Transcript Slide 1
Class 7, Chap 9 - Appendix B Purpose: Gain a deeper understanding of duration and its properties and weaknesses Properties of duration Hedging with duration Weaknesses of duration Convexity 1. Duration increases with maturity but at a decreasing rate 2. Duration decreases as the yield to maturity increases 3. Duration decreases as the coupon payments or interest rate increases Calculate the duration for bonds of several maturities with an 8% coupon paid semiannually, $1,000 face value and yield to maturity of 12%. Duration & Time to Maturity 10 9 8 Adding a year means: • The big payment occurs 1 year later • Adds 1 year to the weighted average • Because there is not a lot of discounting, the weight on the additional year is large Duration 7 6 5 4 823.78 925.60 3 37.74 37.74 35.60 33.58 2 1 0 0.5 0 1 0.5 1 1.5 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Years When we add a year to a long maturity bond it changes the duration much less than when we add a year to a short maturity bond 27 28 29 30 Calculate the duration for bonds of several maturities with an 8% coupon paid semiannually, $1,000 face value and yield to maturity of 12%. Duration & Time to Maturity 10 9 8 Adding a year means: • The big payment occurs 1 year later • Adds 1 year to the weighted average • There is a lot of discounting so the weight on the additional year is small compared to other years Duration 7 6 5 4 33.42 3 37.74 2 1 0 0.5 35.60 1.53 1.44 1 28 28.5 31.53 37.74 29 0 0.5 35.60 1.36 1.29 1 29 29.5 30 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Years When we add a year to a long maturity bond it changes the duration much less than when we add a year to a short maturity bond 27 28 29 30 0 Lets just look at what happens to the present value of cash flows as the maturity increases 1,040 40 1 2 3 4 5 Time to Maturity = 5 years Duration = 4.14 6 0 Lets just look at what happens to the present value of cash flows as the maturity increases 1,040 40 1 2 3 4 5 6 7 8 9 10 Time to Maturity = 10 years Duration = 6.61 7 0 Lets just look at what happens to the present value of cash flows as the maturity increases 1,040 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time to Maturity = 15 years Duration = 7.91 8 0 Lets just look at what happens to the present value of cash flows as the maturity increases 1,040 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time to Maturity = 20 years Duration = 8.53 9 0 Lets just look at what happens to the present value of cash flows as the maturity increases 1,040 Duration = 8.53 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Time to Maturity = 20 years A large percent of the bond value has been received early-on !!! Total weight (sum) = 48% Total weight (sum) = 75% Total weight (sum) = 86% 10 20 Conclusion: Duration increases with maturity but at a decreasing rate because of two effects: 1. Increasing the maturity adds more years to the bond, which increases duration 2. As we increase the time to maturity (TTM), a smaller and smaller fraction of bond value is being received at a later date. This is because later payments are highly discounted. As a result, a large fraction of bond value is received early on, which stabilizes the duration. Lets just look at what happens to the present value of cash flows as the YTM increases 1,000 0 40 40 40 40 40 40 40 40 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 40 5 Lets just look at what happens to the present value of cash flows as the YTM increases 1,000 0 40 40 40 40 40 40 40 40 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 YTM = 10% Duration = 4.18 40 5 Lets just look at what happens to the present value of cash flows as the YTM increases 1,000 0 40 40 40 40 40 40 40 40 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 YTM = 30% Duration = 3.74 40 5 Lets just look at what happens to the present value of cash flows as the YTM increases 1,000 0 40 40 40 40 40 40 40 40 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 YTM = 50% Duration = 3.23 40 5 Lets just look at what happens to the present value of cash flows as the YTM increases 1,000 0 40 40 40 40 40 40 40 40 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 YTM = 70% Duration = 2.71 40 5 Lets just look at what happens to the present value of cash flows as the YTM increases 1,000 0 40 40 40 40 40 40 40 40 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 YTM = 90% Duration = 2.26 40 5 Lets just look at what happens to the present value of cash flows as the YTM increases 1,000 0 40 40 40 40 40 40 40 40 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 40 5 As we increase the yield to maturity, the present value (and as a result the duration