Objectives:

1. Measuring Interest Rate Risk • Objectives: • Consider the relationship between the yields of different maturity zero coupon bonds: the Expectations Hypothesis. • Understand the interest rate risk of zero coupon bonds and the risk-adjusted Expectations Hypothesis. • Consider the valuation of multiple cashflow fixed-payment and floating-payment bonds and loans. • The risk of multiple cashflow fixed-income securities and the concept of duration. • Apply duration analysis to measure the interest rate risk of a financial intermediary’s net worth.

2. The Prices and Yields of Zero-Coupon Bonds • Interest rate risk refers to uncertain changes in the market price or yield of a default-free fixed-income security. • The simplest fixed-income security is a zero coupon bond: a bond that makes a single future payment at its maturity. • Thus, the main distinguishing feature of default-free zero coupon bonds is differences in their times until maturity. • Let us begin by considering why different maturity zero coupon bonds may have different yields to maturity. • To do this, recall that there are different ways of stating a zero-coupon bond’s annual yield-to-maturity depending on what compounding frequency is assumed.

3. Suppose that a zero coupon bond pays a cashflow of FV at maturity, and this maturity date is  years in the future. • Let PV be the current market price (present value) of this zero coupon bond. Also, let im be this bond’s m-times per year compounded annual yield. For example, i1 is an annualized, annually compounded yield while i4 is an annualized, quarterly compounded yield. • Then the relationship between this zero-coupon bond’s current market price, PV, and is current yield, im, is

4. Re-arranging the previous PV - FV relation, we have where is the cash flow’s growth factor or future value factor. • The yield, im, is analogous to a growth rate: the rate at which the • PV grows to equal the FV at maturity. A growth rate, however, • is not unique but depends on the assumed compounding • frequency. m = 1, 2, 4, 12, 365 corresponds to annual, semi- • annual, quarterly, monthly, and daily compounding, respectively. • Thus, given an asset’s PV and FV, for every assumed frequency, • m, there is a different yield or discount rate, im.

5. A compounding frequency that often simplifies calculations is • continuous compounding. It occurs when m is infinite: where i is the continuously-compounded interest rate and e is the exponential function  2.71828. Thus is the continuously-compounded growth factor. Its reciprocal value, , is the continuously- compounded discount factor: • Interest rates on loans and bank deposits are stated based on an • assumed compounding frequency, e.g., monthly or quarterly • One needs to know what compounding frequency is being • assumed to determine the payoffs of the loan or deposit.

6. Example: BankIllinois offers a 24-month Certificate of Deposit • (CD) that has a quarterly-compounded yield of i4 = 3.00 %. If • $10,000 is invested in this CD today, how much would the • CD be worth when it matures? • The yield having a compounding frequency equal to once per • year, im = i1, is called the annual percentage yield (APY). Let us • see how differences in the assumed compounding frequency affect • the stated yield for the same zero coupon security. • Example: BankIllinois offers a four-year CD having an APY of • 4.25 %. Thus, a deposit of $10,000 would be worth FV= • PV(1.0425)4 = $11,811.48 four years from now. What would • be yields, im, for this CD for different compounding frequencies, • m? That is, what yields give an equivalent growth factor of • FV/PV = (1.0425)4 ?

7. In general, since , then APY = i1 = (1 + im/m)m-1, or im = m[( 1+i1)1/m-1]. • When m = , Yields Corresponding to an APY of 4.25 %

8. Suppose that we quote yields assuming continuous compounding • (m = ). Let i() be the yield on a zero coupon bond (ZCB) that • matures in  years. The yield curve or term structure of interest • rates is a graph of ZCB yields, i(), by their times until maturity, • , at a particular date in time. i() “Inverted” Yield Curve: August 1982 “Humped” Yield Curve: Jan 2000 “Normal” Yield Curve: Jan 1999 Jan 2002 Jan 2001 Jan 2003 

9. The Expectations Hypothesis is a theory of the term structure • that can be understood by the following example. Let i,t() be the • continuously compounded yield at date t on a ZCB that matures in •  years. For short-hand, i,t() = it(). Consider the strategies: Strategy 1: At date t, invest $PV in a ZCB having a maturity of two years. At the end of two years, the bond’s future value will be Strategy 2: At date t, invest $PV in a ZCB having a maturity of one year. At the end to the year (date t+1), re-invest the proceeds into another one year bond yielding it+1(1). Then at the end of two years, the bond’s value is

