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PS1Solutions

# PS1Solutions - Finance 100 Problem Set 1 Alternative...

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Finance 100: Problem Set 1 Alternative Solutions Note: Where appropriate, the “final answer” for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas This section contains the formulas you will need for this problem set: 1. Present Value (PV) Formula (a.k.a. Zero Coupon Bond Formula): V 0 = V N (1 + R/m ) mT (1) where V N is the dollar amount to be received N periods in the future. m is the number of compounding periods per annum, T is the number of years until the money is received and R is the nominal interest rate (a.k.a. APR). Note that this formula is equivalent to: V 0 = V N (1 + i ) N since i = R/m and N = mT . I will use this notation interchangeably throughout. 2. Future Value (FV) Formula: V N = V 0 (1 + i ) N (2) where V 0 is the dollar amount received today. Note that the future value formula is just an algebraic manipulation of the present value formula. 1

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3. Present Value of an Annuity Formula: A 0 = a (1 + i ) + a (1 + i ) 2 + ... + a (1 + i ) N - 1 + a (1 + i ) N = a · 1 - (1 + i ) - N i (3) where a is the amount of the annuity payment and i and N are as defined above. 4. Spot and Forward Rate Relation: (1 + r n + t ) n + t = (1 + r n ) n (1 + f n,t ) t (4) where r n + t is the ( n + t )-period spot rate, r n is the n -period spot rate and f n,t is the t -period forward rate from period n to ( n + t ). 5. DV01 (Dollar Value of One Basis Point): DV 01 = B ( R - 0 . 01%) - B ( R ) (5) where B ( R ) is the bond price at interest rate R . This is method one from the lectures. See the lecture slides for the other two methods of computing DV01. 6. Macaulay Duration: D = (1 + i ) mB 0 · ∂B 0 ∂i = (1 + i ) mB 0 · N X n =1 T n c n (1 + i ) - n (6) where B 0 is the price of the bond and T n is the time in years to the n th cash flow, c n . An alternative representation is: 1 B 0 · [ T 1 · PV ( c 1 ) + T 2 · PV ( c 2 ) + ... + T N · PC ( c N )] (7) where PV ( c n ) is the present value of the n th cash flow. Modified Du- ration adjusts this figure by dividing by (1 + i ). II. Problems 1. The time line of our cash outlays for the college expense are presented in Figure 1. 2
Figure 1: 0 1 2 \$20,000 \$20,000 0 Cash Flow Time Period 1.a The present value of these cash flows, and thus the amount we must invest today, is: \$20 , 000 1 . 04 2 + \$20 , 000 1 . 04 4 = \$ 35,587.21 The discount rate follows from the 8% yield-to-maturity ( R = 0 . 08) and semi-annual compounding ( m = 2), implying i = 4%. 1.b Scenario 1: Unchanged Interest Rates. If we spend \$35,587.21 on 4-year zeros today, this means we own bonds with a total face value (or par amount) of: \$35 , 587 . 21 × (1 . 04) 8 = \$48 , 703 . 55 At the end of year 1, we need \$20,000 for the first payment. Thus we need to sell some fraction of our bond holdings; that fraction is determined by the market in the following manner: par (1 . 04) 6 = \$20 , 000 (8) We are just solving the zero formula in reverse to determine the par value we must sell in the market to receive \$20,000 in year 1. The solution is par = \$25 , 306 . 38. After the sale, we are left with bonds with a total face value of \$48 , 703 . 55 - \$25 , 306 . 38 = \$23 , 397 . 17 At the end of year 2, we need to make another pay- ment of \$20,000. Solving a similar equation to (8): par (1 . 04) 4 = \$20 , 000 (9) 3

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yields a par value of \$23,397.17, exactly equal to the par amount of our remaining bonds. So we simply sell all of the remaining bonds, collect the \$20,000 and make the payment.
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PS1Solutions - Finance 100 Problem Set 1 Alternative...

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