Answer to Question #305681 in Financial Math for Moro

Question

The continuous compounding rate is 13,974% per year. The equivalent nominal rate, compounded quarterly, is

Answer 1

Solution

Let us consider, the equivalent nominal rate is "$x$", then the future value is

$A = P{\left( {1 + \frac{x}{n}} \right)^{nt}}\$

But for the continuous compounding, we have

$A = P \cdot {e^{rt}}$

Therefore,

$P \cdot {e^{rt}} = P{\left( {1 + \frac{r}{n}} \right)^{nt}}\\$

${\left( {{e^r}} \right)^t} = {\left( {1 + \frac{r}{n}} \right)^{nt}}\\$

${e^r} = {\left( {1 + \frac{r}{n}} \right)^n}\\$

${e^{0.13974}} = {\left( {1 + \frac{x}{4}} \right)^4}\\$

$1 + \frac{x}{4} = {\left( {{e^{0.13974}}} \right)^{{\textstyle{1 \over 4}}}}\\$

$\frac{x}{4} = {\left( {{e^{0.13974}}} \right)^{{\textstyle{1 \over 4}}}} - 1\\$

$x = 4 \cdot \left[ {{{\left( {{e^{0.13974}}} \right)}^{{\textstyle{1 \over 4}}}} - 1} \right]\\$

$x = 0.1422095828$

Hence the nominal rate is $14.221 \%$