10. If it+1(1) was known at date t, then the Law of One Price equates • the future values of the two investments (FV1 = FV2), implying • The Expectations Hypothesis recognizes that it+1(1) is uncertain • at date t. It modifies the relationship, replacing it+1(1) with its • expected value at date t, Et[it+1(1)]. This leads to the result • Thus, the current yield on a two year bond equals the average • of the yield on the current one year bond plus the expected yield • on the one year bond that will be issued one year from now.

11. Example: The continuously-compounded yields of U.S. • Treasury strips (zero-coupon bonds) that mature one and two • years from now are 4.25 percent and 4.00 percent, respectively. • What does the Expectations Hypothesis predict will be the • continuously- compounded yield observed one year from now • on a U.S. Treasury strip having one year until maturity? • Et[it+1(1)] also is called the one-year forward rate from • year 1 to 2. If the actual rate, it+1(1), turns out to exceed this • forward rate, an investor would have been better off rolling over • one-year bonds (Strategy 2). Otherwise, the two-year bond is • best (Strategy 1).

12. Using this same logic, the Expectations Hypothesis can be • generalized to explain the yield on a ZCB with  years until • maturity, it(). so that a the yield on a long-maturity ZCB is the average of the expected short-maturity yields that will occur during the life of the long-maturity ZCB. • The Expectations Hypothesis is useful for explaining the shape • of yield curves. A downward sloping (inverted) yield curve • implies that short-term (one-year) interest rates are expected to fall. • Conversely, a literal interpretation of the Expectations Hypothesis • suggests that when the yield curve is upward sloping, then short- • term interest rates are expected to rise in the future.

13. ZCB Risk and Return • What is the relationship between expected returns and risk of • different maturity default-free ZCB? Note the following three • characteristics of bonds' returns: 1) Different maturity bonds' returns are highly correlated over short periods of time. Correlations between Monthly Returns of U.S. Treasury Securities, 1970-1996 Maturity in Years 0.5 1 2 4 6 8 10 12 14 0.5 1 1 0.9183 1 2 0.7892 0.9588 1 4 0.6777 0.8782 0.9684 1 6 0.6187 0.8244 0.9252 0.9874 1 8 0.5765 0.7818 0.8848 0.9627 0.9928 1 10 0.5421 0.7448 0.8476 0.9347 0.9768 0.9951 1 12 0.5114 0.7103 0.8122 0.9049 0.9554 0.9825 0.9959 1 14 0.4821 0.6761 0.7767 0.8729 0.9293 0.9633 0.9842 0.9961 1

14. 2) Long term bond prices display greater return volatility (capital risk) than short term bond prices. Annualized Standard Deviations of Monthly Returns of U.S. Treasury Securities, 1970-1996 Maturity in Years 0.5 1 2 4 6 8 10 12 14 Std. Dev.(%) 1.3131 2.2493 3.8839 6.3937 8.6447 10.8346 12.9890 15.1258 17.2817 3) There's a positive interest rate risk premium. Annualized Means of Monthly Returns of U.S. Treasury Securities, 1970-1996 Maturity in Years 0.5 1 2 4 6 8 10 12 14 Means (%) 7.6936 8.0023 8.4718 9.0316 9.3276 9.5225 9.6719 9.7965 9.9083

15. Thus, longer-term bonds have more interest rate risk than shorter • term bonds and require an interest rate risk premium. The • Risk-Adjusted Expectations Hypothesis implies where rp()  0 and increases with .

16. Prices and Yields of Multiple Cashflow Bonds • Having considered the relationship between yields (or prices) of different maturity ZCBs, we consider the relationship between ZCB yields or prices and the price (market value) of a multiple-cashflow bond. • Let P( ) be the price of a ZCB that pays $1 in  years, and let im() be this ZCB’s m-times per year compounded yield. The relationship between this bond’s price and its yield is • Now if we can observe the prices of different maturity ZCBs, we can use them to value a multiple-cashflow fixed-income security.

