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Class 7, Chap 9 - Appendix B

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Purpose: Gain a deeper understanding of duration and its properties and weaknesses Properties of duration Hedging with duration Weaknesses of duration Convexity

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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

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Calculate the duration for bonds of several maturities with an 8% coupon paid semiannually, $1,000 face value and yield to maturity of 12%. 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 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

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Calculate the duration for bonds of several maturities with an 8% coupon paid semiannually, $1,000 face value and yield to maturity of 12%. 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 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

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6 Lets just look at what happens to the present value of cash flows as the maturity increases 1, Time to Maturity = 5 years Duration = 4.14

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7 Lets just look at what happens to the present value of cash flows as the maturity increases 1, Time to Maturity = 10 years Duration = 6.61

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8 Lets just look at what happens to the present value of cash flows as the maturity increases 1, Time to Maturity = 15 years Duration = 7.91

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9 Lets just look at what happens to the present value of cash flows as the maturity increases 1, Time to Maturity = 20 years Duration = 8.53

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10 Lets just look at what happens to the present value of cash flows as the maturity increases 1, Time to Maturity = 20 years Duration = 8.53 Total weight (sum) = 48% A large percent of the bond value has been received early-on !!! Total weight (sum) = 75%Total weight (sum) = 86%

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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.

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1, Lets just look at what happens to the present value of cash flows as the YTM increases

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1, Lets just look at what happens to the present value of cash flows as the YTM increases YTM = 10% Duration = 4.18

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1, Lets just look at what happens to the present value of cash flows as the YTM increases YTM = 30% Duration = 3.74

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1, Lets just look at what happens to the present value of cash flows as the YTM increases YTM = 50% Duration = 3.23

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1, Lets just look at what happens to the present value of cash flows as the YTM increases YTM = 70% Duration = 2.71

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1, Lets just look at what happens to the present value of cash flows as the YTM increases YTM = 90% Duration = 2.26

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1, Lets just look at what happens to the present value of cash flows as the YTM increases 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

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1, 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 Coupon = 10% Duration = 4.04

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1, 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 Coupon = 40% Duration = 3.29

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1, 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. Coupon = 70% Duration = 3.07

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1, 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 Coupon = 100% Duration = 2.97

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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

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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

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Hedge With Duration 25

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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

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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% = $ 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

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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

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Step#1 Find the coupon Coupon = (1,000)*.08 = $80 Draw the cash flows Step#2 Find present values Step#4 Find duration weights Step#5 Find duration 1, The 6 year bond is also a viable option for the hedge 29

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The company will owe in 5 years So the company wants to receive $ (for sure) in 5 years to be completely hedged Each bond pays 1000 in 5 years so they need to buy /1000 = zero coupon bonds Cost: The price of the zero coupon = 1000/(1.08) 5 = The company needs of them so the total cost is ( )(680.58) = $1,000 The full amount of their loan 30

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The company is perfectly hedged!!!! After purchasing the zero coupon bonds, the company has locked-in a positive cash flow in five years no matter what interest rates do!!! , , Bond Loan

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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. The YTM stays at 8% b. The YTM instantaneously increases to 9% c. The YTM instantaneously decreases to 7% Why 1 bond? – if we find the value of all cash flows at time 5 years (1000)( ) =$1, If this was not the case, we would need to buy more or less than one bond But if this was not the case the bond would not have a 5 year duration 32

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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 1, Reinvest for 4 years Reinvest for 3 years Reinvest for 2 years Reinvest for 1 years Collect coupon & sell bond 33

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Case 1 YTM increases to 9% The company will hold the bond for 5 years The coupon will be reinvested at the YTM 1, Reinvest for 4 years Reinvest for 3 years Reinvest for 2 years Reinvest for 1 years Collect coupon & sell bond 34

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Case 2 YTM decreases to 7% The company will hold the bond for 5 years The coupon will be reinvested at the YTM 1, Reinvest for 4 years Reinvest for 3 years Reinvest for 2 years Reinvest for 1 years Collect coupon & sell bond 35

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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. The duration of the bond (used to hedge) will change 2. The YTM of the bond used to hedge could change 36 What kind of risk would the coupons be subject to?

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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: Duration 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% 1. Duration Change 37

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Suppose that after the first two payments the interest rate increases to 9% So what’s the point? This seems really ineffective – why am I not teaching you how to fully resolve this problem? 1, The company no longer has enough money to repay its loan of $ IT IS HARD!!! 2. Reinvestment risk 38

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Difficulties with Duration 39

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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. Every time the interest rate changes the hedging portfolio must be rebalanced 2. One decision managers have to make is how often to rebalance and weigh the cost of doing so 3. 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

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 41 Price a 20 year bond with coupon of 30% and semiannual payments

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 42 Price a 20 year bond with coupon of 30% and semiannual payments

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 43 Price a 20 year bond with coupon of 30% and semiannual payments

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 44 Price a 20 year bond with coupon of 30% and semiannual payments

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 45 Price a 20 year bond with coupon of 30% and semiannual payments

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 46 Price a 20 year bond with coupon of 30% and semiannual payments

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 47 Price a 20 year bond with coupon of 30% and semiannual payments

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 48 Price a 20 year bond with coupon of 30% and semiannual payments

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 49

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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

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 51 13% This is what duration says the graph (relationship) should look like When we do the duration calculation, we find a point on this line Duration is the derivative. It is the slope of the tangent line

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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? What is the actual price? 52

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 53 13% 3%

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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? What is the actual price? 54

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Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch 55 13% 3% 23% Asymmetric Pricing Errors!

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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

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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

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1. Calculate the duration weights 2. Multiply the weights by the time period squared plus and the same time period 3. Sum values and divide by (1+ YTM) 2 to get convexity 58

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Consider a 6 year bond with an 8% coupon paid annually the YTM is 6%. Face value of Calculate the convexity of the bond 1000 Step #1 find the present value of payments Step #2 calculate weights

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Consider a 6 year bond with an 8% coupon paid annually the YTM is 6%. Face value of Calculate the convexity of the bond 60 Measures the curvature of the YTM bond price relationship – larger values = more curvature Step #3 calculate the convexity

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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

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We can use it to adjust the accuracy of the duration calculation!! 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). 62

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63 Price implied by Duration Price implied by Duration & Convexity

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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

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