# FINANCIAL AND INVESTMENT INSTRUMENTS. Lecture 6: Bonds and Debt Instruments: Valuation and Risk Management

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1 AIMS FINANCIAL AND INVESTMENT INSTRUMENTS Lecture 6: Bonds and Debt Instruments: Valuation and Risk Management After this session you should Know how to value a bond Know the difference between the term structure and the yield curve Know the main risks associated with bond investments Know what is meant by duration and how it can be used to estimate the effect of interest rate changes on bond values. Be able to calculate convexity, and use convexity to calculate the effect of interest rates changes on bond prices Be able to calculate the duration and yield to maturity of a bond portfolio Be able to immunise a bond portfolio against the effects of interest rate changes. Appreciate the limitations of duration measures Be able to compare and implement active and passive approaches to bond portfolio management Required Reading: Investment Analysis and Portfolio Management, 7th edition, Frank K. Reilly and Keith C. Brown (Thomson South-Western, 00: Chapters 19 & 0. Solnik, B. and D. McLeavey (00, International Investments, 5th ed, Addison Wesley Chapter 7

2 Valuing a Risk Free Bond A risk free bond can be valued by calculating the Present Value. To value a fixed Coupon Bond. If R 1 is the interest rate for 1 period, R the interest rate for periods, R N the interest rate for N periods then PV = (1 C R (1 C R (1 C R C n (1 1 R n Example: C= 5, R 1 = 6%, R = 7%, N =. => PV = \$96.4. The interest rates R 1, R R N are spot rates of interest on an N-period pure discount bond, and collectively known as the Term Structure of Interest Rates. R 1 is the annualised interest rate on a one-period pure discount bond, R is the annualised interest rate on a two-period pure discount bond, etc. The Expectations Theory of the Term Structure argues that R will be greater than R 1 if interest rates are expected to rise in the second period. If R 1 = R =. = R N = R, then the term structure is said to be FLAT. Assuming a flat term structure: Formula for an N Period Bond with a coupon of C% every period and a nominal value of \$100, and risk free rate of R PV = C C R R C R C n R e.g. C = 5%, N = 10, R = 0.06 then P = 9.6. Comparing Bonds This looks complicated because bonds have different coupons and different maturities. Usually 1 number is quoted: the yield to maturity. The Yield to Maturity, Y, is the value of R, which makes the market price, P, of a bond equal to its present value. It is also called Redemption Yield or Internal Rate of Return, Pr C C C ice= Y Y Y C n Y For example: the price of a 10-year 5% \$100 nominal value bond is \$9.64 Then the yield to maturity is 6%. (Usually found by trial-and-error The yield of 6 % may be compared to the yields on other bonds with a similar maturity so see if it offers a competitive return.

3 R t Term Structure Years Y t Yield Curve Years

4 Yield Curve The graph of yields to maturity against time to maturity is known as the YIELD CURVE of different bonds. This may be upward sloping (often thought to be the usual case downward sloping or flat. NOTE THERE IS A DIFFERENCE BETWEEN THE YIELD CURVE AND THE TERM STRUCTURE. When the term structure is flat, the term structure and yield curves are the same. We illustrate the difference between the yield curve and the term to maturity, with the following example Example A 6% treasury bond with six months to maturity has a price of \$980.95, and a second bond with a coupon of 15% and a maturity of one year has a price of \$ Calculate the yield to maturity on both these bonds, and calculate the term structure of interest rates for the next two six month periods from these bond prices. Term Structure: Bond 1: Yield Curve: Bond 1: PV PV C = = = R 1 = 0.05 R 1 R 1 C = = = Y = 0.05 Y Y Term Structure: Bond : PV C C = 99.07= = R n =0.08 R (1 (1.05 (1 1 R R n Yield Curve: C C Bond : PV = 99.07= = Y= Y Y Y Y This example illustrates there is a small difference between the yield curve and the future spot rates, though there is a relationship between the yield and the term structure: the redemption yield on a bond is a non-linear weighted average of the spot rates, where the weights are related to the size of the coupon payments. To price a bond we should use the spot rates implicit in the term structure: this is called the arbitrage-free method of valuing bonds, though as an approximation, the yield on a similar bond is also used. Yield Spreads Yield spreads are the differences between yields on different types of bonds. The term spread is the difference between the yields on different maturities ie long-term and shortterm bonds; the credit spread (or default spread is the difference in yields on bonds with

