Tài chính doanh nghiệp - Topic 10: Bond valuation, ytm, duration, convexity, and bond var

Financial ModelingTopic #10: Bond Valuation, YTM, Duration, Convexity, and Bond VaRL. Gattis1Learning ObjectivesValue bonds and compute yield to maturityCompute the sensitivity of bond prices with respect to changes in interest rates using duration and convexity and VaRCompute the VaR of a bond using duration and convexityUnderstand the duration and convexity of mortgagesUse Excel’s built-in bond functions23Bond Fundamental Value (V)Where V = fundamental bond value (today)i = time period t = Maturity of bond in yearsf = Annual frequency of bond payments (2: 2 pmts per year)tf = Number of bond payments (e.g.: 10yr bond, f=2; tf=20 pmts)CFi = cashflow at time i, usually consisting of coupon payments at each i (Par*CouponRate/f), and par at maturityr = discount rate for all cashflows (a.k.a., YTM)Note: Many bonds make coupon payments 2 times per year, where each coupon payment is ½ the annual coupon and the annual discount rate is divided by 24Bond Valuation5Bond Valuation FunctionFunction bondval(cr, par, t, freq, r)n = t * freq 'number of pmts For i = 1 To n bondval = bondval + (cr * par / freq) / (1 + r / freq) ^ i Next i bondval = bondval + par / (1 + r / freq) ^ nEnd Function6Yield-to-MaturityYTM, or promised yield, is the discount rate that equates the market price of the bond with the present value of contractual “promised” cashflows.YTM is also the IRR of the bond investment (buying at market price and receiving promised cashflows)Calculate the YTM of bonds 1 and 2 if the are selling for 1,045 and 1,047?7YTM8Interest Rate RiskThere is an inverse relationship between bond value and yieldsLong-term bonds are more sensitive to changes in yieldDuration is a measure of the interest rate risk of bonds, however there are several duration measures that are usedMacaulay Duration: “Weighted Avg. Maturity”Modified Duration: “Price Sensitivity”Effective Duration: Discussed Later9Macaulay DurationMacaulay duration is a measure of the “average” maturity of the stream of payments associated with a bond. Specifically, it is a weighted-average of the length of time until the bond’s remaining payments are made, with the weights equal to the present value of each cash flow relative to the present value of all the bonds cash flows. Assuming frequency = 110Duration Spreadsheet – frequency fModified Duration11Modified Duration with Semiannual Payment (frequency = f pmts per year)Where f = 2 for semiannual payment, (e.g., n=10 for 5 year semiannual payment bond)12Modified Duration FunctionFunction moddur(cr, par, t, freq, r)price = bondval(cr, par, t, freq, r) For i = 1 To (t * freq) moddur = moddur + ((cr * par / freq) / (1 + r / freq) ^ i) * i Next imoddur = moddur + (par / (1 + r / freq) ^ (t * freq)) * (t * freq)moddur = moddur / price / (1 + r / freq) / freqEnd FunctionFunction bondval(cr, par, t, freq, r)n = t * freq 'number of pmts For i = 1 To n bondval = bondval + (cr * par / freq) / (1 + r / freq) ^ i Next i bondval = bondval + par / (1 + r / freq) ^ nEnd FunctionUsing a user-defined function in a functionThe modified duration is Macaulay’s duration divided by one plus the yield to maturity. (for frequency=1 bond) It can be shown that this is equal to -slope of the price-yield function divided by the price.Duration is thus a measure of interest rate risk for bonds. Specifically, it’s the % change in price for a 100 bps change in the YTM13Modified DurationBYTMDmod=-Slope/BΔBΔY14Duration and Actual Price ChangeActual price-yield function is curved away from origin (convex function)Duration provides a linear estimate of ΔPGraphical Interpretation of Convexity PriceYieldBond ValueSlope of the tangent line is modified duration. For small changes in yield, duration predicts price changes well. For larger changes in yield, a correction for convexity (curvature) must be made.y0y1B0Duration Estimate of B1acbDuration Estimate15ConvexityBond price is a convex function of yield. This means that as the yield changes, the bond price will be above the price that would be predicted using the duration measure. On top of the price change estimate using duration, the price will change by the additional amount (based on convexity):The more convex a bond’s price-yield relationship, the more valuable the bond is, given a specific coupon rate. Bonds that are not callable are said to have “positive convexity”. Callable bonds (including Mortgage Backed Securities) have negative convexity.