CHAPTER TWO: Time Value of Money and Term Structure of Interest1Yes ! is the expected rate of return, i.e., the mean of the discount rates for different termsLetNo ! is the discount rate that cannot be used for so long period?Discounted Cash Flow Formula2Term Structure of Interest Rates Our objective is to value riskless cash flows. Given the rich set of fixed-income securities traded in the market, their prices provide the information needed to value riskless cash flows at hand.3Forms of Interest Rates In this market, this information on the time value of money is given in several different forms: Spot interest rates Price of discount bonds (e.g., zero-coupon bonds and STRIPS) Prices of coupon bonds Yield-to-maturity (an average of spot interest rates) Forward interest rates The form in which this information is expressed depends on the particular market.4Determination of Interest Rate Capital production ability the more the capitals expected return, the higher the interest rates and vice versa. Uncertainty of capital production ability the more the uncertainty, the higher the risk premium required and the higher the interest rates and vice versa. Time preference of consumption the stronger preference to current consumption, the higher the risk premium required and the higher the interest rates and vice versa. Risk aversion the more the risk aversion, the higher the risk premium required and the lower the risk- free interest rates. Four basic factors5Theory of Real Interest Rates Real interest rates are determined by supply and demand of funds in the economy. 3 factors in determining real interest rates: Aggregate endowments Aggregate investment opportunities Aggregate preferences for different consumption path6 Consider a representative investor: Has endowment of ( e0, e1) Faces a bond market with interest rate r.7 He maximizes his utility over his consumption now and later:Where b is the bond holding, u0 and u”08 The optimality condition is Relative risk aversion coefficientThus, the real interest rate is given by9Nonlinear technologyTime 0Time 1-(1+RC)Investment opportunity setbb(1+r)10Linear technologyTime 0Time 1b(1+r)b -(1+RC)11 More generally, consider consumption grow at random rate. Investors maximize their expected utility over many periods. Where is his holdings of discount bonds, is future endowments, is future consumption, both can be uncertain.12The Benchmark of Interest Yield to Maturity (YTM) YTM varies with different financial instruments, because the exposure of financial instruments are quite different and the required risk premiums differ from each other.Risk-free interest varies with terms . Its called the term structure of interests. Risk-free interests?No!Yes!13 Nominal and real interest rates nominal interest rate = real interest rate+ inflation real interest rate = pure time value+ risk premium Compound interest interest earned on interest already earned Continuously compounding simple rate of return annually times of interest payments annually compounding rate of interest payments annuallyLetContinuously compounding14Financial Risks and Risk-free Security Default risk Liquidity risk Purchase power risk Interest risk Foreign exchange risk Other market risks Basic financial risks: Risk-free security: Substitute in reality: Treasury 15 Treasury Yield Curve Treasury yield curve usually has three forms: upward, flat and downward. Zero-coupon rates set Bills are zero coupon while notes and bonds have coupons. Zero-coupon rates set can be obtained by conversion.16 Conversion example:Treasury maturity par coupon rate current price1 year10%0A 2 years1,0001,000910.50982.10B17 Shapes of Yield Curveupwardflatdownward Some theories for the shapes of yield curve Unbiased expectations theory Liquidity preference theory Market segment theory Preferred habitat theory18. One years risk-free rate Forward Interest There is a no-dividend stock and its expected return is 15%. The current price is A mini case:. What is one years forward price of this stock?19Position Immediate Cash Flow Cash Flow in the FutureReplicating Stock Using risk-free bond and forward contractSuppose forward price F = $106 per shareShort sell $100 risk- free bondShort sell one stock forward at $106 per shareBuy one stock at $100 per share+$100 $1050106 S1 $100S1 Net Cash Flow0$1ArbitrageStock forward price = $105 per share20Forward price of a risky asset is not the expectation of the future spot price of the asset.Proposition!21The Forward Price for a Traded Asset The forward price for a traded asset without interim income is: F=SerT The forward price for a traded asset with deterministic dividend rate is: F=Se(r-q)T The above equ