Portfolio Management 3-228-07 Albert Lee Chun Duration, Convexity and Bond Portfolio Management Strategies Lecture 10 27 Nov 2008 0 Bond Portfolio Management The major source of risk facing bond portfolio managers has to do with shifts in interest rates. In this session: 1. We examine how bond prices respond to changes in this source of risk. 2. We discuss ways of constructing bond portfolios to insulate against this risk. Albert Lee Chun Portfolio Management 1 Today Review of Bond Fundamentals Term Structure of Interest Rates Interest Rate Risk Sensitivity of Bond Prices to Interest Rates - Duration - Convexity Portfolio Immunization Yield Curve Strategies Some Fun Examples Albert Lee Chun Portfolio Management 2 Review of Bond Fundamentals Albert Lee Chun Portfolio Management 3 Bond Definition Definition: A bond is a debt security that a corporation or a government issues to borrow on a long-term basis. Normally an interest-only loan (when issued at par) - Borrower pays only the interest but none of the principle is paid until the end of the loan. Interest is paid in the form of a periodic coupon. Albert Lee Chun Portfolio Management 4 Components of Bond Bond prices depend on 4 factors 1. Par or Face Value Coupon Rate Number of years to maturity Yield to Maturity (YTM) The “interest rate” that makes the discounted present value of the bond’s coupons equal to its market price (also called internal rate of return of a bond). 2. 3. 4. Albert Lee Chun Portfolio Management 5 Yield to Maturity The yield to maturity (YTM) is an interest rate such that present value of the bond’s coupons and the face value equals the current market price. Often simply called the bond’s yield as in “The yield on the 10 year bond is 5%.” Important: YTM is quoted as an Annual percentage rate (APR). Bond prices are inversely related to yields. Albert Lee Chun Portfolio Management 6 Equation for Bonds Suppose a bond pays 1 coupon per year 1 F PV 1 t t r (1 r ) (1 r ) C Present Value = PV(coupon payments) + PV(face value) = PV of an annuity + PV of F F = face value C = annual coupon amount = F × coupon rate r = yield to maturity t = number of years until maturity Albert Lee Chun Portfolio Management 7 Equation for Bonds For a bond paying a semi-annual coupon (2 times a year) 1 F PV 1 t2 t2 r/2 (1 r / 2 ) (1 r / 2 ) C /2 F = face value C = yearly coupon amount = F × coupon rate r = yield to maturity t = number of periods until maturity Albert Lee Chun Portfolio Management 8 Equation for Bonds More generally, if a bond pays m coupons per year 1 F PV 1 tm tm r/m (1 r / m ) ( 1 r / m ) C /m F = face value C = yearly coupon amount = F × coupon rate r = yield to maturity m = number coupons per year t = number of years until maturity Albert Lee Chun Portfolio Management 9 Bond Details The coupon rate, the face value (or par value) and the maturity date are all determined by the bond issuer (corporation or government). Treasury Bills – One year and less; no coupons Treasury Notes – Between 2 and 10 years; coupons. Treasury Bonds – Longer than 10 years; coupons. Albert Lee Chun Portfolio Management 10 Bond Example Eggbert’s Egg Company Simple Example: Eggbert’s Egg Co. issues a bond with time to maturity of 7 years; the yield to maturity is 10% APR. The firm pays an interest rate in the form of $40 every 6 months for 7 years, and a $1,000 principal at the end of 7 years. Albert Lee Chun Portfolio Management 11 Example: Valuing a Bond Eggbert’s Egg Co. issues a semiannual coupon paying bond with a coupon rate of 8% and a face value of $1,000 that matures in 7 years. If we assume, that the yield to maturity is 10%, what is the price of this bond? The bondholder receives a payment of $40 every six months (a total of $80 per year or 8% per year) 1 1 - 1.05 14 1, 000 Bond Price 40 901 . 01 14 0 . 05 1 . 