A savings bank offers long-term savings certificates at 7 1/2% per year, compounded continuously. If a 10-year certificate costs $1000, what will be its value at maturity? Compare with the value that would be obtained if the interest were compounded annually rather than continuously.

Question

A savings bank offers long-term savings certificates at [latex]7\frac{1}{2}[/latex]% per year, compounded continuously. If a 10-year certificate costs $1000, what will be its value at maturity? Compare with the value that would be obtained if the interest were compounded annually rather than continuously.

Accepted Answer

From (5.2),

[latex]F/P = e^m[/latex] (5.2)

F = P × [F/P, r%, n] = $1000e[latex]^{(0.075)(10)}[/latex] = 42117.00

This problem can also be solved using Appendix C. Since a table is not available for a nominal interest rate of 7[latex]\frac{1}{2}[/latex]% per year, however, it will be necessary to interpolate between the 7% and 8% values.

[F/P, 7%, 10] = 2.0138 [F/P, 8%, 10] = 2.2255

and [latex]\left[F/P, 7\frac{1}{2}\%, 10\right] = 2.0138 + \frac{7.5 - 7.0}{8.0 - 7.0}[/latex] (2.2255 - 2.0138) = 2.1197

The future worth of the savings certificate can now be obtained as

F ≈ $1000(2.1197) = $2119.70

If the interest were compounded annually rather than continuously, the future worth would be

F = $1000(1+ 0.075)[latex]^{10}[/latex] = $2061.00

or $56 less than the amount that is obtained with continuous compounding.

Question 5.2: A savings bank offers long-term savings certificates at 7 1/......

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