5 different credit ratings, coupon spread is the difference in yields on bonds with different coupons. Yield spreads are used in active portfolio bond management (see below. Risks Associated with Debt Instruments The main risk is interest rate risk, but there are a number of other risks that we may identify: call and repayment risk; default risk. In addition liquidity risks and currency risks are the risk that we have already encountered in discussing equity markets. Duration and Interest Rate Sensitivity of a Bond What Determines the Price Volatility for Bonds? Bond prices move inversely to bond yields Bond prices volatility is directly related to term to maturity Bond price volatility increases at a diminishing rate as term to maturity increases Bond price movements are not symmetrical Bond price volatility is inversely related to coupon Key Concept the Longer the Bond the More Interest Rate Sensitive it is. Example: A 10 year zero coupon bond is more interest rate sensitive than a 5 year zero coupon bond. The length of a bond with coupon payments is accurately measured by its DURATION. Duration is a measure of "how long" a bond is. Duration is average number of periods we have to wait for cash flow from the bond. The duration of a pure discount bond is its time to maturity. But for coupon bonds, it is necessary to take into account the size of the coupon, since a high early coupon payment is more valuable than the same payment nearer maturity. For example 8% Treasury 01 sells for 100p, and 4% Exchequer 01 sells for 60p. So x Treasury costs 00p and so does 5 x Exchequer, but x Treasury pays coupons of 4p which is more than 5 x Exchequer which pays 0p so the Treasury stock looks better, is it? The duration of a bond is the weighted maturity of a bond, where the weights are given by the present values of the cash flows. Duration is the weighted average of payments to bondholders. To calculate duration, D, we take each period and weight it by the proportion of the present value of the bond we receive at the end of that period. Formula for (Macaulay Duration of an N year Bond: where D = W 1 W W.... NW N Present value of coupon ct 1 W t = = Price of bond p Y t

6 where Y is bond's yield to maturity N is bond's maturity c t is coupon payment in period t p is bond s price Example: A 10 year 5% bond with nominal value \$100, price = \$9.64, yield to maturity = 6% (assume annual interest payments = 9.64 ( ( ( (1.06 D 10 D = 8.0 years Properties of Duration 1 Duration is measured in units of time Duration of a coupon bond is always less than its time to maturity - Zero coupon bonds have duration equal to maturity There is a positive relationship between term to maturity and duration. 4 For a given maturity, as coupon increases or as the yield increases, duration falls - There is an inverse relationship between coupon and duration - There is a inverse relationship between YTM and duration 5 Duration is a measure of the elasticity of the price of a bond with respect to changes in interest rates, and is important for immunisation strategies. 6 Sinking funds and call provisions cause decline in duration Price of a bond is p= c1 c Y Y c Y c N F... Y N Differentiate w.r.t. Y dp = - dy c1 Y c - Y c - Y 4 c N F...- N Y N1 Defining elasticity of bond prices to interest rates as the percentage change in prices over the percentage change in one plus the yield to maturity

7 dp p i. dy c1 = - p 1 - Y c 1 p Y - c 1 p Y c..- N N F p 1 Y N = D and modified duration D mod = D/Y For small interest rate changes dp di p D p Y mod = Dmod * p % Change in Value of Bond = - (D/(1 Y x (change in Y approximately for a small change in the yield Y. Example If the yield rises 1% on the 5% 10 year bond above then % change in bond value = -(8.0/1.06(0.01 = or 7.6% Change in bond \$ value = * \$9.64 = - \$7.01 So new bond price is \$85.6 However: if we set yield of 0.07 into present value formula, new present value (price of bond is \$85.95: so duration only gives an approximate measure of interest rate sensitivity In summary, if we want to know whether long term bonds are more or less sensitive to interest rate movements, in general the answer is ambiguous and will depend on the size of the coupon payments of the bonds. Duration analysis answers the question unambiguously: for a given change in yields the volatility of bond prices will be higher the higher is duration Limitations of Duration: a. The percentage change estimates using modified duration are good only for small-yield changes b. Difficult to determine the interest rate sensitivity of a portfolio of bonds when there is a change in interest rates and the yield curve experiences a nonparallel shift c. Initial assumption that cash flows from the bond are not affected by yield changes

8 bond price Duration Actual Price-yield relationship yield Convexity Modified duration provides an approximate estimate of the change in bond price for a given change in interest rates. However it is only approximate, because the relationship between interest rates and bond prices in the present value formula is non-linear: duration is a linear approximation of the relationship between bond prices and interest rates. Convexity is a measure of the curvature of the price-yield relationship. The price-yield relationship is not a straight line but a curvilinear relationship (i.e., convex This relationship can be applied to a single bond, a portfolio of bonds, etc. The convex relationship will differ depending on the nature of the cash flows, that is, coupon and maturity Convexity = (d p/di /p c1 c c c4 = i i i i 6 d p d i... d p d i 1 = i c1 c 6 i i c 1 i c4 0 i 4...

