Future value and Compounding:

Future value refers to the amount of money an investment will grow to over some period of time at some given interest rate. The process of earning interest on a loan or other fixed-income instrument where the interest can itself earn interest. In other words, it is the process in which an asset’s earnings, from either capital gains or interest, are reinvested to generate additional earnings over time.

Future value for a single period investment:

Value after one year = Initial investment ( 1 + i )

Future value for a multiple period investment:

FV = PV ( 1 + i )^n

where,

FV= Future vale at n years

PV= Present value

i= rate of interest

n= number of time periods or years

Alternatively, with the use of PVIF in case of even cash flows or annual payments,

FV = PV * FVIF ( i %, n yrs )

where, 

PV= present value

FVIF (i %, n yrs) = future value interest factor at interest rate 'i' and time period 'n' years

Annuity:

An annuity is defined as a series of payments of a fixed amount at each equal interval of time for a given number of periods.

It can be an ordinary annuity or annuity due. In the case of an ordinary annuity, each equal payment is made at the end of each interval of time throughout the period. And for the case of an annuity due, each payment occurs at the beginning of each equal interval throughout the periods.

FVA = PMT * FVIFA (i %, n years)

where, 

FVA = Futue value annuity

PMT=  annual payments

FVIF (i %, n yrs) = future value interest factor annuity at interest rate 'i' and time period 'n' years

Future value of an annuity due:

FVA (due)= PMT * FVIFA (i %, n years) * (1+i)

where, 

FVA (due) = Future Value Annuity due

i = rate of interest

Alternatively, we also have a tabular solution where we use the same formula but we calculate for each successive year and do summation at last to get the desired FVA and FVA (due). 

Tabular solution for FVA and FVA (due):

Q. If the firm gets the payment of Rs. 1000 for 3 years at the end of each year, find the FVA for an 8% rate of interest.

Solution:

We know, 

FVIF = (1+ i)^n

Since the payments are made at the end of the year,

So, for the 1st year payment, it will be compounded for 2 years, so FVIF = (1+.08)^2 = 1.1664

Similarly, for the 2nd year payment, it will be compounded for 1 year, so FVIF = (1+.08)^1 = 1.08

And for the 3rd year payment, it is not compounded, so FVIF = (1+0)^1 = 1.0.

End of Year	Payment (PMT)	FVIF 8%	FV= PMT * FVIF
1	1000	1.1664	1166.4
2	1000	1.08	1080.0
3	1000	1.00	1000.0
			Rs. 3246.4

Future value of annuity (FVA)= Rs. 3246.4

Calculation of FVA using Time Line:

 

Q. If the firm gets the payment of Rs. 1000 for 3 years at the beginning of each year, find the FVA due for an 8% rate of interest.

Solution:

We know, 

FVIF = (1+ i)^n

Since the payments are made at the beginning of the year,

So, for the 1st year payment, it will be compounded for 3 years, so FVIF = (1+.08)^3 = 1.2597

Similarly, for the 2nd year payment, it will be compounded for 2 year, so FVIF = (1+.08)^2 = 1.08

And for the 3rd year payment, it is compounded for only 1 year, so FVIF = (1+0.08)^1 = 1.08

Beginning of Year	Payment (PMT)	FVIF 8%	FV= PMT * FVIF
1	1000	1.2597	1259.7
2	1000	1.1664	1166.4
3	1000	1.08	1080.0
			Rs. 3506.1

Future value of annuity (due)= Rs. 3506.1

Calculation of FVA (due) using Time Line: