sent to Winterfell, but now she seemed certain that Tyrion was the killer . . . perhaps because the dwarf was here, while the queen was safe behind the walls of the Red Keep, hundreds of leagues to the south. Catelyn almost wished she had burned her sister’s letter before reading it. Ser Rodrik tugged at his whiskers. â€Å"Poison, well . . . that could be the dwarf’s work, true enough. Or Cersei’s. It’s said poison is a woman’s weapon, begging your pardons, my lady. The Kingslayer, now . . . I have no great liking for the man, but he’s not the sort. Too fond of the sight of blood on that golden sword of his. Was it poison, my lady?† Catelyn frowned, vaguely uneasy. â€Å"How else could they make it look a natural death?† Behind her, Lord Robert shrieked with delight as one of the puppet knights sliced the other in half, spilling a flood of red sawdust onto the terrace. She glanced at her nephew and sighed. â€Å"The boy is utterly without discipline. He will never be strong enough to rule unless he is taken away from his mother for a time.† â€Å"His lord father agreed with you,† said a voice at her elbow. She turned to behold Maester Colemon, a cup of wine in his hand. â€Å"He was planning to send the boy to Dragonstone for fostering, you know . . . oh, but I’m speaking out of turn.† The apple of his throat bobbed anxiously beneath the loose maester’s chain. â€Å"I fear I’ve had too much of Lord Hunter’s excellent wine. The prospect of bloodshed has my nerves all a-fray . . . â€Å" â€Å"You are mistaken, Maester,† Catelyn said. â€Å"It was Casterly Rock, not Dragonstone, and those arrangements were made after the Hand’s death, without my sister’s consent.† The maester’s head jerked so vigorously at the end of his absurdly long neck that he looked half a puppet himself. â€Å"No, begging your forgiveness, my lady, but it was Lord Jon who—† A bell tolled loudly below them. High lords and serving girls alike broke off what they were doing and moved to the balustrade. Below, two guardsmen in sky-blue cloaks led forth Tyrion Lannister. The Eyrie’s plump septon escorted him to the statue in the center of the garden, a weeping woman carved in veined white marble, no doubt meant to be Alyssa. â€Å"The bad little man,† Lord Robert said, giggling. â€Å"Mother, can I make him fly? I want to see him fly.† â€Å"Later, my sweet baby,† Lysa promised him. â€Å"Trial first,† drawled Ser Lyn Corbray, â€Å"then execution.† A moment later the two champions appeared from opposite sides of the garden. The knight was attended by two young squires, the sellsword by the Eyrie’s master-at-arms. Ser Vardis Egen was steel from head to heel, encased in heavy plate armor over mail and padded surcoat. Large circular rondels, enameled cream-and-blue in the moon-and-falcon sigil of House Arryn, protected the vulnerable juncture of arm and breast. A skirt of lobstered metal covered him from waist to midthigh, while a solid gorget encircled his throat. Falcon’s wings sprouted from the temples of his helm, and his visor was a pointed metal beak with a narrow slit for vision. Bronn was so lightly armored he looked almost naked beside the knight. He wore only a shirt of black oiled ringmail over boiled leather, a round steel halfhelm with a noseguard, and a mail coif. High leather boots with steel shinguards gave some protection to his legs, and discs of black iron were sewn into the fingers of his gloves. Yet Catelyn noted that the sellsword stood half a hand taller than his foe, with a longer reach . . . and Bronn was fifteen years younger, if she was any judge. They knelt in the grass beneath the weeping woman, facing each other, with Lannister between them. The septon removed a faceted crystal sphere from the soft cloth bag at his waist. He lifted it high above his head, and the light shattered. Rainbows danced across the Imp’s face. In a high, solemn, singsong voice, the septon asked the gods to look down and bear witness, to find the truth in this man’s soul, to grant him life and freedom if he was innocent, death if he was guilty. His voice echoed off the surrounding towers. When the last echo had died away, the septon lowered his crystal and made a hasty departure. Tyrion leaned over and whispered something in Bronn’s ear before the guardsmen led him away. The sellsword rose laughing and brushed a blade of grass from his knee. Robert Arryn, Lord of the Eyrie and Defender of the Vale, was fidgeting impatiently in his elevated chair. â€Å"When are they going to fight?† he asked plaintively. Ser Vardis was helped back to his feet by one of his squires. The other brought him a triangular shield almost four feet tall, heavy oak dotted with iron studs. They strapped it to his left forearm. When Lysa’s master-at-arms offered Bronn a similar shield, the sellsword spat and waved it away. Three days growth of coarse black beard covered his jaw and cheeks, but if he did not shave it was not for want of a razor; the edge of his sword had the dangerous glimmer of steel that had been honed every day for hours, until it was too sharp to touch. Ser Vardis held out a gauntleted hand, and his squire placed a handsome double-edged longsword in his grasp. The blade was engraved with a delicate silver tracery of a mountain sky; its pommel was a falcon’s head, its crossguard fashioned into the shape of wings. â€Å"I had that sword crafted for Jon in King’s Landing,† Lysa told her guests proudly as they watched Ser Vardis try a practice cut. â€Å"He wore it whenever he sat the Iron Throne in King Robert’s place. Isn’t it a lovely thing? I thought it only fitting that our champion avenge Jon with his own blade.