I am presented with an investment opportunity where I am given #481,000 on day 1. Thereafter, every 10 days, I am required to give back #50,000 every for 100 days (10 * 50000 = 500000).
How do I calculate the interest rate I am paying?
I am guessing I have to use the present value of annuity problem to find out the interest rate.
Am I wrong in my reasoning below?
Solving for i(interest rate) using the present value formula in wolfram alpha, I get 0.7107%. Since the period is 10 days, I divide by 10 and multiply by 365 to get 25.94%.
I know that it has to be greater than 14% but I am a little surprised that it is 26%.
Thanks guys.
Let the interest rate for using some money for ten days be $p$. Then after the $i^{th}$ back payment due every tenth day you still owe
$i=1\ \ $: $481,000\cdot(1+p)-50,000$
$i=2\ \ $: $481,000\cdot(1+p)^2-50,000\cdot(1+p)-50 000$
$i=3\ \ $: $481,000\cdot(1+p)^3-50,000\cdot(1+p)^2-50 000\cdot(1+p)-50 000$
$\vdots$
$i=10\ $:$481,000\cdot(1+p)^{10}-50,000\cdot\sum_{j=0}^9(1+p)^j=0$
since you have paid back by the $10^{th}$ payment all the capital and the accumulated interest.
Now, let
$$Q=\sum_{j=0}^9(1+p)^j.$$
So, $$(1+p)Q=\sum_{j=0}^9(1+p)^{j+1}=\sum_{j=1}^{10}(1+p)^{j}=\sum_{j=1}^{9}(1+p)^{j}+(1+p)^{10}.$$
With this
$$(1+p)Q-Q=pQ=(1+p)^{10}-1,$$ hence $$Q=\frac{(1+p)^{10}-1}p$$
That is, we have to solve the following equation for $p$.
$$481,000\cdot(1+p)^{10}-50,000\cdot\frac{(1+p)^{10}-1}p=0$$
or
$$\frac p{1-\frac1{\ (1+p)^{10}}}=\frac{50}{481}$$
then
$$p\approx 0.00711\%.$$ So, we pay an interest rate of $0.711\%$ for ten days. The yearly counterpart is $$\sim 26\%.$$