# FIN 684 Fixed-Income Analysis Approximating Price Changes: From Duration to Convexity Professor Robert B.H. Hauswald Kogod School of Business, AU

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1 FIN 684 Fied-Income Analysis Approimating Price Changes: From Duration to Conveity Professor Robert B.H. Hauswald Kogod School of Business, AU From Yields to Their Analysis Review: discrete and continuous spot rates Yield and bond price volatility duration conveity Introduction to the yield curve revisit yields and their calculation the term structure of interest rates Yield curve strategies /8/06 Duration and Conveity Robert B.H. Hauswald

2 Semi-annual Compounding Semi-annual compounding: twice a year the bond holder receives a coupon payment To obtain the semi-annual compounding rate we have: r t, T = Z t, T T t where Z t, T is a discount factor /8/06 Duration and Conveity Robert B.H. Hauswald 3 More Frequent Compounding Let the discount factor Z t, T be given, and let rn. t,. T denote the annualized n- times compounded interest rate. Then rn is defined by the equation rn t, T = n Z t, T n T t Rearranging for Z t, T, we obtain Z t, T = n T t r t, T + n n t, T /8/06 Duration and Conveity Robert B.H. Hauswald 4

3 Continuous Compounding The continuously compounded interest rate is obtained by increasing the compounding frequency n to infinity The continuously compounded interest rate r t, T. obtained from rn t, T for n that increases to infinity, is given by the formula: ln Z t, T r t, T = T t Solving for Z t, T we obtain: r t, T T t, Z t T = e where ln. := log. denotes the natural logarithm Remember the properties of log s? /8/06 Duration and Conveity Robert B.H. Hauswald 5 Discount Factors and Interest Rates Note that independently of the compounding frequency, discount factors are the same otherwise, we would have what? Thus some useful identities are: rn t, T r t, T = n ln + n r t, T n rn t, T = n e /8/06 Duration and Conveity Robert B.H. Hauswald 6

4 Bond Yield to Maturity The price of S of an option-free bond as a function of the bond s yield to maturity y is n c P = + + y i +q i= + y n + q where q = days from settlement to net coupon date days in current coupon period /8/06 Duration and Conveity Robert B.H. Hauswald 7 Differences in YTM Different bonds have different yields Term differences Risk differentials quality spread Sector spreads e.g. industrials v. utilities Optionality Ta differences Benchmark: Treasuries Risky bonds priced at a spread to Treasuries /8/06 Duration and Conveity Robert B.H. Hauswald 8

5 Yield Curve A graph of bond yields to maturity by time to maturity is called a yield curve. 7.00% 6.50% 6.00% 5.50% 5.00% 4.50% 4.00% 3mo 6mo yr yr 3yr 5yr 0yr 30yr /8/06 Duration and Conveity Robert B.H. Hauswald 9 Bond Price Volatility Consider only IR as a risk factor Longer TTM means higher volatility Lower coupons means higher volatility Floaters have a very low price volatility Price is also affected by coupon payments Price value of a Basis Point PVBP= price change resulting from a change of 0.0% in the yield. /8/06 Duration and Conveity Robert B.H. Hauswald 0

6 Price-Yield for Option-Free Bonds price yield /8/06 Duration and Conveity Robert B.H. Hauswald Zero-coupon Eample P y 00 y + y = T T P = + P P y = T T P0 + y + y + y 00 T /8/06 Duration and Conveity Robert B.H. Hauswald

7 Lessons? y=0%, y=0.5% T P 0 P P% % % % /8/06 Duration and Conveity Robert B.H. Hauswald 3 Approimating Price Changes \$ We need to find some approimation of price changes in yield: How would we go about this task? How about a linear approimation? y /8/06 Duration and Conveity Robert B.H. Hauswald 4

8 Focus: Parallel Yield-Curve Shifts r upward move downward move Current TSoIR /8/06 Duration and Conveity Robert B.H. Hauswald 5 T Maturity and Bond-Price Sensitivity Bond Value Par Consider two otherwise identical bonds: The long-maturity bond will have much more volatility with respect to changes in the yield Larger change wrt discount factor at new rate! Z t, T = n rn t, T + n Short Maturity Bond T t C Discount Rate Long Maturity Bond /8/06 Duration and Conveity Robert B.H. Hauswald 6

9 Duration F. Macaulay 938 Better measurement than time to maturity. Weighted average of all coupons with the corresponding time to payment. T YTM bond price Ct Macaulay Duration P y = D MC i= t t= Ct + y suggested weight of each cash-flow date: w t = C t /+y t /Bond Price: what is the sum of all w t? = t= /8/06 Duration and Conveity Robert B.H. Hauswald 7 P + y T t t t = P T w t t From Price to Duration P C C = + + K+ + y + y C + y n + M + y n Macaulay Duration = C C nc + + K+ P + y + y + y n nm + + y n /8/06 Duration and Conveity Robert B.H. Hauswald 8

10 Price Sensitivity: Taking Derivatives C C C P = + + K+ + y + y + y dp dy C + y C + y nc + y /8/06 Duration and Conveity Robert B.H. Hauswald 9 n M + + y nm + y = + + K+ + 3 n+ n+ dp C C nc nm = + + K+ + n n dy + y + y + y + y + y n Duration and IR Eposure P P = D MC + y + y The bond price volatility is proportional to the bond s duration. Thus duration is a natural measure of interest rate risk eposure. /8/06 Duration and Conveity Robert B.H. Hauswald 0

11 Macaulay Duration: Interpretation D = T T t t wt = t t t= t= + y Price CF Duration is the cash-flow weighted sum of times to maturities of each coupon and final cash flow What is the duration of a zero coupon bond? /8/06 Duration and Conveity Robert B.H. Hauswald Duration and IR Sensitivity /8/06 Duration and Conveity Robert B.H. Hauswald

12 Modified Duration DMC DM = + y P P = D y The percentage change in bond price is the product of modified duration and the change in the bond s yield to maturity. /8/06 Duration and Conveity Robert B.H. Hauswald 3 M \$ Duration and Yield Sensitivity dp dy = P Modified Duration Duration is related to the rate of change in a bonds price as its yield changes: linear first-order approimation of price changes in yield - let s see what the problem is /8/06 Duration and Conveity Robert B.H. Hauswald 4 y

13 Predicting Price Changes. Find Macaulay duration of bond. Find modified duration of bond 3. When interest rates change, the change in a bond s price is related to the change in yield according to P P 00 = D * m y 00 Find percentage price change of bond Find predicted dollar price change in bond Add predicted dollar price change to original price of bond Predicted new price of bond or use XLS /8/06 Duration and Conveity Robert B.H. Hauswald 5 A Tale of Two Bonds Coupon bond with duration.8853 Price at 5% for 6M = 0.5% p.a. is \$ If IR increase by bp to 5.0% = in p.a.-terms?, its price will fall to \$964.94, or 0.359% decline. Zero-coupon bond with equal duration must have.8853 years to maturity: why? At 5% semiannual its price is \$,000/ =\$ If IR increase to 5.0%, the price becomes: \$,000/ =\$ % decline. /8/06 Duration and Conveity Robert B.H. Hauswald 6

14 Duration Determinants Duration of a zero-coupon bond equals maturity. Duration of a perpetuity is +y/y. Holding time-to-maturity constant, duration is higher when coupons are lower. Holding coupon rate constant duration does not always increase in time-to-maturity! Holding other factors constant, duration is higher when YTM is lower. /8/06 Duration and Conveity Robert B.H. Hauswald 7 Duration D Zero coupon bond 5% coupon, YTM = 5% 0 3m 6m yr 3yr 5yr 0yr 30yr Maturity /8/06 Duration and Conveity Robert B.H. Hauswald 8

15 Eample A bond with 30-yr to maturity Coupon 8%; paid semiannually YTM = 9% P 0 = \$897.6 D =.37 Yrs if YTM = 9.%, what will be the price? P/P = - y D* P = - y D*P = -\$9.36 P = \$ \$9.36 = \$ /8/06 Duration and Conveity Robert B.H. Hauswald 9 Usefulness of Duration Simple summary statistic of effective average maturity Measures sensitivity of bond price to interest rate changes Measure of bond price volatility Measure of interest-rate risk Risk management by issues, banks, borrowers: match the duration of assets and liabilities hedge the interest rate sensitivity of an investment /8/06 Duration and Conveity Robert B.H. Hauswald 30

16 Duration of a Portfolio Duration of a portfolio is a weighted average of the duration of assets, weights correspond to the percentage of the portfolio invested in a given security The duration of a portfolio of n securities is n DW = wi Di i= where w i is the fraction of the portfolio in security i, and D i is the duration of security i /8/06 Duration and Conveity Robert B.H. Hauswald 3 Cash Flow Matching and Immunization An institution may consider two alternatives to hedge interest rate risk: Cash Flow Matching: buy a set of securities that payoff the eact required cash flow over the period Immunization: portfolio of securities with the same present value and duration of the cash flow needed Immunization is preferred to other strategies allows an institution to choose bonds in terms of liquidity and transaction costs benefits generates the desired stream of cash flows /8/06 Duration and Conveity Robert B.H. Hauswald 3

17 Benefits of Immunization Think of two etremes and the effect of a movement in interest rates: 00% in long term bonds: loose money when interest rates go up as bond prices decline 00% investment in cash: loose money when interest rates go down Best strategy is in the middle: through duration immunization chooses the adequate mi of the two strategies to hedge against changes in interest rates /8/06 Duration and Conveity Robert B.H. Hauswald 33 Immunization Two kinds of Interest rate risk Price risk when interest rate rises Reinvestment risk when interest rate falls Immunization offsets these two risks If the duration is set equal to the investment horizon, the two effects eactly cancel out Matching the durations of asset and liability eliminates interest rate risk. Valid for small changes in interest rates Requires rebalancing regularly or when necessary 34 /8/06 Duration and Conveity Robert B.H. Hauswald

