Tawny makes a deposit into a bank account which credits interest at a nominal interest rate of $10\%$ per annum, convertible semiannually. At the same time, Fabio deposits $1000$ into a different bank account, whic is credited with simple interest. At the end of $5$ years, the forces of interest on the two account are equal, and Fabio's account has accumulated to $Z$. Determine $Z$.
The following is the way I tried.
Tawny has compound interest, semiannual, $10\%$ per annum. So her force of interest is
$$\delta_T = \ln {1.1}$$
Fabio's simple interest has a force dependent on time $t$, so
$$\delta_F = \frac{d}{dt}(1+it) = \frac{i}{1+it}$$
At the end of $5$ years, the forces are equal to each other, so
$$\ln{1.1}=\frac{i}{1+5i}$$
implies
$$i\approx 0.1821$$
So I get the value of $Z$ as
$$Z = 1000(1+5i) \approx 1910.40$$
However, the answer is supposedly $1953$. Can someone explain to me how they got that or what I am doing wrong?
Best Answer
A nominal rate of 10% per annum, compound semiannually, means a rate of 5% per 6 month period. Therefore, the annual effective rate is $1.05^2 - 1 = 10.25\%$, not 10%.