weights) of the earlier payments increase relative to the PV (duration weights) of the later payments That is, the percentage of value [PV(future cash flows)] received early in the bond’s life increases – so the later payments (more interest rate sensitive) are not as important Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate. 1,000 0 50 50 50 50 50 50 50 50 50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Coupon = 10% Duration = 4.04 50 5 Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate. 1,000 0 200 200 200 200 200 200 200 200 200 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Coupon = 40% Duration = 3.29 200 5 Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate. 1,000 0 350 350 350 350 350 350 350 350 350 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Coupon = 70% Duration = 3.07 350 5 Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate. 1,000 0 500 500 500 500 500 500 500 500 500 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Coupon = 100% Duration = 2.97 500 5 Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate. As we increase the coupon rate the present value of early cash flows (duration weights) increases relative to later payments That is, the percentage of value [PV(future cash flows)] received early in the bonds life increases – so the later payments (more interest rate sensitive) are not as important 1. Duration increases with maturity but at a decreasing rate 2. Duration decreases as the yield to maturity increases 3. Duration decreases as the coupon payments increase 4. You need to have a basic understanding of why duration behaves this way Hedge With Duration 25 We have seen that duration measures the sensitivity of assets to changes in interest rates Now lets see how we can use that to manage interest rate risk Basic idea: by taking an offsetting position in an asset/liability with a matched duration an investor can hedge interest rate risk 26 Suppose a company has 5 years left on a loan: The company wants to pay back the loan today but there are stiff prepayment penalties. So, the company decides to offset the loan with another asset. The loan is a balloon payment loan - it is paid back in one lump sum payment in five years – no interim interest payments Current value of the loan is $1,000 at 8% = $1469.33 due in 5 years The company wants to hedge against changes in interest rates and can choose from the following instruments: A 3 year 3% coupon bond with $1,000 face value A five year zero coupon bond with an 8% YTM and face value = 1000 A six year bond with an 8% coupon paid annually and face value = 1,000 and YTM = 8% 27 The company can manage its interest rate risk by matching durations The duration of the 3 year bond will definitely be too short The five year zero coupon bond has a duration of 5 years The six year coupon can not be ruled out so we need to calculate the duration 28 Step#1 Find the coupon Coupon = (1,000)*.08 = $80 Draw the cash flows 1,000 80 80 80 80 80 1 2 3 4 5 80 6 Step#2 Find present values P 80 80 80 80 80 1080 1.08 1.082 1.083 1.084 1.085 1.086 P 74.04 68.59 63.51 58.80 54.45 680.58 1,000 The 6 year bond is also a viable option for the hedge Step#4 Find duration weights w1 74.04 1000 w2 68.59 1000 w3 63.51 1000 w4 58.80 1000 w5 54.45 1000 w6 680 .58 1000 Step#5 Find duration D (0.07404)(1) (0.06859)(2) (0.06351)(3) (0.05880)(4) (0.05445)(5) (0.68058)(6) 4.9927 29 The company will owe 1469.33 in 5 years So the company wants to receive $1469.33 (for sure) in 5 years to be completely hedged Each bond pays 1000 in 5 years so they need to buy 1469.33/1000 = 1.46933 zero coupon bonds Cost: The price of the zero coupon = 1000/(1.08)5 = 680.58 The company needs 1.46933 of them so the total cost is (1.46933)(680.58) = $1,000 The full amount of their loan 30 The company is perfectly hedged!!!! After purchasing the zero coupon bonds, the company has locked-in a positive1469.33 cash flow in five years no matter what interest rates do!!! 1,469.33 Bond 0 1 2 3 4 5 Loan - 1,469.33 31 We saw that the 6 year bond had a duration of 5 years so lets try using it to hedge. To hedge the company can buy one 5 year duration bond for a cost of $1,000 Consider three cases : a. b. c. Why 1 bond? – if we find the value of all cash The YTM stays at 8% flows at time 5 years (1000)(1.085) =$1,469.33. The YTM instantaneously to 9% • If this increases was not the case, we would need to buy more or less than one bond The YTM instantaneously 7%the bond would not • But if decreases this was not thetocase have a 5 year duration 32 Base case: Show that the company is hedged if the YTM = 8% The company will hold the bond for 5 years The coupon will be reinvested