18 Eample: Immunization Suppose an insurance company has the following Liability: Payment of \$9,487 in seven years Assets: The fund manager decides to hold 3-year zero-coupon bond, and Perpetuity Assume market interest rate = 0% Solve for the portfolio weights for immunization Duration of liability = 7 years Duration of a portfolio = weighted average of duration of assets in the portfolio = w 3 + w duration of the perpetuity = +0./0. = years Match the portfolio duration to 7 years: w 3 + w = 7 w = yr zero, and -w=0.5 perpetuity After one year, do you have to rebalance it to maintain the immunization strategy? assume interest rate is still 0% What should be the new weights for each asset? /8/06 35 Duration and Conveity Robert B.H. Hauswald Summary An introduction to duration analysis Interest rate risk: changes in bond prices arising from fluctuating interest rates varying YTMs. Ceteris paribus, the longer the time to maturity, the greater the interest rate risk. Ceteris paribus, the lower the coupon rate, the greater the interest rate risk. Repayment Risk: debtor might repay the amount owed before maturity Default risk: covenants and sinking funds. rating agencies: downgrades and upgrades fied-income analysis: etracting default probabilities and recovery rates from stock and bond prices /8/06 Duration and Conveity Robert B.H. Hauswald 36

19 Appendi The mathematics of duration analysis derivatives approimating functions Taylor series and approimation Correcting for pricing errors second order approimations conveity /8/06 Duration and Conveity Robert B.H. Hauswald 37 Derivatives Difference quotients: taking the limit of m = f = lim f + h f h h 0 h one obtains the derivative of f: f Computing f is called differentiation 38

20 /8/06 Duration and Conveity Robert B.H. Hauswald 39 F Derivatives /8/06 Duration and Conveity Robert B.H. Hauswald 40 Properties of Derivatives ' ' ' ' ' ' ' ' ' ' g g f g f d d g g f g f g f d d g f g f g f g f g f = = + = + = +

21 Bond Price Change Modified duration D * D = + MC y Bond-price changes as first-order or higher order approimations? Pro and con? P = D * P y + C P y +K /8/06 Duration and Conveity Robert B.H. Hauswald 4 Taylor Epansion To measure the price response to a small change in risk factor we use the Taylor epansion. Initial value y 0, new value y, change y: y = y 0 + y f y = f y0 + f ' y0 y + f " y0 y +K /8/06 Duration and Conveity Robert B.H. Hauswald slide 4

22 Pricing Error and Conveity Price Pricing error due to Conveity P P = D M y Duration Yield 43 /8/06 Duration and Conveity Robert B.H. Hauswald \$ The Second Order: Conveity d P C = dy Correcting for curvature: secondorder approimation of the price-yield relation to improve precision of pricechange prediction: addresses one problem with duration which one is left? /8/06 Duration and Conveity Robert B.H. Hauswald 44 r

23 Conveity d P Curv = dy T d P t C = = dy P P 0 t + C t + y + t= y t /8/06 Duration and Conveity Robert B.H. Hauswald 45 Conveity Pitfalls A common pitfall is to consider conveity as the change in duration, it is not Many think that since the duration of a zero coupon bond is a constant, regardless of the interest rate, its conveity is zero this can be somewhat true for dollar conveity which is related to the change in dollar duration the dollar conveity of a zero coupon is not constant, since it includes the price of the bond /8/06 Duration and Conveity Robert B.H. Hauswald 46

24 Measuring Price Changes The mathematical principle: Taylor epansion of price-yield function P dp d P dp = dy + dy + error dy dy Operational implementation: modified duration D and conveity Conv dp Conv error = Ddy + dy + P P /8/06 Duration and Conveity Robert B.H. Hauswald 47 Eample 0 year zero coupon bond with a semiannual yield of 6% 00 P = \$ = The duration is 0 years, the modified duration is: * 0 D = = The conveity d 00 is C = = P dy + y /8/06 Duration and Conveity Robert B.H. Hauswald 48

25 Eample If the yield changes to 7% the price change P is = 9.7 \$ \$ = \$ \$0.74 = \$5.0 P = = 0 0 \$5. /8/06 Duration and Conveity Robert B.H. Hauswald 49 Properties of Conveity There are two general rules on conveity: The higher the yield to maturity, the lower the conveity, all else being equal The lower the coupon, the greater the conveity, all else being equal To minimize the adverse effects of interest rate volatility for a given portfolio duration seek higher conveity while meeting the other constraints in bond portfolios /8/06 Duration and Conveity Robert B.H. Hauswald 50

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