05 Albert Lee Chun Portfolio Management 12 PAR Bonds The price of a par bond is equal to F. A par bond has YTM = coupon rate. Why? Let’s take YTM = coupon rate = r. F = 1000. Suppose at time t, P(t) = 1000(1+r). At time t-1, the value of the bond is: =1000(1+r)/(1+r) =1000 By induction, since this is true at time T-1, this is true for all t. Hence, when YTM = coupon rate, PV = 1000. Most bonds are issued at par, with the coupon rate set equal to the prevailing market yield. Albert Lee Chun Portfolio Management 13 Example: Valuing a “Par Bond” Suppose you are looking at a bond that has a 10% coupon rate and a face value of $100. There are 10 years to maturity and the yield to maturity is 10%. What is the price of this bond? Using the formula: P = PV of annuity + PV of F P = 5[1 – 1/(1.05)20] / .05 + 100/ (1.05)20 P = 100.00 Price = F=100. It’s a par bond. Albert Lee Chun Portfolio Management 14 Example: Yield to Maturity The price of Eggbert’s Egg Co. is currently trading at 1,200. The time to maturity is 4-years with a 14% annual coupon. What is the yield to maturity? The equation to solve is: 1,200 140 r 1 1 1 1,000 4 4 ( 1 r ) ( 1 r ) Using Goal Seek in Excel we get YTM = 2 x r = 7.96%. Albert Lee Chun Portfolio Management 15 Term Structure of Interest Rates Albert Lee Chun Portfolio Management 16 The World is Not Flat In earlier courses, you were presented with a rather simplified view of interest rates, with a constant single interest rate used to discount all cash flows. The yield to maturity was assumed to be the same for all bonds. In reality, bond of different maturities will have different yields to maturity. The yield curve is not flat. Albert Lee Chun Portfolio Management 17 Term Structure of Interest Rates The term structure of interest rates gives the relation between the time to maturity and the yield to maturity of a bond. Yield curve – graphical representation of the term structure. Normal – upward-sloping, long-term yields are higher than short-term yields Inverted – downward-sloping, long-term yields are lower than short-term yields. Albert Lee Chun Portfolio Management 18 Canada Yield Curve November 2002 Albert Lee Chun Portfolio Management 19 Canada Yield Curve May 2006 Canada Yield Curve as of May 2006 4.60% 4.50% Interest Rate 4.40% 4.30% 4.20% 4.10% 4.00% 3.90% 0.25 2 3.75 5.5 7.25 9 10.75 12.5 14.25 16 17.75 19.5 21.25 23 24.75 26.5 28.25 30 Maturity Albert Lee Chun Portfolio Management 20 Interest Rate Risk and the Yield Curve The U.S. central bank sets the short term interest rate – the fed funds rate. Long term interest rates are the functions of expected short-term interest rates, in addition there is a riskpremia associated with uncertainty about the underlying economy forces, such as the growth rate of the economy, inflation, etc. Fed Chairman: Ben Bernanke Albert Lee Chun Portfolio Management 21 Links to Macro-economy The central bank will respond to macroeconomic forces in setting the interest rate. Thus, the yield curve embeds information about current and future macroeconomic conditions. Conversely, it can serve as a barometer of the economy. Inverted yield curves are a leading indicator of recessions. Albert Lee Chun Portfolio Management 22 Price Sensitivity to Interest Rates Albert Lee Chun Portfolio Management 23 Interest Rate Risk Interest Rate Risk ↑ as Time to Maturity ↑ Interest Rate Risk ↑ as Coupon Rate ↓ Interest Rate Risk ↑ as Yield to Maturity ↓ Albert Lee Chun Portfolio Management 24 Interest Rate Risk and Maturity Longer maturity bond prices are more sensitive to changes in yields than shorter maturity bonds. Interest rate risk is larger for longer maturity bonds Albert Lee Chun Portfolio Management 25 Intuition We can see that the slope of the price-yield curve as a function of the interest rate is much steeper for the 30-year bond than for the 1-year bond. The steeper price-yield curve, the more sensitive the bond is to interest rate changes. A large portion of a bond’s value is due to the payment of the face value. If the bond has a longer maturity, the cash flow from the payment of the face value is realized further in the future, this makes it more sensitive to changes in interest rates. Albert Lee Chun Portfolio Management 26 Intuition Why? Because even a small change in the interest rate can have a significant effect if it is compounded over a longer time. Let’s see an example: 20 year bond: Price = 1000x(1+r)-20 10 year bond: Price = 1000x(1+r)-10 10 years 20 years 7% 508.3493 258.419 8% 463.1935 214.5482 % Change 0.097488 0.20448 Albert Lee Chun Portfolio