† The engraved silver blade was beautiful beyond a doubt, but it seemed to Catelyn that Ser Vardis might have been more comfortable with his own sword. Yet she said nothing; she was weary of futile arguments with her sister. â€Å"Make them fight!† Lord Robert called out. Ser Vardis faced the Lord of the Eyrie and lifted his sword in salute. â€Å"For the Eyrie and the Vale!† Tyrion Lannister had been seated on a balcony across the garden, flanked by his guards. It was to him that Bronn turned with a cursory salute. â€Å"They await your command,† Lady Lysa said to her lord son. â€Å"Fight!† the boy screamed, his arms trembling as they clutched at his chair. Ser Vardis swiveled, bringing up his heavy shield. Bronn turned to face him. Their swords rang together, once, twice, a testing. The sellsword backed off a step. The knight came after, holding his shield before him. He tried a slash, but Bronn jerked back, just out of reach, and the silver blade cut only air. Bronn circled to his right. Ser Vardis turned to follow, keeping his shield between them. The knight pressed forward, placing each foot carefully on the uneven ground. The sellsword gave way, a faint smile playing over his lips. Ser Vardis attacked, slashing, but Bronn leapt away from him, hopping lightly over a low, moss-covered stone. Now the sellsword circled left, away from the shield, toward the knight’s unprotected side. Ser Vardis tried a hack at his legs, but he did not have the reach. Bronn danced farther to his left. Ser Vardis turned in place. â€Å"The man is craven,† Lord Hunter declared. â€Å"Stand and fight, coward! † Other voices echoed the sentiment. Catelyn looked to Ser Rodrik. Her master-at-arms gave a curt shake of his head. â€Å"He wants to make Ser Vardis chase him. The weight of armor and shield will tire even the strongest man.† She had seen men practice at their swordplay near every day of her life, had viewed half a hundred tourneys in her time, but this was something different and deadlier: a dance where the smallest misstep meant death. And as she watched, the memory of another duel in another time came back to Catelyn Stark, as vivid as if it had been yesterday. They met in the lower bailey of Riverrun. When Brandon saw that Petyr wore only helm and breastplate and mail, he took off most of his armor. Petyr had begged her for a favor he might wear, but she had turned him away. Her lord father promised her to Brandon Stark, and so it was to him that she gave her token, a pale blue handscarf she had embroidered with the leaping trout of Riverrun. As she pressed it into his hand, she pleaded with him. â€Å"He is only a foolish boy, but I have loved him like a brother. It would grieve me to see him die.† And her betrothed looked at her with the cool grey eyes of a Stark and promised to spare the boy who loved her. That fight was over almost as soon as it began. Brandon was a man grown, and he drove Littlefinger all the way across the bailey and down the water stair, raining steel on him with every step, until the boy was staggering and bleeding from a dozen wounds. â€Å"Yield!† he called, more than once, but Petyr would only shake his head and fight on, grimly. When the river was lapping at their ankles, Brandon finally ended it, with a brutal backhand cut that bit through Petyr’s rings and leather into the soft flesh below the ribs, so deep that Catelyn was certain that the wound was mortal. He looked at her as he fell and murmured â€Å"Cat† as the bright blood came flowing out between his mailed fingers. She thought she had forgotten that. That was the last time she had seen his face . . . until the day she was brought before him in King’s Landing. A fortnight passed before Littlefinger was strong enough to leave Riverrun, but her lord father forbade her to visit him in the tower where he lay abed. Lysa helped their maester nurse him; she had been softer and shyer in those days. Edmure had called on him as well, but Petyr had sent him away. Her brother had acted as Brandon’s squire at the duel, and Littlefinger would not forgive that. As soon as he was strong enough to be moved, Lord Hoster Tully sent Petyr Baelish away in a closed litter, to finish his healing on the Fingers, upon the windswept jut of rock where he’d been born. The ringing clash of steel on steel jarred Catelyn back to the present. Ser Vardis was coming hard at Bronn, driving into him with shield and sword. The sellsword scrambled backward, checking each blow, stepping lithely over rock and root, his eyes never leaving his foe. He was quicker, Catelyn saw; the knight’s silvered sword never came near to touching him, but his own ugly grey blade hacked a notch from Ser Vardis’s shoulder plate. The brief flurry of fighting ended as swiftly as it had begun when Bronn sidestepped and slid behind the statue of the weeping woman. Ser Vardis lunged at where he had been, striking a spark off the pale marble of Alyssa’s thigh. â€Å"They’re not fighting good, Mother,† the Lord of the Eyrie complained. â€Å"I want them to fight.† â€Å"They will, sweet baby,† his mother soothed him. â€Å"The sellsword can’t run all day.