EXPLAIN
Chapter 5 – Time Value of Money
Essentials of Financial Management 4th Edition by Brigham, F., Houston, F., Hsu, J., Kong, Y &
... [Show More] Bany-Ariffin, AN. (2019). Pg. 155 - 188
w
n – compounding period
Example: Find the future value of a 3-Year Ordinary Annuity of $100 at 4%.
FVOA= 100( ((1+0.04)^3-1)/0.04)
FVOA = 312.16
Future Value of An Annuity Due
Just like the present value of an annuity due, future value of an annuity due begins the first payment at the start of each period, giving it an additional period of compounding. Therefore, the formula is just an extension of the future value of ordinary annuity, multiplied by the additional compounding period, summarized below:
FVAD= 100( ((1+0.04)^3-1)/0.04) (1+0.04)
FVOA = 312.16 (1.04)
FVOA = 324.65
DISCOUNTING vs. COMPOUNDING
Compounding method is used to know the future value of present money. Conversely, discounting is a way to compute the present value of future money. In other words, they are the reverse of each other. To be technically correct, when asked to compute for the present value, we refer to the process as discounting. Compounding on the other hand is appropriate term for solving future values.
Perpetuity
A stream of level cash payments that never ends, in other words, FOREVER.
Present value of a Perpetuity is determined using the following formula:
PV=PMT/i
Where: PV is the present value
PMT is the regular cash flow
i is the interest rate
Example: In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?
Solution:
PV = 100,000 / 0.10
PV = 1,000,000
Example: Continuing the preceding example, if the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?
Solution:
PV = 1,000,000 (1+.10)-3
PV = 751,315
Uneven Cash Flow Streams
In certain instance, cash flows are not even per period. If this is the case, computation of either the present value or future value will use the simple formula, and not the annuity. Consider the following example:
Year Cash Flow
0
100
300
300
-50
The present value of the series of cash flow above at 4% interest is determined as follows: PV = FV (1+i)-n
100 (1+.04)-1 = 96.15
300 (1+.04)-2 = 277.37
300 (1+.04)-3 = 266.70
-50 (1+.04)-4 = -42.74
Total 597.48 (Present Value)
Using the same example, the future value of the series of cash flow above at 4% interest is determined as: FV = PV (1+i)n
100 (1+.04)3 = 112.49
300 (1+.04)2 = 324.48
300 (1+.04)1 = 312.00
-50 (1+.04)0 = (50.00)
Total 698.97 (Future Value)
Note: The “n” in the equation is determined by the distance of each cash flow to the end point. In the case of present value, the end point is Year 0. Hence, 100 will travel one period back, 300 will travel two years back, and so on. Notice also that the exponent is negative since it goes back to year zero. For the future value, the end point is Year 4. So 100 in the first year will move three periods to reach year 4, the 300 in year two by two periods, the 300 in year three by one period and the -50 by 0 since, it is on the same period.
Compounding/Discounting Period
The “n” in the formula does not necessarily indicate the time. It is the number of compounding periods. Time is normally measured in years and the timing of cash flow may not always be annually. It is therefore necessary to identify the compounding period or the number of times in a year cash flow is received or paid. For ease of understanding, the following are provided:
Timing No. of Compounding
Annual 1
Semi-Annual 2
Quarterly 4
Monthly 12
Bi-monthly 24
Weekly 52
Daily 365 [Show Less]