Here's the full problem:
Xiang and Dmitry are friends. They agree that Xiang will pay Dmitry $\mathbb{$}800$ immediately and another $\mathbb{$}200$ at the end of three years. In return, Dmitry will pay Xiang $\mathbb{$}K$ in exactly one year and again at the end of exactly two years. Find $\mathbb{$}K$ if the transaction is governed by compound interest with a nominal discount rate of $6\%$ convertible monthly.
Alright so here was my attempt:
From what I've deduced from the above information is that we have an annual period of $T=3$. Then we have a time equation of value of $$0=800(1+i)^3-K(1+i)^2+(200-K)$$ $$\implies K=\frac{800(1+i)^3 +200}{1+(1+i)^2}$$ where $i$ denotes our annual interest. Denote nominal discount convertible annually as $d^{(m)}$. We are given $d^{(12)}=.06$. We wish to find the value of 1+i, so we use conversion formulas. Recalling that $$i^{(m)}=\frac{d^{(m)}}{1-(d^{(m)}/m)}$$, we obtain $i^{(12)}=.0603015075$. Further recalling that $$1+i=(1+ \frac{i^{(m)}}{m})^{m}$$ we obtain $1+i=1.061996367$. Plugging this back into our K equation, we obtain $K=544.31$.
The actual answer is $\mathbb{$}528.90$. So my questions are: Where did I go wrong? And how can I get to the correct answer?
At time $t = 3$ years, the accumulated value of Xiang's payments to Dimitry is $$800(1+i)^3 + 200.$$ At this same time, the accumulated value of Dimitry's payments to Xiang is $$K(1+i)^2 + K(1+i).$$ These must be equal if there is no outstanding balance for either party at the end of three years.