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What is Lump Sum and Multiple Cash Flows ?

Lump Sum Cash Flow: It is a single cash flow on which interest earn for fixed period.

Multiple Cash Flows: It is a fixed or uneven series of cash flows for fixed period.

Formulas for Lump Sum and Multiple Cash Flows:

Variables
Formula
PV
FV/ (1+r) n
FV
PV (1+r) n
N
In(FV / PV)/In (1+r)
R
(FV /PV) 1/n-1

Where,
FV =Future Value
PV =Present Value
N = Number of periods
R = rate of interest

Example 1: Mr. Sharma deposit Rs.20000 for 12 years @7% compounded quarterly into saving a/c. Find out the future value of investment?

FV = PV (1+r) n
= 20000 (1+0.0175) 48
= 45991.97

Example 2: If someone wants to have Rs.10, 00,000 after his retirement then how much amount he would have to invest now @8% per annum compounded annually for 20 years?

PV = FV / (1+r) n
= 10, 00,000 / (1+0.08) 20
= 10, 00,000 / 4.66095714
= 214548.20

Example 3: Nikhil borrow Rs.500, 000 @ 9% compounded monthly and he will have to pay Rs.560, 000 to bank at the end of the loan period. Find out the loan period (n).

N =In (FV/PV) /In (1+r)
= In (560000/500000) / In (1+0.0075)
= In (1.12) /In (1.0075)
= 0.11332/0.00747
=15

Example 4: Mr. A provide loan of Rs.200, 000 to Mr. B for 4 years and after completion of loan period employee have to pay Rs.320, 000.Find out the interest rate charged by Mr. A?

I = (FV/PV) 1/n-1
= (360,000/200,000) 1/4-1
= (1.8) 1/3-1
= 15.8%

Problems of Multiple Cash Flows:

Example 1: suppose Company A purchase machinery at the cost of Rs50, 000 @ 6% compounded half yearly and the machine provides following returns in 5 years: 
1.       25000
2.       22000
3.       10000
4.       5000
5.       2000
Find out the net  present value?

PV = FV/ (1+r) n

Year
Formula
Present  Value
1
25000/(1+0.03) 8
19735.23
2
22000/(1+0.03) 6
18424.65
3
10000/(1+0.03) 4
8884.87
4
5000/(1+0.03) 2
4712.97
5
2000/(1+0.03) 0
2000
TOTAL
53757.72

NPV= present value of investment-Initial investment
= 53757.72-50000
= 3757.72

Example 2: A person deposits Rs.10000, Rs.10000, Rs.20000 and Rs.25000 for 4 years respectively @ 8% compounded annually .Find out the future value?

FV=PV (1+r) n

Year
Formula
Future Value
1
10000(1+0.08) 3
12597.12
2
10000(1+0.08) 2
11664.00
3
20000(1+0.08)1
21600.00
4
25000(1+0.08) 0
25000.00
TOTAL
Rs. 70861.12


*If cash deposits at the end of a year then “n” is 3 and if it deposits at the beginning of a year then “n” is 4. 

Example: Neha wants to invest for his son’s education for 10 years at the beginning of each  year @ 8.2% compounded annually to get Rs. 12, 000 ,Rs.  10, 000 , Rs. 11, 000, Rs. 8, 500, Rs. 7, 600,Rs.  8, 000 , Rs. 13, 000, Rs. 9, 000 , Rs. 12, 100 ,Rs. 6, 000  in the future find out the total present value of future cash flows.
Solution:

Year
Formula = FV / (1+r) n
Present Value
1
12, 000 /(1+0.082) 10
5, 455
2
10, 000 / (1+ 0.082) 9
4, 926
3
11, 000 / (1+ 0.082) 8
5, 851
4
8, 500 / (1+ 0.082) 7
4, 885
5
7, 600 / (1+ 0.082) 6
4, 750
6
8, 000 / (1+ 0.082) 5
5, 405
7
13, 000 / (1+ 0.082) 4
9, 489
8
9, 000 / (1+ 0.082) 3
7, 087
9
12, 000 / (1+ 0.082) 2
10, 256
10
6, 000 / (1+ 0.082) 1
5, 545
Total

Rs. 63, 649



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