Question

Help Please!! I can't find the answer

Derek established his own retirement account 10 years ago. He has discovered that he can obtain a better rate for the next 10 years at 12 percent interest compounded semiannually. Consequently, Derek established a new ordinary annuity account (beginning amount $0.00) and he will contribute $7,000.00 semiannually into the account for the next 10 years. What will be the value of this account at the end of the 10-year period?

A)$83,652.59
B)$257,502.00
C)$244,707.61
D)$264,501.86

Answer 1

We have:

No. of payments n = 10x 4 = 40

Periodic payment pmt = $7000

Period rate r= 12%/4 = 3%

We can use the following formula to calculate the Future value:

`FV=pmt ((1+r)^(n)-1)/(r)`

plugging the values in the formula, we get:

`FV=7000 ((1+0.03)^(40)-1)/(0.03) = 7000 * 75.40126 =$527808.82`

Answer 2

Answer : The calculated answer of FVA is $ 2,57,499.14. This is closest to $257,502.00 - option B.

We follow these steps to arrive at the answer:

We use the Future Value of Annuity formula to arrive at the answer to this question, as the new account begins at $0.00 and there is no mention of the amount in the previous account.

The formula for Future Value of annuity is:

`FVA =P * (\left (1+r\right )^n - 1)/(r)`

where

P = constant periodic contribution

r = rate per period

n = number of periods.

In the question above, P = $7,000.

The given interest rate is 12% p.a. Since the contributions are made semi-annually (twice a year), we need to find the rate per period with the following formula:

`\ r = (Interest rate per year)/(No. of compounding periods per yr)`

So, we get

r =
`\ r = (0.12)/(2)`

r = 0.06

Since there are two compounding periods per year, we get number of compounding periods 'n', by

`\ n = No. of years * No. of compounding periods`

So,
`n = 10 * 2`

n = 20

Substituting the values of P, n and r in the FVA equation we get,

`FVA =3,500 * (1.06^(20)- 1)/(0.06)`

`FVA =3,500 * (2.207135472)/(0.06)`

`FVA =3,500 * 36.7855912`

`FVA = $ 2,57,499.14`