at the YTM Reinvest Reinvest Reinvest ReinvestCollect coupon & for 4 yearsfor 3 years for 2 yearsfor 1 years sell bond 1,000 80 80 80 80 80 1 2 3 4 5 (80)(1.08) 4 (80)(1.08)3 (80)(1.08) 2 (80)(1.08) 80 6 1080 80 1.08 CF5 yr 108.84 100.78 93.31 86.40 (80 1000) 1469.33 33 Case 1 YTM increases to 9% The company will hold the bond for 5 years The coupon will be reinvested at the YTM Reinvest Reinvest Reinvest ReinvestCollect coupon & for 4 yearsfor 3 years for 2 yearsfor 1 years sell bond 1,000 80 80 80 80 80 1 2 3 4 5 (80)(1.09)4 (80)(1.09)3 (80)(1.09)2 80 6 (80)(1.09) 80 1080 1.09 CF5 yr 112.93103.60 95.05 87.20 (80 990.83) 1469.33 34 Case 2 YTM decreases to 7% The company will hold the bond for 5 years The coupon will be reinvested at the YTM Reinvest Reinvest Reinvest ReinvestCollect coupon & for 4 yearsfor 3 years for 2 yearsfor 1 years sell bond 1,000 80 80 80 80 80 1 2 3 4 5 (80)(1.07)4 (80)(1.07)3 (80)(1.07)2 (80)(1.07) 80 6 1080 80 1.07 CF5 yr 104.86 98.00 91.59 85.60 (80 1009.35) 1469.33 35 If the company offsets its assets or liabilities with an instrument of the same duration the position will be immune to changes in interest rates Do you think this really works? It could, but we run into two problems 1. 2. The duration of the bond (used to hedge) will change The YTM of the bond used to hedge could change What kind of risk would the coupons be subject to? 36 1. Duration Change Lets calculate the duration of the bond right after the second coupon is paid – there are four years (coupons) left. Assume they YTM = 8% Weights: P 80 80 80 1080 1000 1.08 1.082 1.083 1.084 P 74.04 68.59 63.51 793.83 1,000 w1 74.04 1000 w2 68.59 1000 w3 63.51 1000 w4 793 .83 1000 Duration D (0.07404)(1) (0.06859)(2) (0.06351)(3) (0.079383)4 3.577years Loan: the loan still has 3 years to maturity so the durations no longer match – this is ok as long as the coupons have and can continue to be reinvested at 8% 37 2. Reinvestment risk Suppose that after the first two payments the interest rate increases to 9% 1,000 80 80 80 80 80 1 2 3 4 5 (80)(1.08) 4 (80)(1.08)3 (80)(1.09)2 80 The company no longer has enough money to repay its loan of $1469.33 6 (80)(1.09) 80 1080 1.09 CF5 yr 108.84 100.7 95.05 87.20 (80 990.83) 1462.69 So what’s the point? This seems really ineffective – why am I not teaching you how to fully resolve this problem? IT IS HARD!!! 38 Difficulties with Duration 39 1. Reallocating large quantities of assets or liabilities to attain the needed durations for assets and liabilities can be very costly 2. Immunization is a dynamic problem 1. 2. 3. Every time the interest rate changes the hedging portfolio must be rebalanced One decision managers have to make is how often to rebalance and weigh the cost of doing so Convexity- Duration only works for small changes in the interest rate 1. For large changes in rates duration will not accurately predict the percent change in the price of a security 40 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch Price a 20 year bond with coupon of 30% and semiannual payments 6000 5000 Bond Price 4000 3000 2000 1000 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Yield To Maturity 41 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch Price a 20 year bond with coupon of 30% and semiannual payments 6000 YTM 1.5% 5000 Bond Price 4000 40 150 1000 t (1 .015/ 2) 40 t 1 (1 .015/ 2) 3000 $5,908.69 2000 1000 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Yield To Maturity 42 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch Price a 20 year bond with coupon of 30% and semiannual payments 6000 YTM 4% 5000 Bond Price 4000 40 150 1000 t (1 .04 / 2) 40 t 1 (1 .04 / 2) 3000 $4,556.21 2000 1000 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Yield To Maturity 43 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch Price a 20 year bond with coupon of 30% and semiannual payments 6000 YTM 8% 5000 Bond Price 4000 40 150 1000 t (1 .08 / 2) 40 t 1 (1 .08 / 2) 3000 $3,177.21 2000 1000 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Yield To Maturity 44 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch Price a 20 year bond with coupon of 30% and semiannual payments 6000 YTM 13% 5000 Bond Price 4000 40 150 1000 t (1 .13 / 2) 40 t 1 (1 .13 / 2) 3000 $2,202.37 2000 1000 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Yield To Maturity 45 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch Price a 20 year bond with coupon of 30% and semiannual payments 6000 YTM 20% 5000 Bond Price 4000 40 150 1000 t (1 .20 / 2) 40 t 1 (1 .20 / 2) 3000 $1,488.95 2000 1000 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Yield To Maturity 46 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch Price a 20 year bond with coupon of 30% and semiannual payments 6000 YTM 30% 5000 Bond Price 4000 40 150 1000 t (1 .30 / 2) 40 t 1 (1 .30 / 2) 3000 $1,000 2000 1000 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Yield To Maturity 47 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch Price a 20 year bond with coupon of 30% and semiannual payments 6000 YTM 45% 5000 Bond Price 4000 40 150 1000 t (1 .45 / 2) 40 t 1 (1 .45 / 2) 3000 666.77$ 2000 1000 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Yield To Maturity 48 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 6000 5000 4000 Bond Price 3000 2000 1000 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Yield To Maturity 49 What does duration say about this relation? Duration is the derivative of the bond pricing formula with respect to the interest rate at a specific point on the graph What does the derivative look like on the graph? 50 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch Duration is the derivative. It is the slope of the tangent line • This is what duration says the graph (relationship) should look like • When we do the duration calculation, we find a point on this line 13% 51 Calculate the duration of the bond if the YTM is 13% D = 7.23 years If the YTM dropped to 3% what price would the duration predict? P R .1 D 7.23 0.7880 P (1 R) 1 0.13 / 2 P (0.7880)(2202.37) 1735.47 Pt 1 Pt P 2202.13 1735.47 3937.84 What is the actual price? 40 150 1000 5038.64 t 40 (1 .03/ 2) t 1 (1 .03 / 2) 5038.64 3937.84 1100.80 52 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 5,038.64 1100 .80 3937 .84 3% 13% 53 Calculate the duration of the bond if the YTM is 13% D = 7.23 years If the YTM jumped to 23% what price would the duration predict? P R .1 D 7.23 0.7880 P (1 R) 1 0.13/ 2 P (0.7880)(2202.37) 1,735.47 Pt 1 Pt P 2202.131735.47 466.39 What is the actual price? 40 150 1000 1300.44 t 40 (1 .23/ 2) t 1 (1 .23 / 2) 466.39 1300.44 834.05 54 Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch Asymmetric Pricing Errors! 1100 .80 1300 .44 834 .05 666 .39 3% 13% 23% 55 The larger the convexity the more curvature there is in the line Duration will work better for bonds with low convexity We will calculate convexity next 56 1. Duration is only accurate for small changes in interest rates 2. Duration will predict lower than actual values 3. The under prediction error is greater when interest rates fall then when they increase 4. Duration will change depending on the interest rate!!!!!!!!! 57 1. Calculate the duration weights 2. Multiply the weights by the time period squared plus and the same time period C W1 t12 t1 W2 t 22 t 2 ... Wn t n2 t n 3. Sum values and divide by (1+ YTM)2 to get convexity 58 Consider a 6 year bond with an 8% coupon paid annually the YTM is 6%. Face value of 1000. Calculate the convexity of the bond 1000 80 80 80 80 80 80 Step #1 find the present value of payments pv (CF1 ) 80 1.06 pv (CF2 ) 80 1.06 2 pv (CF3 ) 80 1.06 3 pv (CF4 ) 80 1.06 4 pv (CF5 ) 80 1.06 5 pv (CF6 ) 1080 1.06 6 pv(CF1 ) 75.47 pv(CF2 ) 71.20 pv(CF3 ) 67.17 pv(CF4 ) 63.37 pv(CF5 ) 59.78 pv(CF6 ) 761.36 Step #2 calculate weights w1 75.47 1098 .35 w1 0.069 w2 71.20 1098 .35 w2 0.065 w3 67.17 1098 .35 w3 0.061 w4 63.37 1098 .35 w4 0.058 w5 59.78 1098 .35 w5 0.054 w6 761 .36 1098 .35 w6 0.693 59 Consider a 6 year bond with an 8% coupon paid annually the YTM is 6%. Face value of 1000. Calculate the convexity of the bond Step #3 calculate the convexity 2 2 2 1 (0.0687)(1 1) (0.0648)(2 2) (0.0612)(3 3) C 2 2 2 2 (1.06 ) (0.0577)(4 4) (0.0544)(5 5) (0.6932)(6 6) C 1 33 .16 2 1.06 C 29.54 Measures the curvature of the YTM bond price relationship – larger values = more curvature 60 Calculate the convexity of a 1.5 year 4% coupon bond with semiannual payments and face value of 5,000 if the risk free rate is currently 5% and the YTM is 9% 61 We can use it to adjust the accuracy of the duration calculation!! R 1 P D C ( R 2 ) P 1 R 2 Example: Estimate the expected percent change in the price of the bond from the previous example (FV = 5000, coupon = 4%, TTM = 1.5yrs, semiannual compounding) if interest rates are expected to increases from 9% to 11.4% (the duration of the bond 1.47yrs). 0.024 1 P 0(.03375 3.346 )(. 024 2 ) 0.03279 1.47 P 1 0.09 2 2 62 Price implied by Duration Price implied by Duration & Convexity 63 3 properties of duration Duration increases with maturity but at a decreasing rate Duration decreases as the yield to maturity increases Duration decreases as the coupon payments or interest rate increases Hedging by matching duration The hedge is only perfect if YTM remains constant over the life of the hedge Convexity Concept Calculation 64