17. A bond that promises multiple cashflows can be viewed as • a set or portfolio of ZCBs, with each ZCB maturing at the date • that a cash flow is paid. Thus, if a bond makes n payments with • the ith payment being made i years in the future and equaling • a cashflow of CFi, then the no arbitrage value of this bond is which, if written in terms of ZCB yields rather than prices, is • This is an application of the Law of One Price: the portfolio • of ZCBs pays the same cashflows as the multiple-payment • bond, so their present values must be the same.

18. Arbitrage enforces the Law of One Price. In the Treasury security market, this occurs by bond traders “stripping” or “reconstituting” Treasury bonds when there is a discrepancy between a Treasury bond’s price and the value of its cashflows as given by Treasury ZCB “strips.” • Prices of U.S. Treasury strips, July 29, 2004.

19. Example: A default-free Eurodollar bond has exactly 3 years until maturity. It pays a face value (principal) of $100 at maturity and annual coupon payments of $8 at the ends of each of the next three years. If the current market prices of zero coupon bonds that pay $1 at the ends of each of the next three years are P(1) = $0.97, P(2) = $0.92, and P(3) = $0.87, respectively, what is the no-arbitrage price of this Eurodollar bond?

20. Example: A default-free Eurodollar bond has exactly 2 years until maturity. It pays a face value (principal) of $100 at maturity and annual coupon payments of $8 at the ends of each of the next two years. If the current market price of this Eurodollar bond is $100 and the annually-compounded yield on a two-year zero coupon bond is 8.1 percent, what must be the annually-compounded yield on a one-year zero-coupon bond for there to be no arbitrage opportunities? which implies i1(1) =5.56 %.

21. Note that, when valuing multiple cash flow bonds based on • the observed yield curve of ZCBs, each cash flow is discounted • by a different yield, im(i). Only if the ZCB yield curve • happened to be “flat” would we discount by the same yield. • Hence, in general, finding the no-arbitrage value of multiple • payment bonds requires using yields (or prices) for many • different ZCBs. • A fundamentally different exercise is to solve for a multiple • payment bond’spromised yield-to-maturity (YTM). To do • this, one must already know this bond’s current market price.

22. Recall that a bond’s YTM is definedas the single discount rate • that equates the bond’s discounted cash flows to its current • market price. Thus, for an n-year coupon bond paying annual • coupons of PMT and a face value of FV, its annually- • compounded YTM,denoted R, is: • A coupon bond’s yield-to-maturity should not be confused • with its coupon rate, which for a bond making annual • payments is PMT/FV. Neither should it be confused with the • bond’s current yield, which is defined as PMT/PV.

23. Depending on a coupon bond’s current market price, PV, • relative to its face value, FV, certain relationships exist between • the bond’s coupon rate, YTM, and its current yield. • If PV = FV, then the bond is selling at par. In this case a bond’s • YTM and current yield both equal its coupon rate: • If, instead, PV > FV, the bond is a premium bond. In this case: • Finally, if PV < FV, the bond is a discount bond. In this case:

24. Example: A 7-year Eurobond making annual payments has PMT=8, FV=100, and PV = 96.55. Hence, it is a discount bond. Solving numerically for the bond’s YTM, R= rate(7,8,-96.55,100) = 0.0868 > PMT/PV =0.0829 > PMT/FV = 0.08. • A coupon bond’s YTM should not be interpreted as the return • one would earn by investing in the bond, even if the bond is held • to maturity. When future interest rates are uncertain, the returns • earned from re-investing the bond’s coupons are uncertain. • Also, YTM should not be used to judge the relative benefits of • investing in two different coupon bonds, even if they have the • same maturity. If the Law of One Price holds (no-arbitrage), • equivalent maturity bonds with different coupons will have • different YTMs when the ZCB yield curve is not “flat.”

