Investments Session 10. Managing Bond Portfolios EPFL - Master in Financial Engineering Philip Valta Spring 2010 Bond Portfolios (Session 10) Investments Spring 2010 1 / 54
Outline of the lecture Duration Convexity Managing Bond Portfolios Immunization Bond Portfolios (Session 10) Investments Spring 2010 2 / 54
Bond Price Duration Yield to Price Relationship Yield to Price Relationship 160 140 120 100 80 60 40 20 0 5 10 15 20 Yield to Maturity (%) Bond Portfolios (Session 10) Investments Spring 2010 3 / 54
Duration Interest Rate Risk Interest Rate Risk Interest rates can uctuate substantially. As a results, bondholders experience capital gains and losses. Thus, even though coupon and principal payments are guaranteed (Treasuries), xed income investments are risky. Why? Why do bond prices respond to interest rate uctuations? Bond Portfolios (Session 10) Investments Spring 2010 4 / 54
Duration Change in Bond Price as a Function of Change in YTM Change of Bond Price as a Function of Change in YTM Bond Portfolios (Session 10) Investments Spring 2010 5 / 54
Duration Bond Price as a Function of YTM Bond Price as a Function of YTM Bond prices and yields are inversely related: as yields increase, bond prices fall; as yields fall, bond prices rise. An increase in a bond s yield to maturity results in a smaller price change than a decrease in yield of equal magnitude. Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of short-term bonds. Bond Portfolios (Session 10) Investments Spring 2010 6 / 54
Duration Bond Price as a Function of YTM Bond Price as a Function of YTM The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases. In other words, interest rate risk is less than proportional to bond maturity. Interest rate risk is inversely related to bonds coupon rate. Prices of low-coupon bonds are more sensitive to changes in interest rates than prices of high-coupon bonds. The sensitivity of a bond s price to a change in its yield is inversely related to the yield to maturity at which the bond currently is selling. Bond Portfolios (Session 10) Investments Spring 2010 7 / 54
Duration Changes in Interest Rates Changes in Interest Rates We have seen how to use interest rates to calculate the value of a zero coupon bond, coupon bond, etc. Since interest rates vary over time, it is important to know what happens to the value of a bond when interest rates change. Suppose you are managing a USD 100 Mio. bond portfolio. How does the value of your portfolio change when interest rates increase by one basis point? Two possibilities: 1 Recompute the value of your portfolio using the new interest rate. 2 Short cut: if changes in interest rates are not too large, use duration and convexity. Bond Portfolios (Session 10) Investments Spring 2010 8 / 54
Duration Duration Duration If changes in y are not too large, we can use a Taylor expansion to approximate the change in the price of a bond. As the degree of the Taylor series rises, it approaches the correct function. Suppose a bond has an initial value of P 0 and the initial yield is y 0. The yield then changes to y 1 = y 0 + 4y. The new price of the bond can be written as P 1 = P 0 + dp dy 4y + 1 d 2 P (4y) +... 2 dy 2 For xed-income instruments, the derivatives dp dy and d 2 P dy 2 important that they have been given special names. are so Bond Portfolios (Session 10) Investments Spring 2010 9 / 54
Duration Duration Duration The duration of a bond measures the sensitivity of a bond price to changes in interest rates. The duration is also a measure of how long on average the holder of the bond has to wait before receiving cash payments. Duration assume a at term structure. Let P(y) denote the price of a bond with a yield to maturity y and cash- ow C t at time t The price of a bond is given by the usual P(y) = T C t t=1. Then, (1+y ) t we have: P 0 (y) = T t=1 tc t (1+y ) t+1 = 1 1+y T t=1 tc t (1+y ) t Bond Portfolios (Session 10) Investments Spring 2010 10 / 54
Duration Duration Duration Therefore, we have: P(y + y) = P(y) 1 1+y T t=1 tc t (1+y ) t y The change in the bond price (the return on the bond caused by a shift in interest rates) is given by: P (y + y ) P (y ) = T tct t=1 (1+y ) t P (y ) P (y ) y 1+y D = T t=1 tct (1+y ) t P (y ) is called the Macaulay Duration of the bond. Bond Portfolios (Session 10) Investments Spring 2010 11 / 54
Duration Duration Duration The duration is a weighted average of the times when payments are made, with the weight applied to time t being equal to the proportion of the bond s total present value provided by the cash ow at time t: D = T t=1 tct (1+y ) t P (y ) = T t( C t /(1+y ) t T ) = P (y ) tw t t=1 t=1 Thus the bond price change due to a small change in interest rates can be approximated by: dp P = D dy 1+y D = D/(1 + y) is called the Modi ed Duration. Bond Portfolios (Session 10) Investments Spring 2010 12 / 54
Duration Example Example Consider a 10-year 6% (annually) coupon bond with yield to maturity of 5%. We can calculate the duration as follows: Date Cash Flow Weight (2) (3) 1 6 0.0530467 0.053047 2 6 0.0505207 0.101041 3 6 0.0481149 0.144345 4 6 0.0458238 0.183285 5 6 0.0436417 0.218208 6 6 0.0415635 0.249381 7 6 0.0395843 0.277090 8 6 0.0376993 0.301595 9 6 0.0359041 0.323137 10 106 0.6041010 6.041010 Total 1.000 7.892150 Bond Portfolios (Session 10) Investments Spring 2010 13 / 54
Duration Example Example Suppose interest rates increase by y = 0.75%. By how much would the bond price change? We have: P P = D y 1+y = 7.89 0.0075 1+0.05 = 5.6373% Bond Portfolios (Session 10) Investments Spring 2010 14 / 54
Duration Properties of Duration Properties of Duration 1 The duration of a zero-coupon bond equals its time to maturity. 2 The bond s duration is lower when the coupon is higher. 3 The bond s duration generally increases with time to maturity. Duration always increases with maturity for bonds selling at par or at a premium. 4 The duration of a coupon bond is higher when the bond s yield to maturity is lower. 5 The duration of a perpetuity is D = (1 + y)/y. Bond Portfolios (Session 10) Investments Spring 2010 15 / 54
Duration Properties of Duration Properties of Duration Maturity Coupon YTM Price Duration Bond 1 1 7 6 100.94 1.00 Bond 2 1 6 6 100.00 1.00 Bond 3 5 7 6 104.21 4.40 Bond 4 5 6 6 100.00 4.47 Bond 5 10 4 6 85.48 8.28 Bond 6 10 8 6 114.72 7.45 Bond 7 20 4 6 77.06 13.22 Bond 8 20 8 7 110.59 11.05 Bond 9 50 6 6 100.00 16.71 Bond 10 50 0 6 5.43 50.00 Bond Portfolios (Session 10) Investments Spring 2010 16 / 54
Duration Properties of Duration Properties of Duration Bond Portfolios (Session 10) Investments Spring 2010 17 / 54
Duration Linearity of Duration Linearity of Duration Consider a portfolio of K bonds with annual payments each representing a fraction π k of the total value. The duration of the portfolio is: K D Π = (1 + y) Π0 (y ) Π(y ) = π k D k k=1 Here Π denotes the total value of the portfolio and D k denotes the duration of the k-th bond in the portfolio.! The duration of a portfolio is the weighted average of the duration of the securities in the portfolio. Bond Portfolios (Session 10) Investments Spring 2010 18 / 54
Convexity Convexity Convexity Consider a 30-year bond with 8% coupon and initial YTM = 8% Bond Portfolios (Session 10) Investments Spring 2010 19 / 54
Convexity Convexity Convexity Modi ed duration is the appropriate measure of interest rate risk. Duration only measures the rst-order (linear) e ect. But the relationship between bond prices and yields is not linear. Duration is good only for small changes in yields. When there are large changes in yields, duration is not a su cient measure of interest rate exposure. It might therefore be necessary to take into account the curvature (the second-order quadratic term) of the price-yield relation. Notice that duration always understates the value of the bond. Bond Portfolios (Session 10) Investments Spring 2010 20 / 54
Convexity Convexity Convexity By Taylor s theorem we have: P(y + y) = P(y) + P 0 (y) y + 1 2 P 00 (y) ( y) 2 The price of a bond is given by the usual P(y) = T C t t=1. Then, (1+y ) t we have: P 0 (y) = P 00 (y) = T t=1 tc t (1+y ) t+1 = 1 1+y T t(t+1)c t = 1 (1+y ) t+2 (1+y ) 2 t=1 T t=1 tc t (1+y ) t T t(t+1)c t (1+y ) t t=1 Bond Portfolios (Session 10) Investments Spring 2010 21 / 54
Convexity Convexity Convexity Therefore, we have: P(y + y) = P(y) 1 1+y T t=1 tc t (1+y ) t y + 1 2 1 (1+y ) 2 T t(t+1)c t ( y) 2 (1+y ) t t=1 The change in the bond price (the return on the bond caused by a shift in interest rates) is given by: P (y + y ) P (y ) = T t=1 P (y ) P (y + y ) P (y ) P (y ) = D tct (1+y ) t P (y ) y 1+y y 1+y + 1 2 C y 1+y + 1 T t=1 2 2 t(t+1)ct (1+y ) t P (y ) 2 y 1+y Here D denotes the duration and C the convexity of the bond. Bond Portfolios (Session 10) Investments Spring 2010 22 / 54
Convexity Example Example Consider a 10-year 6% (annually) coupon bond with yield to maturity of 5% Suppose interest rates increase by y = 0.75%. By how much would the bond price change? We have: 2 P P = D y 1+y + 1 2 C y 1+y! P 0.0075 P = 7.89 1+0.05 + 1 0.0075 2 2 79.57 1+0.05 = 5.43% Bond Portfolios (Session 10) Investments Spring 2010 23 / 54
Convexity Properties of Convexity Properties of Convexity Maturity Coupon YTM Price Convexity Bond 1 1 7 6 100.94 1.78 Bond 2 1 6 6 100.00 1.78 Bond 3 5 7 6 104.21 22.47 Bond 4 5 6 6 100.00 22.92 Bond 5 10 4 6 85.48 75.89 Bond 6 10 8 6 114.72 65.17 Bond 7 20 4 6 77.06 211.53 Bond 8 20 8 7 110.59 157.93 Bond 9 50 6 6 100.00 440.04 Bond 10 50 0 6 5.43 2269.50 Bond Portfolios (Session 10) Investments Spring 2010 24 / 54
Convexity Properties of Convexity Properties of Convexity Convexity of a portfolio: C Π = K π k C k k=1 Here, π k and C k denote respectively the weight of bond k in the total value of the portfolio and the convexity of bond k. Bond Portfolios (Session 10) Investments Spring 2010 25 / 54
Convexity Properties of Convexity Properties of Convexity Bond Portfolios (Session 10) Investments Spring 2010 26 / 54
Convexity Example Example Consider a 10-year zero coupon bond with a yield of 6% (semiannual) and present value of USD 55.368. What is the bond s duration? What is the bond s modi ed duration? What is the bond s convexity? Suppose the yield goes up to 7%. I I How good is the linear approximation? How good is the linear and convexity approximation? Do investors like convexity? Bond Portfolios (Session 10) Investments Spring 2010 27 / 54
Convexity Callable Bonds Duration and Convexity of Callable Bonds Bond Portfolios (Session 10) Investments Spring 2010 28 / 54
Convexity Analogy An Analogy Are there related concepts to duration and convexity when we talk about stocks and options? How does the relation between the price of a call option and the price of the underlying stock look like? Delta: The rate of change of the option price with respect to the underlying stock. Gamma: The rate of change of delta with respect to the price of the underlying stock. Dynamic delta hedging. Bond Portfolios (Session 10) Investments Spring 2010 29 / 54
Managing Bond Portfolios Managing Bond Portfolios Managing Bond Portfolios Active Bond Management I I Interest rate forecasting Identi cation of relative mis-pricing Passive Bond Management I I I Bond index funds Cash ow matching Immunization of interest rate risk 1 Net worth immunization Duration of assets = Duration of liabilities 2 Target date immunization Holding period matches duration Bond Portfolios (Session 10) Investments Spring 2010 30 / 54
Managing Bond Portfolios Example Example 1 Example: consider an insurance company that issues a guaranteed investment contract (GIC) for 10, 000. The GIC has a 5 year maturity and a guaranteed interest rate of 8%. Then the insurance company is obliged to pay 10, 000 (1.08) 5 = 14, 693.28 in 5 years. Suppose that the insurance company chooses to fund its obligation with 10, 000 of 8% annual coupon bonds, selling at par value with 6 years to maturity.! As long as the market interest rate stays at 8% the company has fully funded the obligation, as the present value of the obligation exactly equals the value of the bonds. Can the bond generate enough income to pay o the obligation 5 years from now regardless of interest rates movements? Bond Portfolios (Session 10) Investments Spring 2010 31 / 54
Managing Bond Portfolios Example Example 1 Bond Portfolios (Session 10) Investments Spring 2010 32 / 54
Managing Bond Portfolios Example Example 1 Fixed income investors face two type of risks: I I Price risk Reinvestment risk Increases/decreases in interest rates cause capital losses/gains but at the same time increase/decrease the rate at which reinvested income will grow. For a horizon equal to the portfolio s duration, price risk and reinvestment risk exactly cancel out. Example: if interest rates fall to 7%, the total funds will accumulate to 14, 694.05$ providing a small surplus of.77$. If rates increase to 9% the fund accumulates to 14, 696.02$, providing a small surplus of 2.74$. Bond Portfolios (Session 10) Investments Spring 2010 33 / 54
Managing Bond Portfolios Example Example 2 An insurance company must make a payment of 19, 478$ in 7 years. The interest rate is 10%, so the present value of the obligation is 10, 000. The portfolio manager wishes to fund the obligation using 3-year zero-coupon bonds and perpetuities paying annual coupons. How can the manager immunize the obligation? Bond Portfolios (Session 10) Investments Spring 2010 34 / 54
Managing Bond Portfolios Example Example 2 Immunization requires that the duration of the portfolio of assets equals the duration of the liability. We then proceed in 3 steps: 1 Calculate the duration of liability: 7 years 2 Calculate the duration of asset portfolio: the duration of the portfolio is the weighted average of durations of each component asset, with weights proportional to the funds placed in each asset. Here we have: D ZC = 3 years and D P = (1 + y)/y = 11 years and the portfolio duration is: D A = w 3 + (1 w) 11 3 Find the asset mix that sets the duration of assets equal to the 7-year duration of liabilities. This requires to solve: w 3 + (1 w) 11 = 7! w = 1/2 Bond Portfolios (Session 10) Investments Spring 2010 35 / 54
Managing Bond Portfolios Example Example 2 Suppose that 1 year has passed and that interest rates are still at 10%. Is the position still fully funded? Is it immunized? The PV of the obligation will have grown to 11, 000$. The manager s funds also have grown to 11, 000$: value of zero-coupon bonds goes from 5,000 to 5,500 with passage of time and the perpetuity has paid 500$ of coupon and remains worth 5, 000$! the obligation is still funded. The portfolio weights must be changed and must satisfy: D A = w 2 + (1 w) 11 = 6! w = 5/9 Immunization based on duration is a dynamic strategy. As time passes the duration and time to maturity changes! need to rebalance the immunized portfolio! Bond Portfolios (Session 10) Investments Spring 2010 36 / 54
