About annuity immediate calculation

Question

Q1: Find $s_{12}$ if the nominal interest rate payable monthly is $5%$ per annum.

What I have done is: $$i^{(12)}=0.05$$ $$1+i=(1+i^{(12)}/12)^{12}$$ which leads to $$i=0.0512$$ $$s_{12}=((1+i)^{12}-1)/i=16.03$$

I wonder whether I am right to find $i$. Or should I use $i[12]$ which equals to $i^{(12)}/12$ instead.

Also,

Q2: A loan of £1000 is to be repaid by a level annuity, payable monthly in arrears for two years and calculated on the basis of an interest rate of $9%$ per annum. Calculate the monthly repayment.

In this question which $i$ should I use? $i$ or $i[12]$? I am quite confused.

Answer 1

I'm not exactly sure what your notation means so I'll just answer the question (Q2) and you can hopefully align the notations. Firstly we are asked for two years, this is $24$ months, the annual interest rate is $R=0.09$ which means that the monthly interest is $r=(1+R)^{(1/12)}-1 \approx 0.0072$. Now we will work with entirely monthly quantities.

Let $B(n)$ be the outstanding balance for month $n$, we have the boundary conditions $B(0)=1000$ and $B(23)=0$. For each month we make a payment of $C$, this covers the interest of $r\,B(n)$ and any excess goes to reducing the balance, thus we have $$B(n+1) = B(n) - (C-r\,B(n)) = B(n)(1+r) - C$$ This implies that $$\begin{align}B(n+2) &= B(n+1)(1+r) - C\\ &= (B(n)(1+r) - C))(1+r) - C\\ &= B(n)(1+r)^2 - C((1+r)-1)\end{align}$$ You should be able to see from this that in general we have $$\begin{align}B(n) &= B(0)(1+r)^n - C \sum_{j=0}^{n-1} (1+r)^j\\ &=B(0)(1+r)^n-C\frac{(1+r)^n-1}{r} \end{align}$$ We can substitute in our two boundary conditions to give $$0=B_0(1+r)^N - C\frac{(1+r)^N-1}{r}$$ $$\Rightarrow C = \frac{B_0\,r\,(1+r)^n}{(1+r^N)-1}$$ Which given the fact for us $B_0 = 1000, N=24$ and $r\approx 0.0072$ gives a payment rate of $$ C = 45.5 (3 \mathrm{s.f.}).$$

Answer 2

The notation is standard actuarial notation.

If the nominal annual interest rate is $i^{(12)} = 5\%$ compounded monthly, then this corresponds to an effective annual rate of $$i = \left( 1 + \frac{i^{(12)}}{12} \right)^{\!12} - 1 \approx 0.0511619.$$ The accumulated value of an annuity-immediate of $1$ per year for $12$ years is therefore $$s_{\overline{12}\rceil i} = \frac{(1+i)^{12} - 1}{i} \approx 16.0246,$$ as claimed.

Regarding the second question, the term "payment in arrears" as it applies to annuities is not common actuarial terminology in the United States; instead, the term "annuity-immediate" is used, and "annuity-due" is used to describe payments that are made at the beginning of each payment period.

The equation of value is $$1000 = K a_{\overline{24}\rceil j},$$ where $j$ is the effectively monthly interest rate: $$(1+j)^{12} = 1+i = 1.09,$$ or $j \approx 0.00720732$. Therefore the monthly present value discount factor is $v = (1+j)^{-1} \approx 0.992844$ and the required level monthly payment $K$ is $$K = 1000 \left( \frac{1 - v^{24}}{j}\right)^{\!-1} \approx 45.5238.$$ This is the actuarial approach.