Here's the full problem:
You have two options to repay a loan. You can repay $\mathbb{$}6000$ now and $\mathbb{$}5940$ in one year, or you can repay $\mathbb{$}12000$ in $6$ months. Find the annual effective interest rate(s) i at which both options have the same present value.
In the book I am using, they define the following:
Discount function $\to v(t)=\frac{1}{a(t)}$ where a(t) represents accumulation function
Discount factor $\to v=\frac{1}{1+i} \implies a(t)=(1+i)^t=v^{-t}$ (recalling compound interest) where $i$ is interest
Present value $\to PV_{a(t)}(\mathbb{$}L \mathbb{\;at\;} t_0)=\mathbb{$}Lv(t_0)$ where $\mathbb{$}L$ is the loan
My attempt at work is as follows, but I'm under the impression that it is not correct:
Equating the two payment options using the present value equation, we obtain
$$(6000+5940)\frac{1}{1+i}=12,000\frac{1}{\sqrt{1+i}}$$
$$\implies \frac{11940}{12,000}=\sqrt{1+i}$$
$$\implies (0.995)^2-1=i$$
$$\implies =i-.0009975$$
Was I thinking correctly at the approach to this problem or was I completely off? I'm fully confident that I should not have a negative interest rate. Or for that matter, an interest rate that small. If anyone could give me aid, I'd be very much grateful!
Best Answer
Outline: The PV in the first option is $6000+\frac{5940}{1+i}$. After all, the PV of the first payment of $6000$ is exactly $6000$, since it is being paid now.
The equation you end up with, after multiplying through by $1+i$, is a quadratic in $\sqrt{1+i}$. Let $x=\sqrt{1+i}$ and solve.