Managing Bond Portfolios Comments Comments Duration and convexity are build on restrictive assumptions I I The yield curve is at Term structure only a ected by parallel shifts F F All bonds have the same yield to maturity Risk on the general level of interest rates Bad news: not only is the term structure not at, but it also changes shape through time! Solutions: I I Principal Component Analysis: sheds light on the dynamics of the yield curve Application of general immunization theory Bond Portfolios (Session 10) Investments Spring 2010 37 / 54
Application of Immunization Theory An Application of Immunization Theory Idea Apply an immunization theory that allows for arbitrary changes in the spot rate structure. These changes include parallel shifts but also changes in the curvature of the term structure. Illustration with a numerical example of how to immunize a portfolio to a radical change in the term structure. Bond Portfolios (Session 10) Investments Spring 2010 38 / 54
Spot rate Application of Immunization Theory Application Initial spot rate curve - continuously compounded 0.06 Initial spot rate curve 0.058 0.056 0.054 0.052 0.05 0.048 0.046 0.044 0.042 0.04 0 5 10 15 20 Maturity (years) Bond Portfolios (Session 10) Investments Spring 2010 39 / 54
Application of Immunization Theory Application Assumptions Suppose that you want to invest today a value of $100 for seven years. Investment horizon is therefore seven years. No AAA zero-coupon bond that would guarantee you a terminal value of $100 e (0.05287) = $144.72 in seven years. However, there are 4 bonds a, b, c, and d. Their characteristics are summarized in the next table. Assume for simplicity that they pay annual coupons. Can an appropriate portfolio using bonds a, b, c, and d guarantee the terminal value of $144.72 in seven years? Bond Portfolios (Session 10) Investments Spring 2010 40 / 54
Application of Immunization Theory Application Main features of bonds Coupon Maturity (years) Par value Initial value Bond a 4 7 100 92.18556 Bond b 4.75 8 100 95.42947 Bond c 7 15 100 111.0813 Bond d 8 20 100 123.5471 Bond Portfolios (Session 10) Investments Spring 2010 41 / 54
Application of Immunization Theory Application Question What should your investments in these bonds be, such that your $100 will be transformed into that terminal value of $144.72, even if there is a dramatic shift in the term structure just after you bought the bonds? For example, the steep spot curve from before turns clockwise to become horizontal at a given level, for instance 5.5%. How should you constitute your bond portfolio in order to secure a value extremely close to $144.72 in seven years? Bond Portfolios (Session 10) Investments Spring 2010 42 / 54
Spot rate Application of Immunization Theory Application New spot rate curve 0.06 Initial and new spot rate curve 0.058 0.056 0.054 0.052 0.05 0.048 0.046 0.044 0.042 0.04 0 5 10 15 20 Maturity (years) Bond Portfolios (Session 10) Investments Spring 2010 43 / 54
Application of Immunization Theory Application De nitions Denote the amounts invested in bonds a, b, c, and d with n a, n b, n c, and n d. Denote H k the investors horizon (or duration) to the power of k. The moment of order k of bond l, mk l, is the weighted average of the kth power of its times of payments, the weights being the shares of the bond s cash ows in the initial bond value: m l k = where s(t) is the spot rate for maturity t. N t k c lte s(t)t t=1 B0 l Bond Portfolios (Session 10) Investments Spring 2010 44 / 54
Application of Immunization Theory Application The Moment of Order k of a Bond Portfolio The moment of order k of a bond portfolio is the weighted average of the kth power of its times of payments, the weights being the shares of the portfolio s cash ows in the initial portfolio value. m P k = = L n l B l N 0 P l=1 0 t k c lte s(t)t t=1 B0 l L l=1 n l B0 l mk l P 0 The moment of order 0 of a portfolio (or a bond) is one, since it is the weighted average of 1 s. The moment of order 1 of a portfolio (or a bond) is its duration. Bond Portfolios (Session 10) Investments Spring 2010 45 / 54
