Week 5 Tutorial Solutions - Solutions to week 5 Tutorial...

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Solutions to week 5 Tutorial Questions *************** Part I: Duration Models ************** 1. What are the two different general interpretations of the concept of duration, and what is the technical definition of this term? How does duration differ from maturity? Duration measures the weighted-average life of an asset or liability in economic terms. As such, duration has economic meaning as the interest sensitivity (or interest elasticity) of an asset’s value to changes in the interest rate. Duration differs from maturity as a measure of interest rate sensitivity because duration takes into account the time of arrival and the rate of reinvestment of all cash flows during the assets life. Technically, duration is the weighted-average time to maturity using the relative present values of the cash flows as the weights. 2. A one-year, $100,000 loan carries a coupon rate and a market interest rate of 12 percent. The loan requires payment of accrued interest and one-half of the principal at the end of six months. The remaining principal and accrued interest are due at the end of the year. a. What will be the cash flows at the end of six months and at the end of the year? CF 1/2 = ($100,000 x .12 x ½) + $50,000 = $56,000 interest and principal. CF 1 = ($50,000 x .12 x ½) + $50,000 = $53,000 interest and principal. b. What is the present value of each cash flow discounted at the market rate? What is the total present value? PV of CF 1/2 = $56,000 1.06 = $52,830.19 PV of CF 1 = $53,000 (1.06) 2 = 47,169.81 PV Total CF = $100,000.00 c. What proportion of the total present value of cash flows occurs at the end of 6 months? What proportion occurs at the end of the year? X 1/2 = $52,830.19 $100,000 = .5283 = 52.83% X 1 = $47,169.81 $100,000 = .4717 = 47.17% d. What is the duration of this loan? Duration = .5283(1/2) + .4717(1) = .7358 OR t CF PVof CF PV of CF x t ½ $56,000 $52,830.19 $26,415.09 1 53,000 47,169.81 47,169.81 $100,000.00 $73,584.91
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Duration = $73,584.91/$100,000.00 = 0.7358 years 3. A six-year, $10,000 CD pays 6 percent interest annually and has a 6 percent yield to maturity. What is the duration of the CD? What would be the duration if interest were paid semiannually? What is the relationship of duration to the relative frequency of interest payments? [Hint: In order to save time, set up a spread sheet to automate some computations.] Six-year CD: Par value = $10,000 Coupon rate = 6% R = 6% Maturity = 6 years Annual payments t CF PV of CF PV of CF x t 1 $600 $566.04 $566.04 2 $600 $534.00 $1,068.00 3 $600 $503.77 $1,511.31 4 $600 $475.26 $1,901.02 5 $600 $448.35 $2,241.77 6 $10,600 $7.472.58 $44,835.49 $10,000.00 $52,123.64 Duration = $52,123.64/$1,000.00 = 5.2124 R = 6% Maturity = 6 years Semiannual payments t CF PV of CF PV of CF x t 0.5 $300 $291.26 $145.63 1 $300 $282.78 $282.78 1.5 $300 $274.54 $411.81 2 $300 $266.55 $533.09 2.5 $300 $258.78 $646.96 3 $300 $251.25 $753.74 3.5 $300 $243.93 $853.75 4 $300 $236.82 $947.29 4.5 $300 $229.93 $1,034.66 5 $300 $223.23 $1,116.14 5.5 $300 $216.73 $1,192.00 6 $10,300 $7,224.21 $43,345.28 $10,000.00 $51,263.12 Duration = $51,263.12/$10,000.00 = 5.1263 Duration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being received more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows. 4. You can obtain a loan of $100,000 at a rate of 10 percent for two years. You have a choice of i) paying the interest (10 percent) each year and the total principal at the end of the second year or ii) amortizing the loan, that is, paying interest (10 percent) and principal in equal payments each year. The loan is priced at par. Which payment option is associated with a smaller duration? Why?
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