Interest-Rate Risk II. Duration Rules Rule 1: Zero Coupon Bonds What is the duration of a zero-coupon bond? Cash is received at one time t=maturity weight.

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Interest-Rate Risk II

Duration Rules Rule 1: Zero Coupon Bonds What is the duration of a zero-coupon bond? Cash is received at one time t=maturity weight = 1 So the duration of a zero coupon bond is just its time to maturity in terms of how we have defined “one period” (usually six months)

Duration Rules Rule 2: Coupon Rates Coupons early in the bond’s life reduce the average time to get payments. Weights on early “times” are higher Holding time to maturity constant, a bond’s duration is lower when the coupon rate is higher.

Duration Rules Rule 3: Time to Maturity Holding the coupon rate constant, a bond’s duration generally increases with time to maturity. – If yield is outrageously high, then higher maturity decreases duration. – Rule 4: Yield to Maturity For coupon bonds, as YTM increase, duration decreases. Rule 5: The duration of a level perpetuity is (1+y)/y

Modified Duration of a Portfolio Banks hold several assets on their balance sheets. Let v i be the fraction of total asset PV attributed to asset i. Suppose the bank holds 3 assets Duration of total assets:

Example Bank Assets: – Asset 1: PV=\$ 8MD*=12.5 – Asset 2: PV=\$38MD*=18.0 – Asset 3: PV=\$ 2MD*= 1.75 Total PV = \$48M – v 1 =8/48=0.17, v 2 =38/48=0.79v 3 =2/48=0.04 Modified Duration of Portfolio: D*=(0.17)(12.5) + (0.79)(18) + (0.04)(1.75)=16.42

Review For zero coupon bonds: – YTM=effective annual return For annual bonds: – Effective annual return = YTM assuming we can reinvest all coupons at the coupon rate For semi-annual bonds – Effective six-month return =YTM/2 assuming we can reinvest all coupons at the coupon rate

Effective Annual Return of a Portfolio Example: Portfolio Value: \$110 Annual Bond 1: PV=\$65, EAR=5% Annual Bond 2: PV=\$45, EAR=3% What is effective return on portfolio? (get/pay-1) Get=65*1.05+45*1.03=114.6 Pay=110 Return=114.6/110-1=4.18% But (65/110)*.05+(45/110)*.03=4.18% Bottom line: the EAR of a portfolio is the weighted sum of the EARs of the individual assets in the portfolio where weights are the fraction of each asset of total portfolio value.

Back to Building a Bank From previous example (Building a Bank) Assets: D*=23.02, PV=100M (YTM=1.8%) Liabilities: D*=0.99, PV=75M (YTM=1%) Equity: 25M Currently a 10 bp increase in rates causes:  A = -23.02*.001*100M = -2.30M  L = -0.99*.001*75M = -0.074M  E =-2.30M-(-0.074M) = -2.23M (drop of 8.8%)

Building a Bank Suppose you want a 10bp increase in rates to cause equity to drop by only 4% (1M). Options: A: Hold D* of assets constant and raise D*of liabilities B: Hold D* of liabilities constant and lower D* assets C: Raise D* of liabilities and lower D* of assets

Building a Bank: Option A Hold D* of assets at 23.02 For any given D* of liabilities, a 10bp increase in rates will cause equity to change as follows:  E = -2.30M- (-D*75M*.001) Given that you want a 10bp increase in rates to cause equity to drop by 1M: -1M= -2.30M- (-D*75M*.001) solve for D* D*=17.333

Building a Bank: Option A How to get D* of liabilities to 17.33? Issue a bond or CD with duration greater than 17.33. Example: Issue a zero-coupon bond that matures in 25 years. Assume YTM=1.5%. – Duration=25 – D* = 25/1.015 = 24.63 How much should you issue?

Building a Bank: Option A You want the D* of your “liability portfolio” to be 17.33. Let v=fraction of liability portfolio in the 25yr zero-coupon bond. The rest of your liabilities will come from short-term deposits. 17.33 = v(24.63)+(1-v)(0.99) solve for v v =.6912

Building a Bank: Option A So make the 25yr bond 69.12% of your liability portfolio. Total liabilities = 75M Issue.6912*75M = \$51.84M in 25yr zero- coupon bonds with D*=24.63 Raise \$23.16M in short-term deposits with D*=0.99

Building a Bank: Option A Checking the approximation: Liabilities: – 51.84 in 25yr zero-coupon bonds (YTM=.015) – 23.16 in deposits (YTM=.01) We use the duration approximation to set the target. How do we know if the approximation works? Let’s find the exact change in equity for a 10bp increase in rates. First, we need to find future values

Building a Bank: Option A Future value of Liabilities: – 51.84 in 25yr zero-coupon bonds (YTM=.015) Future value at expiration (face value) = 51.84*(1.015)^25=75.22 – 23.16M in deposits (YTM=.01) Future value at expiration = 23.16*1.01 = 23.39 Present value if rates jump by 10bp: – Zero-coupon bonds: 75.22/1.016^25=50.58 – Deposits: 23.39/1.011 = 23.14 Change in PV of liabilities if rates jump by 10bp: (50.58M + 23.14) – 75M = -1.28M

Building a Bank: Option A We know (slides last Wed) that if rates jump by 10bp, assets will drop by exactly 2.27M (PV of bonds drops from 100M to 97.73M) Change in equity, given a 10bp increase in rates, will be-2.27M-(-1.28M)= -0.99M Our objective was to have it drop by 1M. So we are very close.

