Question 5.2: A savings bank offers long-term savings certificates at 7 1/......
A savings bank offers long-term savings certificates at 7\frac{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.
From (5.2),
F/P = e^m (5.2)
F = P × [F/P, r%, n] = $1000e^{(0.075)(10)} = 42117.00
This problem can also be solved using Appendix C. Since a table is not available for a nominal interest rate of 7\frac{1}{2}% 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 \left[F/P, 7\frac{1}{2}\%, 10\right] = 2.0138 + \frac{7.5 – 7.0}{8.0 – 7.0} (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)^{10} = $2061.00
or $56 less than the amount that is obtained with continuous compounding.