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PS6Solutions - Finance 100 Problem Set 6 Alternative...

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Finance 100: Problem Set 6 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 homework 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 where 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. Future Value of an Annuity Formula: A N = a · (1 + i ) N - 1 i (4) where a is the amount of the annuity payment and i and N are as defined above. 5. Index Forward Price Formula: F = S 0 e ( r f - d ) T (5) where S 0 is the price of the underlying security at time 0, r f is the risk-free rate, d is the dividend yield and T is the time to maturity. 6. Interest Rate Futures Price: F = 100 - InterestRate (6) where F is the futures price. The interest rate is typically LIBOR. II. Problems 1. 1.a The farmer is going to be selling the wheat in the future so he is concerned that the future spot price will be low, which would result in low revenues from the sale of his wheat. He can hedge this risk by selling futures on wheat. 1 1 An easy way to remember whether one should buy or sell futures contracts to hedge price risk is to simply consider what action will be taken in the future. Since the farmer needs to sell his wheat in the future, he should sell futures contracts. 2
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This short position in the futures contract obligates him to sell the wheat in the future at the delivery price (i.e. the future price at the inception date of the contract). Assuming his expectation is correct, the farmer is going to harvest 60,000 bushels in September. Since each contract is for 5,000 bushels, she will need to sell 60,000/5,000 = 12 contracts . And, since the harvest will take place in September, she should sell September contracts. 1.b Table (1) below considers three scenarios for the future spot price of wheat. The gain from the futures position, assuming a cash settlement, is computed using the formula: ( F - S T ) × Quantity . The quantity in parentheses is the payoff to the seller of a futures contract. We multiply by the quantity since the payoff is per unit, which in this case is a bushel. For example, the $9,600 gain from the futures position when the spot price is $3.25 is computed as ($3 . 41 - $3 . 25) × 60 , 000. The revenue from the sale at the spot price is simply the total number of bushels times the price per bushel in the market, or 60 , 000 × $3 . 25 = $195 , 000.
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