Building a Bank: Option A By switching away from short-term deposits we’ve lowered interest-rate risk. Cost (before rates change): Before we tailored the balance sheet: – Liabilities (75M) YTM=1% – Assets (100M) YTM=1.8% – Profits=1.8M-.75M=1.05M After tailoring the balance sheet – Liabilities: 0.6912*.015+0.3088*.01 = 1.3% – Assets (100M) YTM=1.8% – Profits=1.8M-1.3M=0.50M

Building a Bank: Option B Hold D* of liabilities at 0.99 For any given D* of assets, a 10bp increase in rates will cause equity to change as follows:  E = -D*100M*.001-(-0.074M) Given that you want a 10bp increase in rates to cause equity to drop by only 1M: -1 = -D*100*.001-(-0.074) solve for D* D*=10.74

Building a Bank: Option B How to get D* of liabilities to 10.74? Buy a bond duration less than 10.74 Example: zero-coupon bond than matures in 5 years. Assume YTM=1.2%. – Duration=5 – D* = 5/1.012 = 4.94 How much should you purchase?

Building a Bank: Option B You want the D* of your asset portfolio to be 10.74. Let v=fraction of asset portfolio in the 5yr zero- coupon bond (D*=4.94). The rest of your assets will be in the 30-yr coupon bonds (D*=23.02). 10.74 = v(4.94)+(1-v)(23.02) solve for v v = 0.679

Building a Bank: Option B So make the 5yr zero 67.9% of your assets Total assets = 100M Buy.679*100M = \$67.9M in 5yr zeros Purchase \$32.1M in the 30-year coupon paying bond

Building a Bank: Option B Checking the effect: Assets: – 67.9 in 5yr zero-coupon bonds (YTM=.012) – 32.1M in 30-year coupon bonds (YTM=.018) We want to see how the PV of these assets change as we observe a parallel shift in the yield curve. To do this, we need to find future values.

Building a Bank: Option B Future value of Assets: – 67.9 in 5yr zero-coupon bonds (YTM=.012) Future value at expiration (face value) = 67.9*(1.012)^5 = 72.07 – 32.1 in 30-year bonds (YTM=.018, coupon rate=0.18) Future value at expiration (face value)=32.1 Present value if rates jump by 10bp: – 5yr zeros: 72.07/1.013^5=67.56 – 30-yr bonds: N=30, FV=32.1, pmt=.018*32.1, ytm=0.019 PV=31.37 Change in PV of assets if rates jump by 10bp: (67.56+31.37) – 100 = -1.07 (million)

Building a Bank: Option B We know (from class last Wed) that if rates jump by 10bp, liabilities will drop by exactly 0.074M So, given new structure of assets, given a 10bp increase in rates, equity will change as follows: -1.07M-(-0.074M)= -0.996M Our objective was to have it drop by 1M. So we are very close.

Building a Bank: Option B By switching away from short-term deposits we’ve lowered interest-rate risk. Cost (before rates change): Before we tailored the balance sheet: – Liabilities (75M) YTM=1% – Assets (100M) YTM=1.8% – Profits=1.8M-.75M=1.05M After tailoring the balance sheet – Liabilities (75M) YTM=1% – Assets (100M) YTM=.679*.012+.321*.018=1.4% – Profits=1.4M-.75M=0.65M

Important Facts We hedged only at the present time. As time changes and yields change, modified durations will change. Need to periodically rebalance hedging portfolio, even if yields remain constant, or hedge will become useless.

Building a Bank: Option C You can choose several different combinations of the modified durations of assets and liabilities to accomplish the same objective. Next slide: The possible combinations

Building a Bank: Option C D* of Assets=23.02 D* of Liabilities=17.33 D* of Assets=10.74 D* of Liabilities=0.99

Duration Using only duration can introduce approximation error. Duration matching works best for small changes in yields. Duration allows us to match the slope of the price-curve at a given point. As you move away from this point, the slope will change – the source of approximation error.

Duration

Convexity Convexity is a measure of how fast the slope is changing at a given point. Not very convex.More convex.

Convexity Bond investors like convexity – When yields go down, the prices of bonds with more convexity increase more. – When yields go up, the prices of bonds with more convexity drop less The more convex a bond is, the worse the duration approximation will do. – Possible to incorporate convexity into analysis above.

Appendix: Modified Duration of a Portfolio

Appendix Modified Duration of a portfolio (continued)

Appendix Modified Duration of a portfolio (continued)

Appendix Modified Duration of a portfolio (continued)

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