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# Duration model

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### Duration model

1. 1. <ul><li>Interest Rate Risk </li></ul><ul><li>-Duration model </li></ul>
2. 2. Overview <ul><li>This chapter discusses a market value-based model for assessing and managing interest rate risk: </li></ul><ul><ul><li>Duration </li></ul></ul><ul><ul><li>Computation of duration </li></ul></ul><ul><ul><li>Economic interpretation </li></ul></ul><ul><ul><li>Immunization using duration </li></ul></ul><ul><ul><li>* Problems in applying duration </li></ul></ul>
3. 3. Price Sensitivity and Maturity <ul><li>In general, the longer the term to maturity, the greater the sensitivity to interest rate changes. </li></ul><ul><li>Example: Suppose the zero coupon yield curve is flat at 12%. Bond A pays \$1762.34 in five years. Bond B pays \$3105.85 in ten years, and both are currently priced at \$1000. </li></ul>
4. 4. Example continued... <ul><ul><li>Bond A: P = \$1000 = \$1762.34/(1.12) 5 </li></ul></ul><ul><ul><li>Bond B: P = \$1000 = \$3105.84/(1.12) 10 </li></ul></ul><ul><li>Now suppose the interest rate increases by 1%. </li></ul><ul><ul><li>Bond A: P = \$1762.34/(1.13) 5 = \$956.53 </li></ul></ul><ul><ul><li>Bond B: P = \$3105.84/(1.13) 10 = \$914.94 </li></ul></ul><ul><li>The longer maturity bond has the greater drop in price because the payment is discounted a greater number of times . </li></ul>
5. 5. Coupon Effect <ul><li>Bonds with identical maturities will respond differently to interest rate changes when the coupons differ. </li></ul><ul><li>This is more readily understood by recognizing that coupon bonds consist of a bundle of “zero-coupon” bonds. </li></ul><ul><li>With higher coupons, more of the bond’s value is generated by cash flows which take place sooner in time. Consequently, less sensitive to changes in R. </li></ul>
6. 6. Price Sensitivity of 6% Coupon Bond
7. 7. Price Sensitivity of 8% Coupon Bond
8. 8. Remarks on Preceding Slides <ul><li>In general, longer maturity bonds experience greater price changes in response to any change in the discount rate. </li></ul><ul><li>The range of prices is greater when the coupon is lower. </li></ul><ul><ul><li>The 6% bond shows greater changes in price in response to a 2% change than the 8% bond. The first bond has greater interest rate risk. </li></ul></ul>
9. 9. Duration <ul><li>Duration </li></ul><ul><ul><li>Weighted average time to maturity using the relative present values of the cash flows as weights. </li></ul></ul><ul><ul><li>Combines the effects of differences in coupon rates and differences in maturity. </li></ul></ul><ul><ul><li>Based on elasticity of bond price with respect to interest rate. </li></ul></ul>
10. 10. Duration <ul><li>Duration </li></ul><ul><ul><li>D =  N t=1 [CF t • t/(1+R) t ]/  N t=1 [CF t /(1+R) t ] </li></ul></ul><ul><li>Where </li></ul><ul><li>D = duration </li></ul><ul><li>t = number of periods in the future </li></ul><ul><li>CF t = cash flow to be delivered in t periods </li></ul><ul><li>N= time-to-maturity </li></ul><ul><li>R = yield to maturity. </li></ul>
11. 11. Duration <ul><li>Since the price (P) of the bond must equal the present value of all its cash flows, we can state the duration formula another way: </li></ul><ul><li>D =  N t=1 [t  (Present Value of CF t /P)] </li></ul><ul><li>Notice that the weights correspond to the relative present values of the cash flows. </li></ul>
12. 12. Duration of Zero-coupon Bond <ul><li>For a zero coupon bond, duration equals maturity since 100% of its present value is generated by the payment of the face value, at maturity. </li></ul><ul><li>For all other bonds: </li></ul><ul><ul><li>duration < maturity </li></ul></ul>
13. 13. Computing duration <ul><li>Consider a 2-year, 8% coupon bond, with a face value of \$1,000 and yield-to-maturity of 12%. Coupons are paid semi-annually. </li></ul><ul><li>Therefore, each coupon payment is \$40 and the per period YTM is (1/2) × 12% = 6%. </li></ul><ul><li>Present value of each cash flow equals CF t ÷ (1+ 0.06) t where t is the period number. </li></ul>
14. 14. Duration of 2-year, 8% bond: Face value = \$1,000, YTM = 12%
15. 15. Special Case <ul><li>Maturity of a consol: M =  . </li></ul><ul><li>Duration of a consol: D = 1 + 1/R </li></ul>