† Some of the lords on Lysa’s terrace were making wry jests as they refilled their wine cups, but across the garden, Tyrion Lannister’s mismatched eyes watched the champions dance as if there were nothing else in the world. Bronn came out from behind the statue hard and fast, still moving left, aiming a two-handed cut at the knight’s unshielded right side. Ser Vardis blocked, but clumsily, and the sellsword’s blade flashed upward at his head. Metal rang, and a falcon’s wing collapsed with a crunch. Ser Vardis took a half step back to brace himself, raised his shield. Oak chips flew as Bronn’s sword hacked at the wooden wall. The sellsword stepped left again, away from the shield, and caught Ser Vardis across the stomach, the razor edge of his blade leaving a bright gash when it bit into the knight’s plate. Ser Vardis drove forward off his back foot, his own silver blade descending in a savage arc. Bronn slammed it aside and danced away. The knight crashed into the weeping woman, rocking her on her plinth. Staggered, he stepped backward, his head turning this way and that as he searched for his foe. The slit visor of his helm narrowed his vision. â€Å"Behind you, ser!† Lord Hunter shouted, too late. Bronn brought his sword down with both hands, catching Ser Vardis in the elbow of his sword arm. The thin lobstered metal that protected the joint crunched. The knight grunted, turning, wrenching his weapon up. This time Bronn stood his ground. The swords flew at each other, and their steel song filled the garden and rang off the white towers of the Eyrie. â€Å"Ser Vardis is hurt,† Ser Rodrik said, his voice grave. Catelyn did not need to be told; she had eyes, she could see the bright finger of blood running along the knight’s forearm, the wetness inside the elbow joint. Every parry was a little slower and a little lower than the one before. Ser Vardis turned his side to his foe, trying to use his shield to block instead, but Bronn slid around him, quick as a cat. The sellsword seemed to be getting stronger. His cuts were leaving their marks now. Deep shiny gashes gleamed all over the knight’s armor, on his right thigh, his beaked visor, crossing on his breastplate, a long one along the front of his gorget. The moon-and-falcon rondel over Ser Vardis’s right arm was sheared clean in half, hanging by its strap. They could hear his labored breath, rattling through the air holes in his visor. Blind with arrogance as they were, even the knights and lords of the Vale could see what was happening below them, yet her sister could not. â€Å"Enough, Ser Vardis!† Lady Lysa called down. â€Å"Finish him now, my baby is growing tired.† And it must be said of Ser Vardis Egen that he was true to his lady’s command, even to the last. One moment he was reeling backward, half-crouched behind his scarred shield; the next he charged. The sudden bull rush caught Bronn off balance. Ser Vardis crashed into him and slammed the lip of his shield into the sellsword’s face. Almost, almost, Bronn lost his feet . . . he staggered back, tripped over a rock, and caught hold of the weeping woman to keep his balance. Throwing aside his shield, Ser Vardis lurched after him, using both hands to raise his sword. His right arm was blood from elbow to fingers now, yet his last desperate blow would have opened Bronn from neck to navel . . . if the sellsword had stood to receive it. But Bronn jerked back. Jon Arryn’s beautiful engraved silver sword glanced off the marble elbow of the weeping woman and snapped clean a third of the way up the blade. Bronn put his shoulder into the statue’s back. The weathered likeness of Alyssa Arryn tottered and fell with a great crash, and Ser Vardis Egen went down beneath her. Bronn was on him in a heartbeat, kicking what was left of his shattered rondel aside to expose the weak spot between arm and breastplate. Ser Vardis was lying on his side, pinned beneath the broken torso of the weeping woman. Catelyn heard the knight groan as the sellsword lifted his blade with both hands and drove it down and in with all his weight behind it, under the arm and through the ribs. Ser Vardis Egen shuddered and lay still. Silence hung over the Eyrie. Bronn yanked off his halfhelm and let it fall to the grass. His lip was smashed and bloody where the shield had caught him, and his coal-black hair was soaked with sweat. He spit out a broken tooth. â€Å"Is it over, Mother?† the Lord of the Eyrie asked. No, Catelyn wanted to tell him, it’s only now beginning. â€Å"Yes,† Lysa said glumly, her voice as cold and dead as the captain of her guard. â€Å"Can I make the little man fly now?† Across the garden, Tyrion Lannister got to his feet. â€Å"Not this little man,† he said. â€Å"This little man is going down in the turnip hoist, thank you very much.† â€Å"You presume—† Lysa began. â€Å"I presume that House Arryn remembers its own words,† the Imp said. â€Å"As High as Honor.† â€Å"You promised I could make him fly,† the Lord of the Eyrie screamed at his mother. He began to shake. Lady Lysa’s face was flushed with fury. â€Å"The gods have seen fit to proclaim him innocent, child. We have no choice but to free him.† She lifted her voice. â€Å"Guards. Take my lord of Lannister and his . . . creature here out of my sight. Escort them to the Bloody Gate and set them free. See that they have horses and supplies sufficient to reach the Trident, and make certain all their goods and weapons are returned to them. They shall need them on the high road.