9 N d p ct = ( t t d i i t= 1 i 1 t Price Change Due to convexity = ½ x price x Convexity x ( yield. Returning to the above example: 10 year 5% bond with nominal value \$100, price = \$9.64, yield to maturity = 6%, and duration =8.0 Convexity = ( ( ( ( Convexity = 7.57 Price Change Due to convexity = ½ x 9.64 x 7.57 x (0.01 = 0.6 So price change due to duration and convexity = - \$ = - \$6.67 So new price = \$85.96, which is much closer to the actual price change using the PV formula Properties of Convexity Convexity is a measure of the curvature of the price-yield relationship Factors and bond convexity Inverse relationship between convexity and coupon Direct relationship between convexity and maturity Inverse relationship between convexity and yield

10 Calculating Present Value, Yield, Duration for a PORTFOLIO of Bonds Present value The present value of a portfolio of bonds is just the sum of the present values of the bonds held. Duration The duration of the portfolio is just the weighted average of the durations of the bonds that make up the portfolio. It is the sum of each of the durations multiplied by its portfolio weight. Example We have a \$10 million portfolio of bonds, with \$4 million in two of the bonds and million in the third. The duration of the bonds are 1,, years respectively. Then the portfolio s duration is D = 0.4 x x 0. x = 1.8 years Yield to Maturity The yield to maturity is NOT EXACTLY the portfolio of the yields to maturity. It must be calculated separately using the portfolio s cash flows. Example: Bond A is a year 5% coupon bond. Its yield to maturity is 6%, price = \$98.17 Bond B is a year 6% coupon bond. Its yield to maturity is 7%, price = \$97.8 We form a portfolio buying \$600m nominal value of A and \$400 million of B. Cash Flows YEAR CASH FLOW( \$ mil. 0-6( (97.8 = (5 4(6 = 54 6(5 4(6 600 = 654 4(6 400 = = 54/Y 654/Y 44/Y Y = 6.5%

11 Managing a Bond Portfolio Two general strategic approaches: Passive and Active. A fund manager may choose one approach entirely or she may manage part of her portfolio passively and the rest is managed actively. A. Passive Management This approach constructs a portfolio, which tracks a well-known index e.g. Lehman, Merrill Lynch etc. Immunization Some fund managers especially pension fund managers have liabilities as well as assets to consider. These managers must be careful that the value of their assets always at least matches their liabilities e.g. the payments to pensioners. Immunization constructs a bond portfolio which matches the value of liabilities even if interest rates change. Two methods can be considered: Cash flow matching. Managers buys bonds so that the coupon and maturity payments exactly pay for the cash flows that must be made to pensioners. This is not usually possible because there are not enough bonds available in the market to achieve this. Duration matching This sets the modified durations of assets and liabilities to be equal. A change in yields will affect have the same effect on both assets and liabilities. Problems with this approach occur when all interest rates do not change by the same amount. After a change in interest rates durations will change so that the manager will have to adjust her portfolio. To immunize a liability using duration matching: 1. Compute duration of liability. Decide on assets in which to invest. Compute durations of these assets 4. Compute investment weights on the basis that duration of asset portfolio is equal to duration of liabilities Example using Duration to Immunize A pension fund has a scheme that involves a lump sum contribution now, and then pays out 10K per year for five years. The fund decides to invest the lump sum contribution into two bonds: a 10% 1 year bond and a 4% 4 year bond. The term structure of interest rates is at present 1% for all maturities. How much of the contribution should the fund place into each bond, in order to immunize its liabilities?. 1. Compute present value of liabilities and duration of liabilities Present value of fund = 10/(1.1 10/( /(1.1 5 = 6.046K Duration of fund (liabilities =.7745

13 Comprehension Check Define the terms: bond coupon, floating rate bond, duration, yield to maturity,default risk, bond quality rating, cash matching, immunisation. Know how to value a bond sing the present value formula What is the difference between the term structure and the yield curve List the main risks associated with bond investments What does duration mean and how it can be used to estimate the effect of interest rate changes on bond values. What is convexity, and how can convexity be used to calculate the effect of interest rates changes on bond prices What is the duration and yield to maturity of a bond portfolio How would you immunise a bond portfolio against the effects of interest rate changes. What are the limitations of duration measures What is the difference between active and passive approaches to bond portfolio management

14 Exercise Questions 1. Calculate the yield to maturity and duration of a 4% year government bond, priced at \$90 per \$100 nominal value. How much would you expect its price to change if yields rose 0.5%?. Calculate the nominal amounts purchased, cash flows, duration and yield to maturity of a portfolio of two bonds to be formed as follows: \$500 million invested in Government Bond A, which is a 1 year 5 % coupon bond with a price of \$98 per \$100 nominal. \$1000 million invested in Government Bond B, which is a year 6% coupon bond with a price of \$100 per \$100 nominal.. A bank has a portfolio of liabilities valued at \$ 100 million with a duration of 1.5 years and yield of 6% p.a. Its loan( i.e. asset portfolio also has a value of \$100 million, but has a yield of 9% and duration of years. If yields rise 0.5% will the banks asset/ liability position still be in balance? What do you estimate the new difference between the value of assets and liabilities to be. 4. You are asked to manage actively a portfolio of US government and corporate bonds. What economic factors would you take into account in deciding which general classes of bonds to buy/sell? 5. A year floating rate note has a nominal or face value of \$100 and pays 1 month LIBOR every 1 months, with the rate reset at the beginning of each 1 month period. 1 month LIBOR was last set at 6% 6 months ago. Value the bond. 6. Consider a bond exactly similar to question except that the bond pays LIBOR %. Value this bond.

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