Finance 300 Financial MarketsHousekeepingDuration CalculationDuration & VolatilitySlide 5Slide 6PowerPoint PresentationSlide 8Pricing Error from ConvexityCorrection for ConvexityDuration & Convexity CalcsSlide 12Slide 13Finance 300Financial MarketsLecture 12Professor J. Petry, Fall, 2002©http://www.cba.uiuc.edu/jpetry/Fin_300_fa02/http://webboard.cites.uiuc.edu/2HousekeepingWe will have class on Thursday, October 10thOffice hours:No office hours today; Thursday 4-5 Friday 12-2Monday 12-2.Midterm #2 on Tuesday, October 15th.30-40 multiple choice questionsIn class problems, TTD3Duration CalculationWhat is this bond’s duration? What does it mean?Duration Calculation5 year 9% bond (annual coupon payment)durationyear coupon NPV @ 7% NPV/V NPV/V*t1 90 84.112 0.078 0.0782 90 78.609 0.073 0.1453 90 73.467 0.068 0.2044 90 68.661 0.063 0.2545 1090 777.155 0.718 3.5916 1082.00 1.00 4.274Duration & VolatilityDuration–A measure of the effective maturity of a bond, defined as the weighted average of the times until each payment, with weights proportional to the present value of the payment.–Bond price volatility and duration are directly related.–We frequently want to consider the percentage price impact of a 1% change in yields. We call this Modified Duration. •We found this in the case of our 5% and 9% coupon bonds by calculating all of the changes in bond prices for 7%, then 6% interest rates. We could have found a close approximation by dividing the duration by (1 + yield). •Volatility = Δ price/price = -[duration / (1 + yield)] * Δi•Starting with interest rates at 7% going to 6%, our 5% coupon bond:–-[4.52/(1.065)]*1%=4.244%, also stated as 4.244 years Modified Duration•What is the modified duration of our 9% coupon bond?5Duration & VolatilityDuration–Other useful things about duration:TRADING STRATEGIES•Duration trading strategies would include increasing duration exposure ahead of expected decreases in interest rates, (or decreasing ahead of interest rate increases) to maximize price impact on your holdings.IMMUNIZATION•Duration can be used as a basis for immunizing your portfolio from changes in interest rates.•When interest rates change, there is both good news and bad news for the bond holder. If rates go down, what is the good news?•What is the bad news?•How about if rates go up?•Which of these two impacts has the worst short-term impact? Which has the worst long-term impact?•Matching the duration of your assets and liabilities results in complete immunization from interest rate risk.6Duration & VolatilityDurationIMMUNIZATION (CONT’D)•Assume an insurance company issues a guaranteed investment contract (GIC) for 10,000, which has a 5 year maturity at 8% interest rates. GICs are essentially zero coupon bonds issued by the insurance company to its customers. •The insurance company is saying they will pay the customer 10,000*(1.08)5 = 14,693.28 in five years.•What duration does this instrument have?•Suppose they fund this liability by investing in a 6 year 8% annual coupon bond. (Can you guess what the duration is?) If you can’t you can always calculate it!•What happens if rates stay the same? go down? go up?•What is interest rate risk? Price risk? Reinvestment risk?7Guaranteed Investment Contract, 5 yrs, $10,000 at 8% annYears Remaining Accumulated ValuePayment Number until Obligation of Invested PaymentA. Rates remain at 8%1 4800x(1.08)4 =1,088.392 3800x(1.08)3 =1,007.773 2800x(1.08)2 =933.124 1800x(1.08)1 =864.005 0800x(1.08)0 =800.00Sale of Bond 0 10,800/1.08 = 10,000.0014,693.288B. Rates fall to 7%1 42 33 24 15 0Sale of Bond 0C. Rates increase to 9%1 42 33 24 15 0Sale of Bond 09Pricing Error from ConvexityPriceYieldDurationPricing Error from Convexity10Correction for Convexity)(21duration) modified(2yConvexityyPPAdjust the pricing equation to take this into account:Convexity•Adjusts the duration figure to account for the convex relationship between prices and yields (2nd derivative).•It is a good thing to have in bonds. Always increases the price of the bond, regardless of move in interest rates.•It only becomes a significant factor with large changes in interest rates•It increases with:•Term to maturity, lower coupon rates, lower yields•Even with this correction, the price change resulting from duration and convexity adjustments is still an approximation.11Duration & Convexity CalcsDuration & Convexity Calculation5 year 9% bond (annual coupon payment)duration convexityyear coupon NPV @ 7% NPV/V NPV/V*t NPV/V*t(t+1)1 90 84.112 0.078 0.078 0.1552 90 78.609 0.073 0.145 0.4363 90 73.467 0.068 0.204 0.8154 90 68.661 0.063 0.254 1.2695 1090 777.155 0.718 3.591 21.5486 1082.00 1.00 4.27 24.22Volatility=Δ price/price =-[duration / (1 + yield)] * Δi = 4.27/1.07*1% =-3.9907%Corrected ΔP/P = -modified duration Δy + ½ Convexity Δy2 = -3.9907 x –0.01 + .5 (24.22) (-.01)2 = .039907+.001211=4.1118%Check this against actual price change. Actual = 4.44/108.2=-4.104%12Duration & Convexity CalcsVolatility=Δ price/price =-[duration / (1 + yield)] * Δi = Corrected ΔP/P = -modified duration Δy + ½ Convexity Δy2 = = =Check this against actual price change. Actual = =Duration & Convexity Calculation5 year 5% bond (annual coupon payment)duration convexityyear coupon NPV @ 7% NPV/V NPV/V*t NPV/V*t(t+1)1234513Duration & VolatilityExample–Assume a 30 year bond, 8% coupon and initial yield to maturity of 8%. The bonds duration is 11.37 years. •(What does this mean?) –Convexity for this bond is 212.4. •(What does this mean?)–If yields move from 8% to 10%, how much would you expect the price of this bond to move?–What would the price be at the new interest rate?–Check your answer by re-valuing the bond at the new