Question

Robbie deposits 1000 into a bank account. The bank credits interest at a nominal annual rate of j convertible semiannually for the first 7 years and a nominal annual rate of 2j convertible quarterly for all years thereafter. The accumulated amount in the account at the end of 10.5 years is 1980. Calculate the accumulated amount in the account at the end of 5 years

Answer 1

1276.30

Explanation:

The future value formula is ...

FV = P(1 +r/n)^(nt)

where FV is the future value, P is the principal invested at annual rate r compounded n times per year for t years.

The value at the end of the first 7 years is ...

FV₇ = 1000(1 +j/2)^(2·7)

At the end of the next 3 1/2 years, the future value is ...

FV₁₀ = (FV₇)(1 + 2j/4)^(4·3.5)

= 1000 × (1 +j/2)^(14) × (1 + j/2)^(14)

FV₁₀ = 1000(1 +j/2)^28 = 1980 . . . . given by the problem statement

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This tells us the expression 1+j/2 is the 28th root of FV₁₀/1000 = 1.98^(1/28).

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We are interested in the value after 5 years, so we want the result after 10 compounding periods, not 28:

FV₅ = 1000(1 +j/2)^(2·5) = 1000·(1.98)^(10/28)

FV₅ = 1000·1.980^(5/14) ≈ 1276.30