25. Not all bonds promise to pay fixed-coupons. Floating-rate • bonds and loans pay a fixed principal value, FV, at maturity, • but have coupons, PMTt , tied to the level of a particular short • term interest rate. • Most often, floating rate coupons are tied to the interest rate • on a money market security having a maturity equal to the • bond’s coupon interval. • Example: Many floating-rate bonds paying semi-annual • coupons tie their coupon rates to six-month LIBOR. If we let • i½, t be the semi-annually compounded yield on a six-month • LIBOR investment at date t, then the bond’s coupon payment • made at date t, PMTt equals

26. Hence, if a floating-rate bond makes semi-annual coupon • payments on April 15 and October 15 of every year, a coupon • payment made on April 15, 2006 would be equal to ½ FV • times the six-month LIBOR prevailing on Oct 15, 2005. • Thus, the actual amount of the coupon payment to be made • in April 2006 is unknown until the coupon reset date of Oct 15, • 2005 when the relevant six-month LIBOR becomes known. • Floating rate bonds such as this LIBOR-linked bond will • produce cash flows equivalent to “rolling-over” an investment • in money market securities. • To see this, suppose we invest an amount, FV, in a six-month • LIBOR deposit with a semi-annual compounded yield of i½, t.

27. At the end of six-months, the balance equals [1+½ i½, t]FV . • Let us keep the interest of ½ i½, tFV and re-invest the princi- • pal of FV in a new six-month LIBOR deposit yielding i½, t+½. • At the end of the next six-months, we again keep the interest • of ½ i½, t+½FV and re-invest the principal of FV in another • six-month LIBOR deposit yielding i½, t+1. • By repeating this procedure of rolling-over our principal of FV • into six-month LIBOR deposits while keeping the interest paid, • we produce cash flows over an n year period of PMTt+½,PMTt+1 ,… , PMTt+n-½,PMTt+n+ FV ½ i½, tFV ,½ i½, t+½FV , ... , ½ i½, t+n-1(½)FV , [1+½ i½, t+n-½]FV

28. Clearly, the above series of payments is exactly the same as • those one would obtain from investing in the LIBOR-linked • floating-rate bond having an n year maturity. Thus, the value • and risk of the floating-rate bond investment equals the value • and risk of this series of money market investments. • In particular, the value of this default-free floating rate bond • will equal its principal value, FV, immediately after each • coupon payment is made. • Each coupon payment is uncertain prior to its reset date, but it • is it still possible to determine each coupon’s no-arbitrage value • prior to its reset date. This has practical importance since • floating-rate bonds are sometimes “stripped” and their coupons • are then sold separately.

29. Suppose we wish to value the floating rate coupon to be • made on April 15, 2006. Assume that today’s date is Oct 15 • 2004 and consider the following replication strategy. • On Oct 15 2004: (date t) • Invest in a one year ZCB that pays FV at maturity, which • is an amount equal to FVP(1). • Borrow at a one and one-half year maturity, agreeing to pay • back FV . The amount borrowed is FVP(1½). • On Oct 15 2005: (date t+1) • Re-invest the proceeds of FV into six-month LIBOR at i½, t+1. • On April 15 2006: (date t+1½) • Obtain re-investment cashflow of FV [1+ ½ i½, t+1]. • Repay borrowing of FV . • Note that the net cash flow on April 15, 2006 (date t+1½) is • ½FV i½, t+1, which exactly replicates the floating-rate coupon.

30. Therefore, by the Law of One Price, the value of the floating • rate coupon must equal the initial (date t) cost of creating the • replicating cashflow, which is • FVP(1) - FVP(1½) = FV [P(1) - P(1½)] Example: A floating rate coupon tied to six-month LIBOR and having a principal of $100 is scheduled to be paid exactly 4 years from today. If the coupon reset date is six months prior (3½ years from today) and today’s prices of ZCBs are P(3½ ) = 0.87 and P(4) = 0.85, what is the present value of this coupon? PVcoupon = FV [P(3½) - P(4)] = ($100) [0.87 – 0.85] = $ 2.00

31. The Measurement of Interest Rate Risk Using Duration • Recall that a multiple-fixed cashflow bond can be valued as a • portfolio of ZCBs using the no-arbitrage relation: where PV= the current price of a bond that promises n future payments. i = number of years until the ith payment is received, i = 1,…, n. P(i) = the current price of a ZCB that pays $1 in i years. CFi= the bond’s cash flow to be paid in i years.