Management 27 Interest Rate Risk and the Coupon Rate Bonds with higher coupon rates are less sensitive to changes in yields. Interest rate risk is inversely related to the coupon rate. Why? because their value depends less on the discounted face value. Higher coupon payments move the average maturity of the bond’s cash flows forward in time, making the bond less sensitive to the face value payment. Albert Lee Chun Portfolio Management 28 Coupon Rates and Sensitivity Zero coupon bonds are more sensitive to yield changes. Sensitivity increases with maturity. Albert Lee Chun Portfolio Management 29 Changes in Bond Prices Bond Coupon Maturity Initial YTM A 12% 5 years 10% B 12% 30 years 10% C 3% 30 years 10% D 3% 30 years 6% Change in yield to maturity (%) Albert Lee Chun Portfolio Management A B C D 30 Sensitivity of Prices As the maturity increases from A to B, the bond becomes more sensitive. As the coupon rate decreases from B to C, the bond becomes more sensitive. As the initial yield to maturity decreases from C to D, the bond becomes more sensitive (for coupon paying bonds). At lower yields, the more distant payments have relatively greater present values and account for a greater share of the bond’s total value. Albert Lee Chun Portfolio Management 31 Trading Strategies Suppose you expect that the Federal Reserve will lower interest rates at the next FOMC meeting. You want to build a portfolio of bonds such that you maximize the gain in the value of your portfolio. Would you construct a portfolio with A. Maximum interest rate sensitivity? B. Minimum interest rate sensitivity? Answer: A. You want to build a portfolio that will maximize the price appreciation for a negative change in interest rates, that is one with maximum interest rate sensitivity. Albert Lee Chun Portfolio Management 32 Trading Strategies So how can we construct a portfolio that is most sensitive to interest rates movement? We know that we want to choose: - Portfolio of long maturity bonds over a portfolio of short maturity bonds - Portfolio of low coupon bonds over a portfolio of high coupon bonds In other words, ideally you would want to hold a portfolio of long maturity zero-coupon bonds! Albert Lee Chun Portfolio Management 33 Trading Strategies Suppose we want to hold a bond portfolio but suspect that interest rates are about to rise. What should we do? We would want to hold a portfolio with minimum interest rate sensitivity. Short-maturity bonds with high coupons. Albert Lee Chun Portfolio Management 34 Duration Albert Lee Chun Portfolio Management 35 The Duration Measure n D n Ct (1 r ) t t 1 n t Ct (1 r ) PV (C ) t t t 1 Price t t 1 Developed by Frederick R. Macaulay t = time at which coupon or principal payment occurs Ct = interest or principal payment that occurs at time t r = yield to maturity on the bond Albert Lee Chun Portfolio Management 36 Characteristics of Macaulay Duration A zero-coupon bond’s duration equals its maturity Duration of a bond with coupons is always less than its term to maturity because duration gives weight to these interim payments There is an inverse relationship between duration and coupon size. There is a positive relationship between term to maturity and duration, but duration increases at a decreasing rate with maturity. There is an inverse relationship between YTM and duration. Albert Lee Chun Portfolio Management 37 Bond Pricing with Continuous Compounding Note: This is not done in practice, used mostly in academic models, but it will be useful to illustrate the idea of duration. Continuous Compounding Semi-annual Compounding is the price of the i-th cash flow at time , given the yield to maturity Albert Lee Chun Portfolio Management 38 Continuous Compounding • If we compound more and more frequently... • Every minute, every second, every millisecond... • Then in the limit, we have continuous compounding... Letting x = YTM and taking the inverse, that is 1/exp(x), we have our 1-year continuous compounding discount factor. Albert Lee Chun Portfolio Management 39 Bond Pricing Price of a bond with n coupon payments. In General Continuous Compounding where Albert Lee Chun Portfolio Management 40 Duration (Continuous