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Sylvia invests her money in an account earning interest based on simple discount at a $2$$\%$ annual rate. What is her effective interest rate in the fifth year?

The formula for effective interest rate is $r = (1+\frac in)^n -1$

The way I would do this is find $i$ which is $0.0204$

Then I would plug everything in $r = (1+ \frac {0.0204}{5})^5-1 $ and my answer is $2.056 \%$

However, when looking up solutions, this is what they did

They took $i$ and multiplied it by $5$ which equals to $0.1020$

Then their effective interest rate formula would look like this $r = (1 + 0.1020)^{1/5}-1$ and they get $.0196 \%$

What am I doing wrong?

Allan
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  • how did you find i ? –  Sep 09 '17 at 00:23
  • $d = \frac {i}{i+1}$ $0.2 = \frac {i}{i+1}$ – Allan Sep 09 '17 at 00:28
  • my thought was maybe you were thinking APR versus APY. –  Sep 09 '17 at 00:41
  • In case of simple interest the future value ($n=5$) is $C_0\cdot (1+i\cdot n)=C_0\cdot (1+i\cdot 5)$. $C_0$ denotes the present value. Do you agree ? Then the effective interest rate is $i_{eff}=\left(\frac{C_0\cdot (1+i\cdot 5)}{C_0} \right)^{1/5}-1=\left( 1+i\cdot 5 \right)^{1/5}-1$. Doesn´t it sounds logic. – callculus42 Sep 09 '17 at 00:56
  • $d$ is the rate of discount. See here for an example and explanation. https://math.stackexchange.com/a/1346195/144421 – callculus42 Sep 09 '17 at 01:06
  • @callculus The problem states simple discount! You cannot use compound interest formula... – alexjo Sep 09 '17 at 15:24

1 Answers1

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The annual effective rate of interest for year $t$, which we denote by $i(t)$, is the ratio of the amount of interest earned in a year, from time $t−1$ to time $t$, to the accumulated amount at the beginning of the year (i.e., at time $t−1$): $$ i(t)=\frac{a(t)-a(t-1)}{a(t-1)} $$ For the simple-discount method, we have $a(t)=\frac{1}{1-dt}$ where $d=2\%$ is the simple discount rate. Observing that $a(5)=\frac{1}{1-0.02\times 5}=\frac{1}{0.9}$ and $a(4)=\frac{1}{1-0.02\times 4}=\frac{1}{0.92}$ $$ i(5)=\frac{a(5)-a(4)}{a(4)}\approx 2.22222\% $$

NOTE: this exercise comes from the book Mathematical Interest Theory, 2nd ed. (Leslie Vaaler,James Daniel): ex. Sec 1.8 n. (2) at pag 65, sol. at pag. 441

alexjo
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