A university alumni group wants to provide a semiannual scholarship to current students. If the scholarship amount of $400 is awarded semiannually beginning 6 months from now, and the fund earns interest at a rate of 4% per year compounded continuously, find the amount of the endowment the alumni are required to make now.
Question
A university alumni group wants to provide a semiannual scholarship to current students. If the scholarship amount of $400 is awarded semiannually beginning 6 months from now, and the fund earns interest at a rate of 4% per year compounded continuously, find the amount of the endowment the alumni are required to make now.
Solution
To solve this problem, we need to use the formula for the present value of a continuous income stream, which is:
PV = ∫ from t=0 to T (C * e^(-rt)) dt
where:
- PV is the present value (the amount of the endowment the alumni are required to make now)
- C is the cash flow per period (the scholarship amount of $400)
- r is the interest rate (4% per year, or 0.04)
- t is the time (in years)
- T is the total time period (in years)
Since the scholarship is awarded semiannually, we have two periods per year. Therefore, the cash flow per period is $400, and the total time period is 0.5 years (6 months).
Substituting these values into the formula, we get:
PV = ∫ from t=0 to 0.5 ($400 * e^(-0.04t)) dt
To solve this integral, we can use the rule ∫e^(at) dt = (1/a) * e^(at). Therefore, the integral becomes:
PV = [-$400/0.04 * e^(-0.04t)] from t=0 to 0.5
Evaluating this expression at t=0.5 and subtracting the value at t=0, we get:
PV = -400/0.04 * e^(-0.04*0))
Solving this equation, we find that the present value (the amount of the endowment the alumni are required to make now) is approximately $398.94.
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