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# Supply chain

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Supply chain

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### Supply chain

1. 1. FINA 4310 Lecture Notes 11b: Bond Management
2. 2. Risk Management Using Duration <ul><li>A bond portfolio manager can control interest rate risk by changing the duration of the portfolio </li></ul><ul><ul><li>To reduce interest rate risk, decrease duration by selling long-term bonds and buying short-term bonds (or just keeping more in cash) </li></ul></ul>
3. 3. Example <ul><li>You currently have \$1,000,000 </li></ul><ul><ul><li>\$750,000 invested in a 20-year bond with a duration of 16 years, </li></ul></ul><ul><ul><li>\$250,000 in cash (demand deposits at a bank) . </li></ul></ul><ul><li>The cash component has a duration of zero </li></ul><ul><li>Portfolio weights are .75 and .25 </li></ul><ul><li>Duration of portfolio is </li></ul><ul><li> (.75)(16) + (.25)(0)=12 </li></ul>
4. 4. Example, continued <ul><li>Now, you decide that you are bearing too much interest rate risk, and would like to decrease your duration to 8 years. </li></ul><ul><li>How much bond would you have to sell? </li></ul><ul><li>Solve this equation to find target weight: </li></ul><ul><li>(WB)(16) + (1-WB)(0) = 8 Solution: WB = .50 </li></ul><ul><li>Want to have (.5)(1,000,000) = \$500,000 in bonds </li></ul><ul><li>To get there, you need to sell \$250,000 of bonds </li></ul>
5. 5. Duration of a perpetuity <ul><li>Consider a perpetuity that pays \$1,000 per year forever, and its yield is .08 </li></ul><ul><li>The price of this perpetuity would be: \$1,000/.08 = \$12,500 </li></ul><ul><li>The duration of the perpetuity would be: 1.08/.08 = 13.5 </li></ul>
6. 6. Sample Question <ul><li>You have the following portfolio: </li></ul><ul><ul><li>\$1,000,000 in cash </li></ul></ul><ul><ul><li>\$2,000,000 market value in 28-year treasury bonds with duration 25 years </li></ul></ul><ul><ul><li>Zero-coupon bonds expiring in 5 years with face value \$5,000,000 and current yield 4% </li></ul></ul><ul><ul><li>A perpetuity that pays \$100,000 per year, current yield 5% </li></ul></ul>
7. 7. Sample Question, continued <ul><li>Compute the portfolio value </li></ul><ul><li>Compute the portfolio duration </li></ul><ul><li>Your target duration is ten years. You would like to achieve your target by buying or selling some of the long bonds. How much would you have to buy or sell? </li></ul>
8. 8. Find portfolio value... <ul><li>Portfolio value: </li></ul><ul><li>Cash: \$1,000,000 </li></ul><ul><li>Long Bonds: \$2,000,000 </li></ul><ul><li>5-year Zero: \$4,109,635 [\$5M/(1.04) 5 ] </li></ul><ul><li>Perpetuity: \$2,000,000 [100K/.05] </li></ul><ul><li>Total: \$9,109,635 </li></ul>
9. 9. Find portfolio weights... <ul><li>W1 = 1,000,000/ 9,109,635 = .1098 </li></ul><ul><li>W2 = 2,000,000/9,109,635 = .2195 </li></ul><ul><li>W3 = 4,109,635/ 9,109,635 = .4511 </li></ul><ul><li>W4 = 2,000,000/9,109,635 = .2195 </li></ul>
10. 10. Find Duration <ul><li>Cash: duration = 0 </li></ul><ul><li>Long bond: duration=25 </li></ul><ul><li>Zero: duration = 5 </li></ul><ul><li>Perpetuity: duration = 21 (1.05/.05) </li></ul><ul><li>Portfolio duration: </li></ul><ul><li>.1098(0) + .2195 (25) + .4511(5) + .2195(21) = 12.3525 </li></ul>
11. 11. Target Duration <ul><li>You now want to get your duration to 10. Your current duration is too high. You can lower it by selling long term bonds. </li></ul><ul><li>If you buy \$X worth of long-term bonds for cash, you increase your portfolio duration by: </li></ul><ul><li>[X/ 9,109,635] (25) </li></ul><ul><li>Want to lower duration by 2.3525 </li></ul><ul><li>(from 12.3525 to 10) </li></ul>
12. 12. Algebra <ul><li>-2.3525 = [X/ 9,109,635] (25) </li></ul><ul><li>X = -2.3525 (9,109,635)/25 </li></ul><ul><li>X = -857,217 < Sell this much in bonds </li></ul><ul><li>(You are left with \$1,142,783 in the bonds) </li></ul>
13. 13. Verify Answer <ul><li>New portfolio weights: </li></ul><ul><li>W1 = 1,857,217/ 9,109,635 = .2038 </li></ul><ul><ul><li>W2 = 1,142,783/9,109,635 = .1254 </li></ul></ul><ul><ul><li>W3 = 4,109,635/ 9,109,635 = .4511 </li></ul></ul><ul><ul><li>W4 = 2,000,000/9,109,635 = .2195 </li></ul></ul><ul><li>New duration: </li></ul><ul><li>.1254 (25) + .4511(5) + .2195(21) = 10 </li></ul>
14. 14. Market Timing <ul><li>A bond portfolio manager can attempt to time the market by changing the duration of the portfolio </li></ul><ul><li>Increase duration when you think interest rates are about to fall </li></ul><ul><li>Decrease duration when you think interest rates are about to rise </li></ul>