Outstanding part-way balance for loan compounded quarterly A loan of $R65000$ with an interest of $16\%$ per annum, compounded quarterly, is to be amortized by equal quarterly payments over $3$ years.
Question: What is the outstanding amount on the loan at the start of the third quarter?
My attempt:
$$
S=P(1+i)^n\\
65000\left(1+\frac{0.16}{4}\right)^{36}
$$
 A: A little background. When dealing with these kinds of income streams, it is important to know if the payments begin immediately or after the first interval. With loans, what usually happens is that the principal is borrowed and the first payment is due at the start of the next period. This is called an annuity immediate.
The first thing to note that if the loan is compounded quarterly, we treat it as a simple annual cash flow at the period of the compounding. So a $16\%$ per annum compounded quarterly is treated as a simple $4\%$ per quarter loan. 
Secondly, the question asks for the outstanding amount of the loan, not what the value would be without payments, so the first thing we need to do is to calculate the payment.
It always pays to draw a timeline for these questions, but I'm not sure how to do that here, so forgive its lack. What are the cashflows in this problem? We have an initial loan principle $P = R65,000$ and a quarterly interest rate $i = 4\% = 0.04$. We need to calculate $p$, the payment. What happens? We borrow $P$ at $t_0$. At $t_1$, the end of the first or beginning of the second quarter, the loan gets increased by the interest, and then the payment is removed. Mathematically, this is:
$$
P(1.04) - p
$$
Next quarter, $t_2$, the remaining balance gets the interest charge and the payment is then subtracted. This is:
$$
\left(P(1.04) - p\right)(1.04) - p\\
=P(1.04)^2 - p(1.04) - p
$$
At $t_3$, the end of the third/start of the fourth quarter, we have:
$$
\left(\left(P(1.04) - p\right)(1.04) - p\right)(1.04)-p\\
=P(1.04)^3 - p(1.04)^2 - p(1.04) - p
$$
The pattern continues. In general, for the cash flows to balance we need:
$$
P(1+i)^n = p\sum_{k=0}^{n-1}(1+i)\\
P(1+i)^n = p\frac{1-(1+i)^n}{1-(1+i)}\\
p = \frac{-i\cdot P(1+i)^n}{1-(1+i)^n}
$$
Solving this equation in your case gives us $p \approx R5,578.30$. 
Now the question asked what will the outstanding balance be at the beginning of the third quarter. As we are working under the assumption that all cashflows happen at the end of the quarter, this is the balance at the end of the second quarter. We can see from the above that the value of the outstanding balance at $t_k$ would be:
the value of the loan at $t_k$ would be:
$$
P(1+i)^k - p\sum_{t=0}^{k-1}(1+i)\\
$$
Plugging the values we have into the general formula where $k=2$ we get a balance of $R58,924.27$.