† â€Å"The high road,† Tyrion Lannister said. Lysa allowed herself a faint, satisfied smile. It was another sort of death sentence, Catelyn realized. Tyrion Lannister must know that as well. Yet the dwarf favored Lady Arryn with a mocking bow. â€Å"As you command, my lady,† he said. â€Å"I believe we know the way.† A Game of Thrones Chapter Forty Pale white mists rose off Alyssa’s Tears, where the ghost waters plunged over the shoulder of the mountain to begin their long tumble down the face of the Giant’s Lance. Catelyn could feel the faint touch of spray on her face. Alyssa Arryn had seen her husband, her brothers, and all her children slain, and yet in life she had never shed a tear. So in death, the gods had decreed that she would know no rest until her weeping watered the black earth of the Vale, where the men she had loved were buried. Alyssa had been dead six thousand years now, and still no drop of the torrent had ever reached the valley floor far below. Catelyn wondered how large a waterfall her own tears would make when she died. â€Å"Tell me the rest of it,† she said. â€Å"The Kingslayer is massing a host at Casterly Rock,† Ser Rodrik Cassel answered from the room behind her. â€Å"Your brother writes that he has sent riders to the Rock, demanding that Lord Tywin proclaim his intent, but he has had no answer. Edmure has commanded Lord Vance and Lord Piper to guard the pass below the Golden Tooth. He vows to you that he will yield no foot of Tully land without first watering it with Lannister blood.† Catelyn turned away from the sunrise. Its beauty did little to lighten her mood; it seemed cruel for a day to dawn so fair and end so foul as this one promised to. â€Å"Edmure has sent riders and made vows,† she said, â€Å"but Edmure is not the Lord of Riverrun. What of my lord father?† â€Å"The message made no mention of Lord Hoster, my lady.† Ser Rodrik tugged at his whiskers. They had grown in white as snow and bristly as a thornbush while he was recovering from his wounds; he looked almost himself again. â€Å"My father would not have given the defense of Riverrun over to Edmure unless he was very sick,† she said, worried. â€Å"I should have been woken as soon as this bird arrived.† â€Å"Your lady sister thought it better to let you sleep, Maester Colemon told me.† â€Å"I should have been woken,† she insisted. â€Å"The maester tells me your sister planned to speak with you after the combat,† Ser Rodrik said. â€Å"Then she still plans to go through with this mummer’s farce?† Catelyn grimaced. â€Å"The dwarf has played her like a set of pipes, and she is too deaf to hear the tune. Whatever happens this morning, Ser Rodrik, it is past time we took our leave. My place is at Winterfell with my sons. If you are strong enough to travel, I shall ask Lysa for an escort to see us to Gulltown. We can take ship from there.† â€Å"Another ship?† Ser Rodrik looked a shade green, yet he managed not to shudder. â€Å"As you say, my lady.† The old knight waited outside her door as Catelyn summoned the servants Lysa had given her. If she spoke to her sister before the duel, perhaps she could change her mind, she thought as they dressed her. Lysa’s policies varied with her moods, and her moods changed hourly. The shy girl she had known at Riverrun had grown into a woman who was by turns proud, fearful, cruel, dreamy, reckless, timid, stubborn, vain, and, above all, inconstant. When that vile turnkey of hers had come crawling to tell them that Tyrion Lannister wished to confess, Catelyn had urged Lysa to have the dwarf brought to them privately, but no, nothing would do but that her sister must make a show of him before half the Vale. And now this . . . â€Å"Lannister is my prisoner,† she told Ser Rodrik as they descended the tower stairs and made their way through the Eyrie’s cold white halls. Catelyn wore plain grey wool with a silvered belt. â€Å"My sister must be reminded of that.† At the doors to Lysa’s apartments, they met her uncle storming out. â€Å"Going to join the fool’s festival?† Ser Brynden snapped. â€Å"I’d tell you to slap some sense into your sister, if I thought it would do any good, but you’d only bruise your hand.† â€Å"There was a bird from Riverrun,† Catelyn began, â€Å"a letter from Edmure . . . â€Å" â€Å"I know, child.† The black fish that fastened his cloak was Brynden’s only concession to ornament. â€Å"I had to hear it from Maester Colemon. I asked your sister for leave to take a thousand seasoned men and ride for Riverrun with all haste. Do you know what she told me? The Vale cannot spare a thousand swords, nor even one, Uncle, she said. You are the Knight of the Gate. Your place is here.† A gust of childish laughter drifted through the open doors behind him, and her uncle glanced darkly over his shoulder. â€Å"Well, I told her she could bloody well find herself a new Knight of the Gate. Black fish or no, I am still a Tully. I shall leave for Riverrun by evenfall.† Catelyn could not pretend to surprise. â€Å"Alone? You know as well as I that you will never survive the high road. Ser Rodrik and I are returning to Winterfell. Come with us, Uncle. I will give you your thousand men. Riverrun will not fight alone.† Brynden thought a moment, then nodded a brusque agreement. â€Å"As you say. It’s the long way home, but I’m more like to get there. I’ll wait for you below.