Compounding) Duration is the weighted average maturity of a bond's cash flows. Note that for continously compounded yields we have: Cash flow times t n weighted by C exp( rt ) t t present value of t 1 D cash flow. Price where Ct is the cash flow at time t, and r is the yield to maturity Albert Lee Chun Portfolio Management 41 More Generally For the case of a fixed income security with continuous compounding with n cash flows Ci each occurring at time ti is the price, the i-th cash flow at time and is the yield to maturity Albert Lee Chun Portfolio Management 42 Sensitivity of Price to Yield Differentiate price with respect to the yield to maturity. So given continuously compounded yields, duration is the (negative) percentage chance in price to change in the yield. Albert Lee Chun Portfolio Management 43 Modified Duration For yields compounded m time per year, an adjusted measure of duration can be used to approximate the percentage change in price to a change in yield: modified duration Dmod Macaulay duration 1 y m Albert Lee Chun Portfolio Management 44 Modified Duration W e k n o w th a t P C 1 y 1 C 1 y 2 ... C 1 y T F 1 y T w h e re C is th e c o u p o n , F is th e fa c e v a l u e , y is th e yie ld o f th e b o n d , a n d T th e m a tu rity. D iffe re n tia tin g w ith re s p e c t to y w e g e t th a t P y 1 C 2 1 y 1 1 y 2 C 3 1 y 1* C 1 1 y .. . T C T 1 1 y 2*C 1 y 2 ... T F T 1 1 y T *C 1 y T T *F 1 y T D iv id in g w ith P o n b o th s id e s w e g e t th a t P 1 y P 1 1 1 1 Albert Lee Chun y y 1* C 1 1 y 1 1 y t* 2*C 1 C ash * D D y 2 ... T *C 1 y T T *F flo w a t tim e t / 1 y 1 y T 1 P t B o n d P ric e * Portfolio Management 45 Modified Duration and Bond Price Volatility Bond price movements will vary proportionally with modified duration for small changes in yields An estimate of the percentage change in bond prices equals the change in yield multiplied by modified duration P P Dmod y P = change in price for the bond P = beginning price for the bond Dmod = the modified duration of the bond y = yield change Albert Lee Chun Portfolio Management 46 Test your Intuition Again Suppose you are hired to manage Eggbert Kapital Management’s portfolio of bonds, and you expect a significant decrease in interest rates. Should you invest in long maturity bonds with low coupons, or low maturity bonds with high coupons? High duration bond or low duration bonds? Answer: You should invest bonds whose price will go up the most if interest rates fell, so pick bonds with long maturities and low coupon rates. You would want to pick high duration bonds! Albert Lee Chun Portfolio Management 47 Interpretation of Duration 8-year, 9% annual coupon bond 1200 Cash flow 1000 800 Bond Duration = 5.97 years 600 400 200 0 1 Blue: present value of each cash flow. Albert Lee Chun 2 3 5 4 6 7 8 Year Fulcrum Point Portfolio Management 48 Bond Duration in Years Under Different Terms YTM=6% COUPON RATES Years to Maturity 8 1 5 10 20 50 100 0.02 0.04 0.06 0.08 0.995 4.756 8.891 14.981 19.452 17.567 17.167 0.990 4.558 8.169 12.980 17.129 17.232 17.167 0.985 4.393 7.662 11.904 16.273 17.120 17.167 0.981 4.254 7.286 11.232 15.829 17.064 17.167 Source: L. Fisher and R. L. Weil, "Coping with the Risk of Interest Rate Fluctuations: Returns to Bondholders from Naïve and Optimal Strategies," Journal of Business 44, no. 4 (October 1971): 418. Copyright 1971, University of Chicago Press. Albert Lee Chun Portfolio Management 49 Bond Duration vs Maturity Albert Lee Chun Portfolio Management 15-50 50 Modified Duration Dmod P dP dy This is the first derivative of price with respect to the yield. For small changes this will give a good estimate, but this is a linear estimate on the tangent line Albert Lee Chun Portfolio Management 51 Example Eggbert’s Egg Company issued an 8% bond with 3 years to maturity. The yield to maturity is 8% and the bond pays semi-annual coupons. What is the Macaulay duration? What is the Modified Duration? (See Spreadsheet) Albert Lee Chun Portfolio Management 52 Example Year Payment PV Weight Year*Weight 0.5 4 3.846154 0.038462 0.019231 1 4 3.698225 0.036982 0.036982 1.5 4 3.555985 0.03556 0.05334 2 4 3.419217 0.034192 0.068384 2.5 4 3.287708 0.032877 