Question

Suppose $40,000 was invested on January 1, 1980 at an annual effective interest rate of 7% in order to provide an annual (calendar-year) scholarship of $5,000 each year forever, the scholarships paid out each January 1.

(a) In what year can the first $5,000 scholarship be made?
(b) What smaller scholarship can be awarded the year prior to the first $5,000 scholarship?

Answer 1

The first $5,000 scholarship from a $40,000 investment at a 7% interest rate can be awarded on January 1, 1981. The smaller scholarship that can be awarded the year prior is $2,800.

Step-by-step explanation:

This question involves the concepts of perpetuity and compound interest. A perpetuity is a type of annuity that receives an infinite series of payments. In this case, the student wants to fund a perpetual scholarship of $5,000 annually at a 7% effective annual interest rate.

To determine when the first $5,000 scholarship can be made, we use the perpetuity formula:

Perpetuity Payment = (Investment Amount) x (Interest Rate)

So, $5,000 = $40,000 x 0.07, and it follows that since the amount invested and the interest rate are appropriate for the perpetuity payment desired, the first scholarship payment of $5,000 can be made one year after the investment, on January 1, 1981.

Before the first full scholarship payment can be made, the fund will have accumulated interest for one year. Therefore, the equation to use here is:

Accumulated Value = (Investment Amount) x (1 + Interest Rate)

For the smaller scholarship, we would subtract the $5,000 that would be preserved for perpetuity payments moving forward:

Smaller Scholarship = ($40,000 x 1.07) - $40,000

Calculating gives $2,800 as the smaller scholarship available the year prior to the first $5,000 scholarship.

Answer 2

a). The first $5,000 will be paid on January 1, 1982

b). The smaller scholarship that will be received=$2,800

Step-by-step explanation:

a). This can be expressed as follows;

I=PRT

where;

I=Interest amount

P=Initial value of investment

R=Annual interest rate

T=number of years the money is invested for

In our case;

I= $5,000

P=$40,000

R=7%=7/100=0.07

T=t

Replacing;

5,000=40,000×0.07×t

5,000=2,800 t

t=5,000/2,800

t=1.786 years which will be 2 years since although the first amount will be in 1.7 years, the interest has to be paid on January 1 of the next year

1980+2=1982

The first $5,000 will be paid on January 1, 1982

b). What to receive after 1 year

I=PRT

where;

P=$40,000

R=7%=7/100=0.07

T=1

I=(40,000×0.07×1)=2,800

The smaller scholarship that will be received=$2,800