† He went striding off, his cloak swirling behind him. Catelyn exchanged a look with Ser Rodrik. They went through the doors to the high, nervous sound of a child’s giggles. Lysa’s apartments opened over a small garden, a circle of dirt and grass planted with blue flowers and ringed on all sides by tall white towers. The builders had intended it as a godswood, but the Eyrie rested on the hard stone of the mountain, and no matter how much soil was hauled up from the Vale, they could not get a weirwood to take root here. So the Lords of the Eyrie planted grass and scattered statuary amidst low, flowering shrubs. It was there the two champions would meet to place their lives, and that of Tyrion Lannister, into the hands of the gods. Lysa, freshly scrubbed and garbed in cream velvet with a rope of sapphires and moonstones around her milk-white neck, was holding court on the terrace overlooking the scene of the combat, surrounded by her knights, retainers, and lords high and low. Most of them still hoped to wed her, bed her, and rule the Vale of Arryn by her side. From what Catelyn had seen during her stay at the Eyrie, it was a vain hope. A wooden platform had been built to elevate Robert’s chair; there the Lord of the Eyrie sat, giggling and clapping his hands as a humpbacked puppeteer in blue-and-white motley made two wooden knights hack and slash at each other. Pitchers of thick cream and baskets of blackberries had been set out, and the guests were sipping a sweet orange-scented wine from engraved silver cups. A fool’s festival, Brynden had called it, and small wonder. Across the terrace, Lysa laughed gaily at some jest of Lord Hunter’s, and nibbled a blackberry from the point of Ser Lyn Corbray’s dagger. They were the suitors who stood highest in Lysa’s favor . . . today, at least. Catelyn would have been hard-pressed to say which man was more unsuitable. Eon Hunter was even older than Jon Arryn had been, half-crippled by gout, and cursed with three quarrelsome sons, each more grasping than the last. Ser Lyn was a different sort of folly; lean and handsome, heir to an ancient but impoverished house, but vain, reckless, hot-tempered . . . and, it was whispered, notoriously uninterested in the intimate charms of women. When Lysa espied Catelyn, she welcomed her with a sisterly embrace and a moist kiss on the cheek. â€Å"Isn’t it a lovely morning? The gods are smiling on us. Do try a cup of the wine, sweet sister. Lord Hunter was kind enough to send for it, from his own cellars.† â€Å"Thank you, no. Lysa, we must talk.† â€Å"After,† her sister promised, already beginning to turn away from her. â€Å"Now.† Catelyn spoke more loudly than she’d intended. Men were turning to look. â€Å"Lysa, you cannot mean to go ahead with this folly. Alive, the Imp has value. Dead, he is only food for crows. And if his champion should prevail here—† â€Å"Small chance of that, my lady,† Lord Hunter assured her, patting her shoulder with a liver-spotted hand. â€Å"Ser Vardis is a doughty fighter. He will make short work of the sellsword.† â€Å"Will he, my lord?† Catelyn said coolly. â€Å"I wonder.† She had seen Bronn fight on the high road; it was no accident that he had survived the journey while other men had died. He moved like a panther, and that ugly sword of his seemed a part of his arm. Lysa’s suitors were gathering around them like bees round a blossom. â€Å"Women understand little of these things,† Ser Morton Waynwood said. â€Å"Ser Vardis is a knight, sweet lady. This other fellow, well, his sort are all cowards at heart. Useful enough in a battle, with thousands of their fellows around them, but stand them up alone and the manhood leaks right out of them.† â€Å"Say you have the truth of it, then,† Catelyn said with a courtesy that made her mouth ache. â€Å"What will we gain by the dwarf’s death? Do you imagine that Jaime will care a fig that we gave his brother a trial before we flung him off a mountain?† â€Å"Behead the man,† Ser Lyn Corbray suggested. â€Å"When the Kingslayer receives the Imp’s head, it will be a warning to him,† Lysa gave an impatient shake of her waist-long auburn hair. â€Å"Lord Robert wants to see him fly,† she said, as if that settled the matter. â€Å"And the Imp has only himself to blame. It was he who demanded a trial by combat.† â€Å"Lady Lysa had no honorable way to deny him, even if she’d wished to,† Lord Hunter intoned ponderously. Ignoring them all, Catelyn turned all her force on her sister. â€Å"I remind you, Tyrion Lannister is my prisoner.† â€Å"And I remind you, the dwarf murdered my lord husband!† Her voice rose. â€Å"He poisoned the Hand of the King and left my sweet baby fatherless, and now I mean to see him pay!† Whirling, her skirts swinging around her, Lysa stalked across the terrace. Ser Lyn and Ser Morton and the other suitors excused themselves with cool nods and trailed after her. â€Å"Do you think he did?† Ser Rodrik asked her quietly when they were alone again. â€Å"Murder Lord Jon, that is? The Imp still denies it, and most fiercely . . . â€Å" â€Å"I believe the Lannisters murdered Lord Arryn,† Catelyn replied, â€Å"but whether it was Tyrion, or Ser Jaime, or the queen, or all of them together, I could not begin to say.