Question

Problem 3 [20p]: A cash flow the attached diagram (where years). If the interest rate is 18% calculate the uniform series equivale A cash flow over 10 years is shown in gram (where the time axis is in estrate is 18% per year, cash flow. 3.000 52.200 ss. ES 51.65 SHES 5712

3000	3000	3000	3000	3000	2250	1688	1266	949	712	534
0	1	2	3	4	5	6	7	8	9	10

Solve without Using excel/Matlab.
solve using GIVEN formulas whenever APPLICABLE. ((Do not try to solve by assuming each cash flow as a single cash flow at its corresponding year))

Answer 1

We first need to find the PF pf the given CF stream, which we will solve using excel:

Year	CF	Discount Factor		Discounted CF
0	$    3,000.00	1/(1+0.18)^0=	1	1*3000=	$     3,000.00
1	$    3,000.00	1/(1+0.18)^1=	0.847457627	0.847457627118644*3000=	$     2,542.37
2	$    3,000.00	1/(1+0.18)^2=	0.71818443	0.718184429761563*3000=	$     2,154.55
3	$    3,000.00	1/(1+0.18)^3=	0.608630873	0.608630872679291*3000=	$     1,825.89
4	$    3,000.00	1/(1+0.18)^4=	0.515788875	0.515788875151941*3000=	$     1,547.37
5	$    2,250.00	1/(1+0.18)^5=	0.437109216	0.437109216230459*2250=	$         983.50
6	$    1,688.00	1/(1+0.18)^6=	0.370431539	0.370431539178355*1688=	$         625.29
7	$    1,266.00	1/(1+0.18)^7=	0.313925033	0.313925033201996*1266=	$         397.43
8	$        949.00	1/(1+0.18)^8=	0.266038164	0.266038163730505*949=	$         252.47
9	$        712.00	1/(1+0.18)^9=	0.225456071	0.225456070958055*712=	$         160.52
10	$        534.00	1/(1+0.18)^10=	0.191064467	0.191064466913606*534=	$         102.03
				NPV = Sum of all Discounted CF	$   13,591.42

Now we can calculate the equal annual payment by using the 2nd formula from the given list which is of the PV of an annuity

We are given

$(P/A,i,n)=\frac{(1+i)^{n}-1}{i(1+i)^{n}}$

This can be reduced to:

$(P/A,i,n)=\frac{(1+i)^{n}}{i(1+i)^{n}} - \frac{1}{i(1+i)^{n}}$

$(P/A,i,n)=\frac{1}{i} - \frac{(1+i)^{-n}}{i}$

1-(1+i)-n (P/A, i, n) = ?

Where $(P/A,i,n)=$ present value calculated above/Annuity payment, i = interest rate, n = time

We are given the following information:

i	18.00%
n	10
P	$            13,591.42

We need to solve the following equation to arrive at the required PMT:
$P=A \times \frac{1-(1+i)^{-n}}{i}\\\\ 13591.42=\times \frac{1-(1+0.18)^{-10}}{0.18} \\\\ A=3024.30$

So the equal annual payment should be of 3024.30