16. 16. Duration Gap <ul><li>Suppose the bond in the previous example is the only loan asset (L) of an FI, funded by a 2-year certificate of deposit (D). </li></ul><ul><li>Maturity gap: M L - M D = 2 -2 = 0 </li></ul><ul><li>Duration Gap: D L - D D = 1.885 - 2.0 = -0.115 </li></ul><ul><ul><li>Deposit has greater interest rate sensitivity than the loan, so DGAP is negative. </li></ul></ul><ul><ul><li>FI exposed to rising interest rates. </li></ul></ul>
17. 17. Features of Duration <ul><li>Duration and maturity: </li></ul><ul><ul><li>D increases with M, but at a decreasing rate. </li></ul></ul><ul><li>Duration and yield-to-maturity: </li></ul><ul><ul><li>D decreases as yield increases. </li></ul></ul><ul><li>Duration and coupon interest: </li></ul><ul><ul><li>D decreases as coupon increases </li></ul></ul>
18. 18. Economic Interpretation <ul><li>Duration is a measure of interest rate sensitivity or elasticity of a liability or asset: </li></ul><ul><li>[ Δ P/P]  [ Δ R/(1+R)] = -D </li></ul><ul><li>Or equivalently, </li></ul><ul><li>Δ P/P = -D[ Δ R/(1+R)] = -MD × Δ R </li></ul><ul><li>where MD is modified duration. </li></ul>
19. 19. Economic Interpretation <ul><li>To estimate the change in price, we can rewrite this as: </li></ul><ul><li> Δ P = -D[ Δ R/(1+R)]P = -(MD) × ( Δ R) × (P) </li></ul><ul><li>Note the direct linear relationship between Δ P and -D. </li></ul>
20. 20. Semi-annual Coupon Payments <ul><li>With semi-annual coupon payments: </li></ul><ul><li>( Δ P/P)/( Δ R/R) = -D[ Δ R/(1+(R/2)] </li></ul>
21. 21. An example: <ul><li>Consider three loan plans, all of which have maturities of 2 years. The loan amount is \$1,000 and the current interest rate is 3%. </li></ul><ul><li>Loan #1, is a two-payment loan with two equal payments of \$522.61 each. </li></ul><ul><li>Loan #2 is structured as a 3% annual coupon bond. </li></ul><ul><li>Loan # 3 is a discount loan, which has a single payment of \$1,060.90. </li></ul>
22. 22. Duration as Index of Interest Rate Risk
23. 23. Immunizing the Balance Sheet of an FI <ul><li>Duration Gap: </li></ul><ul><ul><li>From the balance sheet, E=A-L. Therefore,  E=  A-  L. In the same manner used to determine the change in bond prices, we can find the change in value of equity using duration. </li></ul></ul><ul><ul><li> E = [-D A A + D L L]  R/(1+R) or </li></ul></ul><ul><ul><li> E  D A - D L k]A(  R/(1+R)) </li></ul></ul>
24. 24. Duration and Immunizing <ul><li>The formula shows 3 effects: </li></ul><ul><ul><li>Leverage adjusted D-Gap </li></ul></ul><ul><ul><li>The size of the FI </li></ul></ul><ul><ul><li>The size of the interest rate shock </li></ul></ul>
25. 25. An example: <ul><li>Suppose D A = 5 years, D L = 3 years and rates are expected to rise from 10% to 11%. (Rates change by 1%). Also, A = 100, L = 90 and E = 10. Find change in E. </li></ul><ul><li> D A - D L k]A[  R/(1+R)] </li></ul><ul><ul><li>= -[5 - 3(90/100)]100[.01/1.1] = - \$2.09. </li></ul></ul><ul><li>Methods of immunizing balance sheet. </li></ul><ul><ul><li>Adjust D A , D L or k. </li></ul></ul>
26. 26. Immunization and Regulatory Concerns <ul><li>Regulators set target ratios for an FI’s capital (net worth): </li></ul><ul><ul><li>Capital (Net worth) ratio = E/A </li></ul></ul><ul><li>If target is to set  (E/A) = 0: </li></ul><ul><ul><li>D A = D L </li></ul></ul><ul><li>But, to set  E = 0: </li></ul><ul><ul><li>D A = kD L </li></ul></ul>
27. 27. *Limitations of Duration <ul><ul><li>Immunizing the entire balance sheet need not be costly. Duration can be employed in combination with hedge positions to immunize. </li></ul></ul><ul><ul><li>Immunization is a dynamic process since duration depends on instantaneous R. </li></ul></ul><ul><ul><li>Large interest rate change effects not accurately captured. </li></ul></ul><ul><ul><ul><li>Convexity </li></ul></ul></ul><ul><ul><li>More complex if nonparallel shift in yield curve. </li></ul></ul>
28. 28. *Duration Measure: Other Issues <ul><li>Default risk </li></ul><ul><li>Floating-rate loans and bonds </li></ul><ul><li>Duration of demand deposits and passbook savings </li></ul><ul><li>Mortgage-backed securities and mortgages </li></ul><ul><ul><li>Duration relationship affected by call or prepayment provisions. </li></ul></ul>
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