† Lysa had named Cersei in the letter she had sent to Winterfell, but now she seemed certain that Tyrion was the killer . . . perhaps because the dwarf was here, while the queen was safe behind the walls of the Red Keep, hundreds of leagues to the south. Catelyn almost wished she had burned her sister’s letter before reading it. Ser Rodrik tugged at his whiskers. â€Å"Poison, well . . . that could be the dwarf’s work, true enough. Or Cersei’s. It’s said poison is a woman’s weapon, begging your pardons, my lady. The Kingslayer, now . . . I have no great liking for the man, but he’s not the sort. Too fond of the sight of blood on that golden sword of his. Was it poison, my lady?† Catelyn frowned, vaguely uneasy. â€Å"How else could they make it look a natural death?† Behind her, Lord Robert shrieked with delight as one of the puppet knights sliced the other in half, spilling a flood of red sawdust onto the terrace. She glanced at her nephew and sighed. â€Å"The boy is utterly without discipline. He will never be strong enough to rule unless he is taken away from his mother for a time.† â€Å"His lord father agreed with you,† said a voice at her elbow. She turned to behold Maester Colemon, a cup of wine in his hand. â€Å"He was planning to send the boy to Dragonstone for fostering, you know . . . oh, but I’m speaking out of turn.† The apple of his throat bobbed anxiously beneath the loose maester’s chain. â€Å"I fear I’ve had too much of Lord Hunter’s excellent wine. The prospect of bloodshed has my nerves all a-fray . . . â€Å" â€Å"You are mistaken, Maester,† Catelyn said. â€Å"It was Casterly Rock, not Dragonstone, and those arrangements were made after the Hand’s death, without my sister’s consent.† The maester’s head jerked so vigorously at the end of his absurdly long neck that he looked half a puppet himself. â€Å"No, begging your forgiveness, my lady, but it was Lord Jon who—† A bell tolled loudly below them. High lords and serving girls alike broke off what they were doing and moved to the balustrade. Below, two guardsmen in sky-blue cloaks led forth Tyrion Lannister. The Eyrie’s plump septon escorted him to the statue in the center of the garden, a weeping woman carved in veined white marble, no doubt meant to be Alyssa. â€Å"The bad little man,† Lord Robert said, giggling. â€Å"Mother, can I make him fly? I want to see him fly.† â€Å"Later, my sweet baby,† Lysa promised him. â€Å"Trial first,† drawled Ser Lyn Corbray, â€Å"then execution.† A moment later the two champions appeared from opposite sides of the garden. The knight was attended by two young squires, the sellsword by the Eyrie’s master-at-arms. Ser Vardis Egen was steel from head to heel, encased in heavy plate armor over mail and padded surcoat. Large circular rondels, enameled cream-and-blue in the moon-and-falcon sigil of House Arryn, protected the vulnerable juncture of arm and breast. A skirt of lobstered metal covered him from waist to midthigh, while a solid gorget encircled his throat. Falcon’s wings sprouted from the temples of his helm, and his visor was a pointed metal beak with a narrow slit for vision. Bronn was so lightly armored he looked almost naked beside the knight. He wore only a shirt of black oiled ringmail over boiled leather, a round steel halfhelm with a noseguard, and a mail coif. High leather boots with steel shinguards gave some protection to his legs, and discs of black iron were sewn into the fingers of his gloves. Yet Catelyn noted that the sellsword stood half a hand taller than his foe, with a longer reach . . . and Bronn was fifteen years younger, if she was any judge. They knelt in the grass beneath the weeping woman, facing each other, with Lannister between them. The septon removed a faceted crystal sphere from the soft cloth bag at his waist. He lifted it high above his head, and the light shattered. Rainbows danced across the Imp’s face. In a high, solemn, singsong voice, the septon asked the gods to look down and bear witness, to find the truth in this man’s soul, to grant him life and freedom if he was innocent, death if he was guilty. His voice echoed off the surrounding towers. When the last echo had died away, the septon lowered his crystal and made a hasty departure. Tyrion leaned over and whispered something in Bronn’s ear before the guardsmen led him away. The sellsword rose laughing and brushed a blade of grass from his knee. Robert Arryn, Lord of the Eyrie and Defender of the Vale, was fidgeting impatiently in his elevated chair. â€Å"When are they going to fight?† he asked plaintively. Ser Vardis was helped back to his feet by one of his squires. The other brought him a triangular shield almost four feet tall, heavy oak dotted with iron studs. They strapped it to his left forearm. When Lysa’s master-at-arms offered Bronn a similar shield, the sellsword spat and waved it away. Three days growth of coarse black beard covered his jaw and cheeks, but if he did not shave it was not for want of a razor; the edge of his sword had the dangerous glimmer of steel that had been honed every day for hours, until it was too sharp to touch. Ser Vardis held out a gauntleted hand, and his squire placed a handsome double-edged longsword in his grasp. The blade was engraved with a delicate silver tracery of a mountain sky; its pommel was a falcon’s head, its crossguard fashioned into the shape of wings. â€Å"I had that sword crafted for Jon in King’s Landing,† Lysa told her guests proudly as they watched Ser Vardis try a practice cut. â€Å"He wore it whenever he sat the Iron Throne in King Robert’s place. Isn’t it a lovely thing? I thought it only fitting that our champion avenge Jon with his own blade.