Finding the present value of the given cashflow.

Question

A loan is repayable by an annuity certain , which is payable annually in arrear for 16 years and calculated at effective rate of interest $5\%$ pa. The payments at t=1 , t=2 , t=3 , t=4 , . . . . . . t=15 , t=16 are given as : (100 , 100 , 120 , 120 , . . . . . . . , 240 , 240) .

We need to find the amount of the loan , or the present value of these payments.

Present value at t=0 is given by ( $100v + 100 v^{2} + 120 v^{3} + 120v^{4} $ . . . . . $+ 240v^{15} + 240v^{16}$) where , $ v = (1+i)^{-1}$.

It isn't solvable right away , so what I did was , combining the consecutive payments , thereby dealing with $8$ payments now , with rate of interest $ i^{'} = 10.25\%$. ( $1+ i^{'} = (1 + 1.05)^{2}$) where $i^{'}$ is the effective rate of interest for two years.

So the cashflow looks like this now : ( 200 , 240 , 280 , . . . . , 480).

I found out its present value as : $200 a_{[8]} + (40v)(Ia)_{[7]}$ , where , $ a_{[8]}$ is the present value of 8 payments of 1 unit for 8 years and $ (Ia)_{[7]}$ is the increasing annuity for 7 years.

Is the above relation OK ?

Answer 1

The cash flow is equivalent to this \begin{matrix} t & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16\\ \hline A & 100 & & 120 & & 140 & & 160 & & 180 & & 200 & & 220 & & 240 & \\ B & & 100 & & 120 & & 140 & & 160 & & 180 & & 200 & & 220 & & 240 \end{matrix} where the payment are made every two periods of compunding.

The interest is then $j=(1+i)^2-1=10.25\%$ and $v=\frac{1}{1+j}=0.907029$. So we have for the cash flow $B$ $$ PV_B=100\,a_{\overline{8}|j}+20\,v\,(Ia)_{\overline{7}|j}= 845.13 $$ and for the cash flow $A$ $$ PV_A=(1+i)PV_B= 887.38 $$ and then the present value for the original cash flow is $PV=PV_A+PV_B=(2+i)PV_B$ that is $$ PV=(2+i)\left[100\,a_{\overline{8}|j}+20\,v\,(Ia)_{\overline{7}|j}\right]= 1,732.51 $$

Answer 2

The cashflow for periods $(2n-1)$ and $2n$ are equal. Let this be $C_n$. These can be combined together into one cashflow in period $2n$ of $$C_n(1+i)+C_n=C_n(2+i)$$. From the cashflows given, $C_n=100+20(n-1)$.

Hence we have cashflows of $$(2+i)\{100,120,\cdots,240\}$$ for years $2,4,6,\cdots,16$.

Taking two years as one period, the interest rate becomes $i'=(1+i)^2-1$.

Defining the discount factor for one period as $w=\dfrac 1{1+i'}=\dfrac 1{(1+i)^2}=v^2$, the present value can then be computed as

$$\begin{align} (2+i)\left(100w+120w^2+\cdots+240w^8\right) &=(2+i)\sum_{n=1}^8 (100+20(n-1))w^n\\ &=(2+i)\left(100a_{{\overline{8|}}i'}+20(Ia)_{{\overline{7|}}i'}\right)\qquad\blacksquare \end{align}$$