0.082193 3 104 82.19271 0.821927 2.465781 100 1 2.725911 Price D = 2.5726 and Dm = 2.621 Albert Lee Chun Portfolio Management 53 Short Cut Formulas Macaulay Duration “Short Cut” Formulas D 1 y y 1 y T c y T c 1 y 1 y D 1 y /2 y Semi-annual coupons 1 y / 2 2T c / 2 y / 2 2T c 1 y / 2 1 y Modified Duration “Short Cut” Formula D * 1 1 T c y / 1 y y c 1 y 1 y T where c = coupon rate per period y = yield per period T = periods remaining Albert Lee Chun Portfolio Management 54 Price-Yield Curve Price Yield to Maturity Albert Lee Chun Portfolio Management 55 First-order Approximation Duration provides a first-order approximation of the price-yield curve. Approximating a curve with a straight line Price Approximation Error slope Dmod P Yield Albert Lee Chun Portfolio Management 56 Convexity Albert Lee Chun Portfolio Management 57 Convexity The convexity is the measure of the curvature and is the second derivative of price with respect to yield (d2P/dy2) divided by price Convexity is the percentage change in dP/dy for a given change in yield 2 d P Convexity dy 2 P Albert Lee Chun Portfolio Management 58 Correction for Convexity We can better approximate a curve using a quadratic function. P 2 1 D y Convexity (y ) 2 P CFt 2 Convexity (t t ) 2 t P (1 y ) t 1 (1 y ) 1 T 2T CFt / 2 1 t t 1 Convexity ( )( ) y 2 t 1 y t 2 2 2 P (1 ) (1 ) 2 2 Albert Lee Chun Portfolio Management Semi-annual coupons 59 Duration plus Convexity The Taylor Series for a function P around y is given by P(y+Δy) = P(y) + P’(y) Δy + ½ P’’(y) (Δy)2 + ... Thus, as a second order approximation P(y+Δy) - P(y) ≈ P’(y) Δy + ½ P’’(y) (Δy)2 Therefore, ΔP ≈ - Dm P Δy + ½ P C (Δy)2 2 d P dy Albert Lee Chun Portfolio Management 2 C P 60 Duration plus Convexity Thus for a small change in the yield Δy, the modified duration and the convexity give the second-order approximation to the price-yield curve. ΔP ≈ - Dm P Δy + ½ P C (Δy)2 Price change due to duration - Dm P Δy Price change due to convexity ½ P C (Δy)2 Albert Lee Chun Portfolio Management 61 Convexity of Bonds Portfolio 0 Duration Duration + Convexity Change in yield to maturity (%) Albert Lee Chun Portfolio Management 62 Duration-Convexity Effects Changes in a bond’s price resulting from a change in yields are due to: Bond’s modified duration Bond’s convexity Relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield change Convexity is desirable Albert Lee Chun Portfolio Management 63 Portfolio Immunization Albert Lee Chun Portfolio Management 64 Protection from Interest Rate Risk Pension funds, insurance companies and other financial institutions hold billions of dollars in fixed income securities. One of the most widely used analytical techniques is known as immunization, as it protects a bond portfolio from interest rate movements. This is of major practical value so let’s learn how to structure a portfolio to protect it against interest rate risk! Albert Lee Chun Portfolio Management 65 2 Scenarios Scenario 1: You want to save money for a major expense one year from now. Scenario 2: You want to save money to pay for your kid’s college tuition 20 years from now. Given scenario 1: If you invest in one year treasury bills, there is very little risk. Investing in 20 year treasuries exposes you to interest rate risk. Given scenario 2: Holding 20 year treasury bonds would provide predictable results, investing in one year T-bills would result in reinvestment risk. Albert Lee Chun Portfolio Management 66 Life Insurance Company Suppose you work for a life insurance company, and you expect to make a series of cash payments. One thing you can do is to purchase zero-coupon bonds, having different maturities so that the principal exactly matches each separate obligation. This may not be the best option as corporate bonds offer higher yields and zero-coupon corporate bonds are rare. Albert Lee Chun Portfolio Management 67 Matching Durations Match the duration of your portfolio with the duration of your obligations. Intuition: If the duration of your bond portfolio matches the duration of your obligation stream, then the present value of both your bond portfolio and your obligation stream will respond (to a first-order approximation) to