† The engraved silver blade was beautiful beyond a doubt, but it seemed to Catelyn that Ser Vardis might have been more comfortable with his own sword. Yet she said nothing; she was weary of futile arguments with her sister. â€Å"Make them fight!† Lord Robert called out. Ser Vardis faced the Lord of the Eyrie and lifted his sword in salute. â€Å"For the Eyrie and the Vale!† Tyrion Lannister had been seated on a balcony across the garden, flanked by his guards. It was to him that Bronn turned with a cursory salute. â€Å"They await your command,† Lady Lysa said to her lord son. â€Å"Fight!† the boy screamed, his arms trembling as they clutched at his chair. Ser Vardis swiveled, bringing up his heavy shield. Bronn turned to face him. Their swords rang together, once, twice, a testing. The sellsword backed off a step. The knight came after, holding his shield before him. He tried a slash, but Bronn jerked back, just out of reach, and the silver blade cut only air. Bronn circled to his right. Ser Vardis turned to follow, keeping his shield between them. The knight pressed forward, placing each foot carefully on the uneven ground. The sellsword gave way, a faint smile playing over his lips. Ser Vardis attacked, slashing, but Bronn leapt away from him, hopping lightly over a low, moss-covered stone. Now the sellsword circled left, away from the shield, toward the knight’s unprotected side. Ser Vardis tried a hack at his legs, but he did not have the reach. Bronn danced farther to his left. Ser Vardis turned in place. â€Å"The man is craven,† Lord Hunter declared. â€Å"Stand and fight, coward! † Other voices echoed the sentiment. Catelyn looked to Ser Rodrik. Her master-at-arms gave a curt shake of his head. â€Å"He wants to make Ser Vardis chase him. The weight of armor and shield will tire even the strongest man.† She had seen men practice at their swordplay near every day of her life, had viewed half a hundred tourneys in her time, but this was something different and deadlier: a dance where the smallest misstep meant death. And as she watched, the memory of another duel in another time came back to Catelyn Stark, as vivid as if it had been yesterday. They met in the lower bailey of Riverrun. When Brandon saw that Petyr wore only helm and breastplate and mail, he took off most of his armor. Petyr had begged her for a favor he might wear, but she had turned him away. Her lord father promised her to Brandon Stark, and so it was to him that she gave her token, a pale blue handscarf she had embroidered with the leaping trout of Riverrun. As she pressed it into his hand, she pleaded with him. â€Å"He is only a foolish boy, but I have loved him like a brother. It would grieve me to see him die.† And her betrothed looked at her with the cool grey eyes of a Stark and promised to spare the boy who loved her. That fight was over almost as soon as it began. Brandon was a man grown, and he drove Littlefinger all the way across the bailey and down the water stair, raining steel on him with every step, until the boy was staggering and bleeding from a dozen wounds. â€Å"Yield!† he called, more than once, but Petyr would only shake his head and fight on, grimly. When the river was lapping at their ankles, Brandon finally ended it, with a brutal backhand cut that bit through Petyr’s rings and leather into the soft flesh below the ribs, so deep that Catelyn was certain that the wound was mortal. He looked at her as he fell and murmured â€Å"Cat† as the bright blood came flowing out between his mailed fingers. She thought she had forgotten that. That was the last time she had seen his face . . . until the day she was brought before him in King’s Landing. A fortnight passed before Littlefinger was strong enough to leave Riverrun, but her lord father forbade her to visit him in the tower where he lay abed. Lysa helped their maester nurse him; she had been softer and shyer in those days. Edmure had called on him as well, but Petyr had sent him away. Her brother had acted as Brandon’s squire at the duel, and Littlefinger would not forgive that. As soon as he was strong enough to be moved, Lord Hoster Tully sent Petyr Baelish away in a closed litter, to finish his healing on the Fingers, upon the windswept jut of rock where he’d been born. The ringing clash of steel on steel jarred Catelyn back to the present. Ser Vardis was coming hard at Bronn, driving into him with shield and sword. The sellsword scrambled backward, checking each blow, stepping lithely over rock and root, his eyes never leaving his foe. He was quicker, Catelyn saw; the knight’s silvered sword never came near to touching him, but his own ugly grey blade hacked a notch from Ser Vardis’s shoulder plate. The brief flurry of fighting ended as swiftly as it had begun when Bronn sidestepped and slid behind the statue of the weeping woman. Ser Vardis lunged at where he had been, striking a spark off the pale marble of Alyssa’s thigh. â€Å"They’re not fighting good, Mother,† the Lord of the Eyrie complained. â€Å"I want them to fight.† â€Å"They will, sweet baby,† his mother soothed him. â€Å"The sellsword can’t run all day.