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Unformatted text preview: Math 105 Word Problems 1 Present Value Recall the formula for continuously compounded interest : if P dollars are invested today at an interest rate r , compounded continuously, then the value A after t years is A = P e rt . Another way of looking at this is that P is the present value of the amount A to be received after t years: P = Ae- rt . A related formula is needed when instead of a single amount invested once, one considers a continuous stream of income . This is usually represented by a function K ( t ), giving the annual rate of income at time t in dollars per year. Then the present value of the continuous stream of income K ( t ) over the period from T 1 to T 2 , with an interest rate r , is: [present value] = T 2 T 1 K ( t ) e- rt dt. In other words, the present value is the price that you should be willing to pay at time t = 0 to have the income stream K ( t ) from time T 1 to T 2 in the future. 1. Printing press Consider a small printing company that does most of its work on one printing press. The firms profits are directly inuenced by the amount of material that the press can produce (assuming that other factors, such as wages, are held constant). We may say that the press is producing a continuous stream of income for the company. Of course, the eciency of the press may decline as it gets older. At time t , let K ( t ) be the annual rate of income from the press. (This means that the press is producing K ( t ) 1 365 dollars per day at time t .) Find a model for the present value of the income generated by the printing press over the next T years, assuming an interest rate r (with interest compounded con- tinuously). 2. A company estimates that the rate of revenue produced by a machine at time t will be 5000- 100 t dollars per year. Find the present value of this continuous stream of income over the next 4 years at a 16% interest rate. 3. Gas reservoir Suppose that a large farm with a known reservoir of gas beneath the ground sells the gas rights to a company for a guaranteed payment at the rate of 10 , 000e . 04 t dollars per year. Find the present value of this perpetual stream of income, assuming an interest rate of 12%, compounded continuously. 4. Find the present value of a stream of earnings generated over the next 2 years at the rate of 50+7 t thousand dollars per year at time t , assuming a 10% interest rate. 5. Find the present value of a continuous stream of income over the next 4 years, where the rate of income is 50e- . 08 t thousand dollars per year at time t , and the interest rate is 12%. 1 Capital Value and Improper Integrals The capital value of an asset such as a machine is sometimes defined as the present value of all future net earnings. The actual lifetime of the asset may not be known, and since some assets may last indefinitely, the capital value of the asset may be written in the...
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