a change in the underlying yield. Specifically, if yields decrease, the present value of your obligations will increase, but the value of your bond portfolio will increase (approximately) by the same amount – so the value of your portfolio will be enough to cover the obligation! Albert Lee Chun Portfolio Management 68 Duration for a Portfolio If the portfolio has price PV PV = V1 + V2 + V3 + ... + Vm D = w1D1 + w2D2 + ... + wmDm where wi = Vi/ PV in a portfolio of m bonds. Vi is the value of Bond i in the portfolio. Albert Lee Chun Portfolio Management 69 Example Suppose Eggbert’s Egg Company has to pay for a new chocolate factory 10 years from now. The cost of the factory is 1 million dollars and it wishes to invest that money now. Suppose no zero-coupon bonds of that maturity are available. Suppose there are 3 corporate bonds (FV=100) to choose from: 1 2 3 Coupon 6% 11% 9% Albert Lee Chun Maturity 30 10 20 Price 69.04 113.01 100.00 Portfolio Management Yield 9% 9% 9% 70 Example Duration of each Bond: D1 = 11.44 D2 = 6.54 D3 = 9.61 Since Bond 1 has duration greater than 10, we must use this bond in our portfolio. Let’s use bonds 1 and 2. (See Spreadsheet) Albert Lee Chun Portfolio Management 72 Example Present Value of the obligation is: $414, 642.86 The duration of the obligation is 10 years. To immunize the portfolio we need to 1. Equate the present value of the portfolio with the present value of the obligation. 2. Equate the duration of the portfolio with the duration of the obligation. Albert Lee Chun Portfolio Management 73 Example The value of total invested in the Bonds 1 and Bond 2 has to equal to PV of the obligation V1 + V2 = PV = $414, 642.86 And the duration of the bond portfolio has to equal the duration of the obligation V1/PV * D1 + V2/PV* D2 = 10 We get: V1/PV * D1 + (PV – V1) /PV* D2 = 10 => One equation and 1 unknown. Albert Lee Chun Portfolio Management 74 Example From the spreadsheet: V1 = 292617.60 V2 = 122025.26 We want to purchase V1/P1 = 4238.20 shares of Bond 1 V2/P2 = 1079.80 shares of Bond 2 Of, course, in practice, you will have to round these numbers. Albert Lee Chun Portfolio Management 75 Example Suppose there is a sudden shift in the yield to either 8% or 10%. The value of the bond portfolio is: Portfolio Obligation Error 8% 457928.3 456386.9 +1541.33 10% 378075.2 376889.5 +1185.69 Due to convexity, the portfolio is always worth more than the obligation. Albert Lee Chun Portfolio Management 76 Homework See if you can replicate the previous example at home, without looking at the spreadsheet. If you look at the spreadsheet answer first, you will think that this is very easy. Try doing it from scratch. Replicate the immunization problem using only Bonds 1 and 3. Albert Lee Chun Portfolio Management 77 Yield Curve Strategies Bullet strategy: maturity of the securities are highly concentrated at one point on the curve. Barbell strategy: maturity of securities included in the portfolio are concentrated at two extreme maturities. Ladder strategy: the portfolio is constructed to have approximately equal amounts of each maturity. Albert Lee Chun Portfolio Management 78 Yield Curve Strategies Bond Coupon Maturity Price YTM Duration_ Convexity A 0.085 5 100 0.085 4.0054435 19.81635 B 0.095 20 100 0.095 8.8815081 124.1702 C 0.0925 10 100 0.0925 6.43409 55.45054 Bullet Portfolio: 100% bond C Barbell Portfolio: 50.2% bond A and 49.8% bond B Albert Lee Chun Portfolio Management 79 Yield Curve Strategies Duration of Bullet: Duration of Barbell: Convexity of Bullet: 6.43409 6.433724 Yield of Bullet: 0.0925 Yield of Barbell: 0.08998 This is the convexity yield 55.45054 0.00252 Convexity of Barbell: Albert Lee Chun Give up yield to get better convexity. 71.78459 Portfolio Management 80 Barbell vs. Bullet The choice between these strategies depends on the magnitude of the shift in yields. Although convexity is preferred, there is the disadvantage of a convexity yield, as the market charges a higher price and offers a lower yield. Thus the benefits from convexity are only realized for large shifts in yields. See the article in the course reader for details. Albert Lee Chun Portfolio Management 81

© Copyright 2022 DropDoc