32. Duration is an indirect measure of a multiple payment bond's • interest rate risk. It can also be used to measure the risk of a • portfolio of fixed-income securities. • The duration of a bond is defined to equal the maturity of a ZCB • that has the same interest rate risk as the bond. Example: Suppose that a multiple payment bond has a rate of return standard deviation of 4 percent per year. Based on the previous graph, we see that a 2-year ZCB also has a standard deviation of 4 percent. Hence, the multiple payment bond’s duration = 2 years. • Formally, if we let M be the standard deviation of the multiple • payment bond (say 4%) and () is the standard deviation of a ZCB • of maturity , then the multiple payment bond’s duration, D,satisfies • M = (D)

33. Duration can be calculated by taking a present value weighted • average of the times until a multiple payment bond's payments • are received. This formula for duration is given by* • Note that P(i) CFi / PVis the ratio of the present value of the • cashflow received in i years to the present value of all cashflows. * There are different duration formulas, such as Macaulay duration, Modified duration, and key rate duration. The above formula is sometimes referred to as Fisher-Weil duration and is one of the more intuitive and accurate measures.

34. Example: A default-free Eurodollar bond has exactly 3 years • until maturity. It pays a face value (principal) of $100 at • maturity and annual coupon payments of $8 at the ends of each • of the next three years. If the current market prices of zero • coupon bonds that pay $1 at the ends of each of the next three • years are P(1) = $0.94, P(2) = $0.88, and P(3) = $0.83, • respectively, what is this bond’s duration?

35. Example: Suppose the current term structure is such that What are the value and duration of 1) a two-year ZCB paying $100 at maturity? 2) a bond paying cash flows of $100 at the ends of both the first and second years?

36. Coupon bonds whose coupons are high relative to their • principal or face value have a greater present value of • payments occurring earlier in their life. Hence, their durations • will be shorter than bonds paying lower coupons. • Duration can also be used to measure the interest rate risk of • liabilities, that is, payments that are made, rather than received. • This can allow us to analyze the interest rate risk of a financial • institution's net worth or equity capital. • What happens to net worth if market interest rates rise • unexpectedly? Fall unexpectedly?

37. The duration (interest rate risk) of a firm’s net worth (equity) • can be calculated from the duration of its assets, DA, and the • duration of its non-ownership liabilities (debt), DL. Let A be the • value of the firm’s assets, and let L be the value of the firm’s • debt liabilities, so that net worth (equity), E, equals A - L. • Then the duration of the firm’s net worth, DE, is given by • Example: A bank has assets of $1.1 billion and debt (deposits) of • $1.0 billion. The durations of its assets and debt are 1.25 years • and 1 year, respectively. What is the duration of its net worth?

38. If a FI manager wishes to make the FI’s net worth be unaffected by market interest rates, then she should set DE = 0. For this to occur, she must manage DA and DL such that • Note that, if instead the FI manager matches the durations of assets and liabilities, that is, DA = DL= D, then • so that the duration of net worth equals that of the assets. The effect is that the FI’s net worth to asset ratio, E/A, is unaffected by market interest rates. This would stabilize the FI’s capital ratio, a result that would please FI (bank) regulators.

39. Duration and Immunization • Immunization is an investment strategy that involves a fixed- • income portfolio. Its objective is to determine the smallest initial • portfolio value that, when managed properly, will have a future • value sufficient to pay a given liability at a specific future date. Example: A pension fund promises to pay workers a retirement benefit of $X in 10 years. What is the minimum that the pension fund needs to invest today, denoted Xt? How should the investment be managed to guarantee that the fund can pay $X in 10 years? 1) One strategy is to buy and hold 10 year ZCBs. If the current market price of a ZCB that pays $1 in ten years is Pt(10), then the minimum amount that the pension fund needs to invest is

40. 2) An alternative immunization strategy would be to purchase a portfolio of coupon bonds of different maturities that are initially worth Xt =Pt(10)X. In selecting this coupon bond portfolio, its initial duration, D, equals 10 years. As time passes, we manage the portfolio so as to set its duration equal to the years remaining until the final payment of $X is made. Thus, after one year, the portfolio’s duration is D = 9 years. After two years, the portfolio’s duration should be D = 8 years, and so on until after 9 years the portfolio’s duration is D = 1 year. • By shrinking the portfolio’s duration as the payment date of the • retirement benefits is approached, the risk of an investment in • original-maturity 10 year ZCBs is replicated. Thus, the two • strategies will produce the same return over the 10 year period.