Answer to Question #305681 in Financial Math for Moro

Question #305681

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


1
Expert's answer
2022-03-08T00:03:02-0500

Solution


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


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


But for the continuous compounding, we have


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


Therefore,

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


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


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


e0.13974=(1+x4)4{e^{0.13974}} = {\left( {1 + \frac{x}{4}} \right)^4}\\


1+x4=(e0.13974)141 + \frac{x}{4} = {\left( {{e^{0.13974}}} \right)^{{\textstyle{1 \over 4}}}}\\


x4=(e0.13974)141\frac{x}{4} = {\left( {{e^{0.13974}}} \right)^{{\textstyle{1 \over 4}}}} - 1\\


x=4[(e0.13974)141]x = 4 \cdot \left[ {{{\left( {{e^{0.13974}}} \right)}^{{\textstyle{1 \over 4}}}} - 1} \right]\\


x=0.1422095828x = 0.1422095828


Hence the nominal rate is 14.221%14.221 \%


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