Application of Immunization Theory Application A General Immunization Theorem Theorem Suppose that the spot rate structure can be expanded into a Taylor series of order m 1 and that it undergoes a variation. Then a su cient condition for a bond portfolio to be immunized against such a variation is the following: 1 Any moment of order k (k = 0, 1,..., 2m 1) of the bond portfolio is equal to the kth power of the investor s horizon H. 2 The moment of order 2m is equal to the 2mth power of H plus a positive arbitrary constant. Bond Portfolios (Session 10) Investments Spring 2010 46 / 54
Application of Immunization Theory Application System of Equations This result leads to the following system of four linear equations with four unknowns n a, n b, n c, and n d : n a B a 0 P 0 m0 a + n bb0 b P 0 m0 b + n c B0 c P 0 m0 c + n d B0 d P 0 m0 d = H0 n a B0 a P 0 m1 a + n bb0 b P 0 m2 b + n c B0 c P 0 m3 c + n d B0 d P 0 m4 d = H1 n a B0 a P 0 m2 a + n bb0 b P 0 m2 b + n c B0 c P 0 m2 c + n d B0 d P 0 m2 d = H2 n a B0 a P 0 m3 a + n bb0 b P 0 m3 b + n c B0 c P 0 m3 c + n d B0 d P 0 m3 d = H3 Bond Portfolios (Session 10) Investments Spring 2010 47 / 54
Application of Immunization Theory Application System of Equations For simplicity de ne I n the row vector of unknowns (n a, n b, n c, n d ) I m the square matrix of terms B 0 l P 0 mk l, for l = a, b, c, d, and k = 0, 1, 2, 3 I h the vector of horizons to the powers of 0, 1, 2, 3 The system can now be written: nm = h and solved for n: n = m 1 h Bond Portfolios (Session 10) Investments Spring 2010 48 / 54
Application of Immunization Theory Application Solution 2 6 4 n a n b n c n d 3 7 5 = 2 6 4 n = m 1 h, 43.6218-11.4694 0.81844-0.0188-48.7516 13.4968-1.0056 0.0237 10.2440-3.3070 0.3078-0.0088-3.2934 1.1062-0.1107 0.0036 3 2 7 6 5 4 1 7 49 343 3 7 5 Bond Portfolios (Session 10) Investments Spring 2010 49 / 54
Application of Immunization Theory Application Solution The solution n is then: I n a = 2.99784 I n b = 4.57423 I n c = 0.82821 I n d = 0.257722 We go long the bonds b and d, and short the bonds a and c. Bond Portfolios (Session 10) Investments Spring 2010 50 / 54
Application of Immunization Theory Application Solution Suppose now that at time ε, immediately after the purchase of portfolio (n a, n b, n c, n d ), the initial spot structure that was increasing turns clockwise to become horizontal at level 5.5% per year (continuously compounded). What happens to the value of our portfolio under the initial and new structure? What is the value of the portfolio in 7 years under the initial and new structure? How e cient is the immunization strategy? Bond Portfolios (Session 10) Investments Spring 2010 51 / 54
Application of Immunization Theory Application Solution # of bonds n a n b n c n d Value at t 0 92.19 95.43 111.08 123.55 Value in portfolio -276.36 436.52-92 31.84 Total = 100 Value at t ε 90.65 94.31 113.37 127.68 Value in portfolio -271.76 431.39-93.90 32.91 Total = 98.637 Value of portfolio in 7 years Initial term structure 100e (0.05287) = 144.72 New term structure 98.637e (0.0557) = 144.96 Bond Portfolios (Session 10) Investments Spring 2010 52 / 54
Application of Immunization Theory Application More changes Suppose that the term structure experiences an even more radical change. From the initial term structure, make it turn clockwise to ten horizontal structures, from 1% to 10%. Bond Portfolios (Session 10) Investments Spring 2010 53 / 54
Application of Immunization Theory Application More changes New spot rate structure Portfolio value in 7 years 0.01 144.7241 0.02 144.7448 0.03 144.791 0.04 144.8524 0.05 145.9218 0.06 145.9952 0.07 145.0708 0.08 145.149 0.09 145.2321 0.10 145.3237 Bond Portfolios (Session 10) Investments Spring 2010 54 / 54