CHAPTER TWO: Time Value of Money and Term Structure of Interest

1. CHAPTER TWO: Time Value of Money and Term Structure of Interest

2. Yes ! is the expected rate of return, i.e., the mean of the discount rates for different terms Discounted Cash Flow Formula ? No ! is the discount rate that cannot be used for so long period Let

3. Term 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.

4. Forms 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.

5. Determination of Interest Rate • Four basic factors • Capital production ability—— the more the capital’s 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.

6. Theory 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 path

7. Consider a representative investor: • Has endowment of ( e0, e1) • Faces a bond market with interest rate r.

8. He maximizes his utility over his consumption now and later: Where b is the bond holding, u’>0 and u”<0

9. The optimality condition is Thus, the real interest rate is given by Relative risk aversion coefficient

10. Nonlinear technology Time 1 b(1+r) -(1+RC) b Time 0 Investment opportunity set

11. Linear technology Time 1 (1+r)b -(1+RC) b Time 0

12. 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.

13. The Benchmark of Interest — Yield to Maturity (YTM) ? No! 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 interests ? Yes! Risk-free interest varies with terms . It’s called the term structure of interests.

14. Nominal and real interest rates • Compound interest — interest earned on interest already earned — nominal interest rate = real interest rate+ inflation — real interest rate = pure time value+ risk premium — Continuously compounding — simple rate of return annually — times of interest payments annually — compounding rate of interest payments annually Let Continuously compounding

15. Financial Risks and Risk-free Security • Basic financial risks: — Default risk — Liquidity risk — Purchase power risk — Interest risk — Foreign exchange risk — Other market risks — Risk-free security: • Substitute in reality: Treasury

16. — 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.

17. Conversion example: Treasury maturity par coupon rate current price A 1 year 1,000 0 910.50 B 2 years 1,000 10% 982.10

18. — Shapes of Yield Curve downward upward flat • Some theories for the shapes of yield curve — Unbiased expectations theory — Liquidity preference theory — Market segment theory — Preferred habitat theory

19. Forward Interest A mini case: —There is a no-dividend stock and its expected return is 15%. The current price is . One year’s risk-free rate . What is one year’s forward price of this stock? ?

20. Suppose forward price F = $106 per share Replicating Stock Using risk-free bond and forward contract Position Immediate Cash Flow Cash Flow in the Future Short sell $100 risk-free bond  $105 +$100 Short sell one stock forward at $106 per share 0 106 – S1 Buy one stock at $100 per share  $100 S1 Net Cash Flow 0 $1 Stock forward price = $105 per share Arbitrage

21. Proposition! Forward price of a risky asset is not the expectation of the future spot price of the asset.

22. The 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 equation can be obtained through the following arbitrage strategy: • Buy spot e-qT of the asset and reinvest income from the asset in the asset. • Short a forward contract on one unit of the asset.

23. F 0 T Se-qT The Forward Price for a Traded Asset • The holding of the asset grows at rate q so that e-qT x eqT ,or exactly one unit of the asset, is held at time T. Under the terms of the forward contract, the asset is sold for F at time T, leading to the following cash flow: Se-qT=Fe-rT F=Se(r-q)T

24. 0 1 2 3 n — Zero-coupon rates & forward interest rates • Forward interest rates are the expectation of future risk-free spot interest rates.

25. Zero-coupon rates Discount factors Forward rates • Zero-coupon rates, discount factors & • forward interest rates

26. Valuation of FRA • An FRA is equivalent to an agreement where interest at a predetermined rate, RK, is exchanged for interest at the market rate, R. • Reference rate R • Interest rate RK to be earned • Time period between T1 and T2 • Notional amount L

27. Valuation Rule of FRA • FRA has the cash flow: L(R- RK)(T2-T1) at T2 • An FRA can be valued by assuming that the forward interest rate is certain to be realized. • The value of the FRA promising RK is: • L(RF -RK)(T2-T1)P(0,T2) • P(0,T2) is the price of zero discount bond maturing at T2 with notional 1. • Is there anything special about this rule?

28. FRA: Cash Flow Decomposition Floating rate deposit Starting t1 ending t2 Buying an FRA Fixed rate Loan Starting t1 ending t2 = +

29. FRA: Cash Flow Decomposition

30. 0 2 n t 1 0 1 2 n t Swap Price — Interest rate swap Cash Flow of Buyer Cash Flow of Seller

31. —Interest Rate Swap • Quotation for LIBOR

32. 0 1 2 … n t — Pricing Par Bonds

33. Par Par*i Par*i Par*i 0 1 2 …. n t Par*i Par*i Par*i Par 0 1 2 … n t Par*fn Par*f1 Par Par*f2 0 1 2 … n t Par*fn Par Par*f1 Par*f2 — Zero-coupon pricing technique Investment Cash Flow Financing Cash Flow

34. Par Par 0 1 2 … n t 0 1 2 … n t 0 1 2 … n t Par*f1 Par*f1 Par*fn Par*fn Par Par*f2 Par*f2 Par Par 0 1 2 … n t Par — Further illustration of composition & decomposition NPV = 0 • Decomposition of finance cash flow = 0 + =

35. Swap as Sequence of FRA • Calculate forward rates for each of the LIBOR rates that will determine swap cash flows. • Calculate swap cash flows on the assumption that the LIBOR rates will equal the forward rate. • Set the swap value equal to the present value of these cash flows.

36. 1 2 0 1 2 n t 3 Swap Decomposition of FRAs

37. P1 i1 i1 i1 P1 0 1 2 …. n t i1 i1 i1 P1 0 1 2 …. n t i2 i2 i2 P2 P2 0 1 2 …. n t i2 i2 i2 P2 Currency Swap • Fixed interest rate currency swap

38. Pricing Currency Swap as Sequence of Currency Forwards • Currency forward contract can be priced as if the forward price of the underlying asset is realized. • Forward price for a foreign currency can be thought of as a stock with price S and paying dividend with known rate of foreign currency interest rate rf • Forward price of a foreign currency is S*exp((rd-rf)T) Where rd is the interest rate for domestic currency, and rfis the interest rate for foreign currency.

39. Summary of Chapter Two • Time Value of Money  Term Structure of Interest • Risk-free Rates are Benchmark and Market Expectation • Forward Price is not the Expectation of Future Spot Price for Risky Assets • Forward Price for traded asset • Replication  Composition & Decomposition