“Convexity & Duration1617Convexity for Semiannual Payment BondConvexity is approximately the second derivative of price with respect to yield Where f = 2 for semiannual payment, (e.g., t=10 for 5 year semiannual payment bond)18Convexity FunctionFunction convexity(cr, par, t, freq, r)price = bondval(cr, par, t, freq, r)For i = 1 To (t * freq) convexity= convexity + (cr * par / freq) * (1 + r / freq) ^ -i * i * (1 + i)Next iconvexity = convexity + par * (1 + r / freq) ^ -(t * freq) * (t * freq) * (1 + t * freq)convexity = convexity / (price * (1 + r / freq) ^ 2) * freq ^ (-2)End Function19ConvexityMortgage (MBS) Duration and Convexity20PriceYieldA 5-year noncallable bond and a 30-year mortgage security both have a duration of around 3-4 years. The mortgage duration is low because homeowners may prepay their mortgage (at par) when rates fall. 5-year non-callable bond30-yr mortgageMortgage (MBS) Duration and Convexity21PriceYieldThe mortgage has negative convexity / Interpretations of negative convexity1. Duration falls as interest rates fall because or prepayments, and mortgage durations get higher when rates increase because homeowners will delay prepayment2. The convexity adjustment is negative (the value is less than duration estimate)3. The appreciation of the mortgage is limited since the mortgage can always be prepaid at par 5-year non-callable bond30-yr mortgage100=parThe Curse of Negative ConvexityNegative convexity (interest rate risk) of mortgages is not attractive to investors (or banks) if rates fall or riseAs rates fallnon-callable bond price will appreciate according to its duration (and even more due to convexity)Mortgage do not appreciate much because the mortgage can always be prepaid at par (limited upside!)Worse yet, as rates fall the mortgage prepays and investors (banks) receive the cash and must reinvest it in new securities with lower rates (reinvestment risk)As rates riseThe mortgage value decreases more than non-callable bonds because the duration gets longer as homeowners plan to hold on to the mortgage longer. This is bad because the mortgages instrument’s low coupon will last longer.2223Duration and Convexity VaRThe change in bond value for a given yield change (Δy) isWhere: B0 = Initial Bond Value, d=Modified Convexity, c = ConvexityIf you assume interest rate changes are normally distributed with a zero mean and expected standard deviation, σ, then the X% Confidence Interval VaR for a bond portfolio value of V$ is:Where Zx is from the normal distribution table Z90% = 1.28; Z95% = 1.65; Z99% = 2.33Duration and Convexity VaR95% confident that losses will not exceed $19.25 (1.75%) in 5 days24Key Rate DurationDuration is a measure of gain or loss from a 100 bps parallel shift in the entire yield curve (1 day – 30 year maturity)Yield curves shift are often not parallel (changing the slope of the yield curve)Key rate durations (say 1- ,5-, and 30-year KRDs) measure the sensitivity to a 100 bps change in that maturity onlyKRDs are used to measure the sensitivity of yield curve re-shaping (a.k.a., twisting or non-pararallel changes).E.g. A 10-year, zero coupon bond is only sensitive to change in 10 year yield. (This affects the measured risk (VaR) of bonds)25YTParallel ShiftYTNon-Parallel ShiftFlatteningExcel Price, Duration, and Yield FunctionsExcel’s bond functions use dates and not maturity in years. Parameters areWe will always use 100Omit Basis for now, addressed in a later topic26Excel Price, Duration, and Yield FunctionsFor this class, if you are given dates, use Excel’s functions. The user-defined functions only work on payment dates27Learning ObjectivesValue bonds and compute yield to maturityCompute the sensitivity of bond prices with respect to changes in interest rates using duration and convexity and VaRCompute the VaR of a bond using duration and convexityUnderstand the duration and convexity of mortgagesUse Excel’s built-in bond functions28