† Some of the lords on Lysa’s terrace were making wry jests as they refilled their wine cups, but across the garden, Tyrion Lannister’s mismatched eyes watched the champions dance as if there were nothing else in the world. Bronn came out from behind the statue hard and fast, still moving left, aiming a two-handed cut at the knight’s unshielded right side. Ser Vardis blocked, but clumsily, and the sellsword’s blade flashed upward at his head. Metal rang, and a falcon’s wing collapsed with a crunch. Ser Vardis took a half step back to brace himself, raised his shield. Oak chips flew as Bronn’s sword hacked at the wooden wall. The sellsword stepped left again, away from the shield, and caught Ser Vardis across the stomach, the razor edge of his blade leaving a bright gash when it bit into the knight’s plate. Ser Vardis drove forward off his back foot, his own silver blade descending in a savage arc. Bronn slammed it aside and danced away. The knight crashed into the weeping woman, rocking her on her plinth. Staggered, he stepped backward, his head turning this way and that as he searched for his foe. The slit visor of his helm narrowed his vision. â€Å"Behind you, ser!† Lord Hunter shouted, too late. Bronn brought his sword down with both hands, catching Ser Vardis in the elbow of his sword arm. The thin lobstered metal that protected the joint crunched. The knight grunted, turning, wrenching his weapon up. This time Bronn stood his ground. The swords flew at each other, and their steel song filled the garden and rang off the white towers of the Eyrie. â€Å"Ser Vardis is hurt,† Ser Rodrik said, his voice grave. Catelyn did not need to be told; she had eyes, she could see the bright finger of blood running along the knight’s forearm, the wetness inside the elbow joint. Every parry was a little slower and a little lower than the one before. Ser Vardis turned his side to his foe, trying to use his shield to block instead, but Bronn slid around him, quick as a cat. The sellsword seemed to be getting stronger. His cuts were leaving their marks now. Deep shiny gashes gleamed all over the knight’s armor, on his right thigh, his beaked visor, crossing on his breastplate, a long one along the front of his gorget. The moon-and-falcon rondel over Ser Vardis’s right arm was sheared clean in half, hanging by its strap. They could hear his labored breath, rattling through the air holes in his visor. Blind with arrogance as they were, even the knights and lords of the Vale could see what was happening below them, yet her sister could not. â€Å"Enough, Ser Vardis!† Lady Lysa called down. â€Å"Finish him now, my baby is growing tired.† And it must be said of Ser Vardis Egen that he was true to his lady’s command, even to the last. One moment he was reeling backward, half-crouched behind his scarred shield; the next he charged. The sudden bull rush caught Bronn off balance. Ser Vardis crashed into him and slammed the lip of his shield into the sellsword’s face. Almost, almost, Bronn lost his feet . . . he staggered back, tripped over a rock, and caught hold of the weeping woman to keep his balance. Throwing aside his shield, Ser Vardis lurched after him, using both hands to raise his sword. His right arm was blood from elbow to fingers now, yet his last desperate blow would have opened Bronn from neck to navel . . . if the sellsword had stood to receive it. But Bronn jerked back. Jon Arryn’s beautiful engraved silver sword glanced off the marble elbow of the weeping woman and snapped clean a third of the way up the blade. Bronn put his shoulder into the statue’s back. The weathered likeness of Alyssa Arryn tottered and fell with a great crash, and Ser Vardis Egen went down beneath her. Bronn was on him in a heartbeat, kicking what was left of his shattered rondel aside to expose the weak spot between arm and breastplate. Ser Vardis was lying on his side, pinned beneath the broken torso of the weeping woman. Catelyn heard the knight groan as the sellsword lifted his blade with both hands and drove it down and in with all his weight behind it, under the arm and through the ribs. Ser Vardis Egen shuddered and lay still. Silence hung over the Eyrie. Bronn yanked off his halfhelm and let it fall to the grass. His lip was smashed and bloody where the shield had caught him, and his coal-black hair was soaked with sweat. He spit out a broken tooth. â€Å"Is it over, Mother?† the Lord of the Eyrie asked. No, Catelyn wanted to tell him, it’s only now beginning. â€Å"Yes,† Lysa said glumly, her voice as cold and dead as the captain of her guard. â€Å"Can I make the little man fly now?† Across the garden, Tyrion Lannister got to his feet. â€Å"Not this little man,† he said. â€Å"This little man is going down in the turnip hoist, thank you very much.† â€Å"You presume—† Lysa began. â€Å"I presume that House Arryn remembers its own words,† the Imp said. â€Å"As High as Honor.† â€Å"You promised I could make him fly,† the Lord of the Eyrie screamed at his mother. He began to shake. Lady Lysa’s face was flushed with fury. â€Å"The gods have seen fit to proclaim him innocent, child. We have no choice but to free him.† She lifted her voice. â€Å"Guards. Take my lord of Lannister and his . . . creature here out of my sight. Escort them to the Bloody Gate and set them free. See that they have horses and supplies sufficient to reach the Trident, and make certain all their goods and weapons are returned to them. They shall need them on the high road.† â€Å"The high road,† Tyrion Lannister said. Lysa allowed herself a faint, satisfied smile. It was another sort of death sentence, Catelyn realized. Tyrion Lannister must know that as well. Yet the dwarf favored Lady Arryn with a mocking bow. â€Å"As you command, my lady,† he said. â€Å"I believe we know the way.†