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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?

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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%.

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  • $\begingroup$ You have no idea how many problems I was doing wrong because of this misunderstanding and it made my life far easier. Thank you so much. $\endgroup$
    – hyg17
    Oct 13, 2014 at 3:48

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