Present Value of a
Single Amount

There are occasions when we need to determine the length of time (n) in a Present Value (PV) calculation. To do this, we need to know the three other components in the PV calculation: present value amount, future cash amount (FV), and the interest rate used for discounting the future cash amount (i).



5. Exercise #5. What is the length of time involved if a future amount of $5,000 has a present value of $1,000, and the time value of money is 8% compounded annually?

The following time line depicts the information we know, along with the unknown component (n):

PV=	$1,000					FV=	$5,000
					.....
0	1	2	3			n=??
n = ??;   i = 8% compounded annually

5a. Calculation Using the Present Value of 1 Table
As we had done in Part 3, we start with the PV formula: PV = FV x [ 1 ÷ (1 + i)ⁿ ]. We then substitute the PV of 1 factor for the expression "[ 1 ÷ (1 + i)ⁿ ]". Our equation now reads:

  PV = FV x [ PV factor ]  

From the information we've been given, we know that the future value is $5,000, the present value is $1,000, and the annual interest rate is 8% compounded annually. Let's plug those numbers into our equation to solve for (n), the number of annual time periods:

	PV   =	FV x [ PV factor for n = ?? years; i = 8% per year ]
	$1,000   =	$5,000 x [ PV factor for n = ?? years; i = 8% per year ]
	$1,000 / $5,000   =	PV factor for n = ?? years; i = 8% per year
	0.20000   =	PV factor for n = ?? years; i = 8% per year
	0.20000   =	PV factor for n = 21 years; i = 8% per year

Our equation tells us that the PV of 1 factor is 0.20000. Since the rate for discounting is 8% compounded annually, we look at the PV of 1 Table in the 8% column and find the amount closest to 0.200; this would be 0.199. We see it is located in the row where n = 21. This tells us that the missing component, length of time, is approximately 21 years.




6. Exercise #6. What is the length of time involved if a future amount of $10,000 has a present value of $3,000, and the time value of money is 10% compounded semiannually?

The following time line depicts the information we know, along with the unknown component (n):

PV=	$3,000					FV=	$10,000
					.....
	← 6 months →	← 6 months →	← 6 months →
0	1	2	3			n
n = ?? semiannual periods;   i = 5% per semiannual period

6a. Calculation Using a PV of 1 Table
As mentioned previously, if you are furnished with a table containing present value factors, the formula PV of 1 = FV times [ 1 ÷ (1 + i)ⁿ ], can be restated to:

  PV = FV x [ PV factor ]  

From the information we've been given, we know that the future value is $10,000 and the present value is $3,000. The annual interest rate is 10% compounded semiannually, which we can restate as 5% per semiannual period. When we plug in the information we know, we can solve for (n), the number of semiannual periods:

	PV   =	FV x [ PV factor for n = ?? semiannual periods; i = 5% per semiannual period ]
	$3,000   =	$10,000 x [ PV factor for n = ?? semiannual periods; i = 5% per semiannual period ]
	$3,000 / $10,000   =	PV factor for n = ?? semiannual periods; i = 5% per semiannual period
	0.30000   =	PV factor for n = ?? semiannual periods; i = 5% per semiannual period
	0.30000   =	PV factor for n = 25 semiannual periods; i = 5% semiannually

Our equation tells us that the PV factor is 0.30000. Since the rate for discounting is 5% per semiannual period, we look at the PV of 1 Table in the 5% column and find the amount closest to 0.300; this would be 0.295. We see it is located in the row where n = 25. This tells us that the missing component, length of time, is approximately 25 semiannual periods (about 12.5 years).



7. Exercise #7. What is the length of time involved if a future amount of $100,000 has a present value of $75,000, and the time value of money is 12% compounded monthly?

The following time line depicts the information we know, along with the unknown component (n):

PV=	$75,000					FV=	$100,000
					.....
	← 1 month →	← 1 month →	← 1 month →
0	1	2	3			n
n = ?? months;   i = 1% per month

7a. Calculation Using a PV of 1 Table
From the information we've been given, we know that the future value is $100,000 and the present value is $75,000. The annual interest rate is 12% compounded monthly, which we can restate as 1% per month. Let's plug in the information we know and solve for (n):

	PV   =	FV x [ PV factor for n = ?? months ; i = 1% per month ]
	$75,000   =	$100,000 x [ PV factor for n = ?? months; i = 1% per month ]
	$75,000 / $100,000   =	PV factor for n = ?? months; i = 1% per month
	0.75000   =	PV factor for n = ?? months; i = 1% per month
	0.75000   =	PV factor for n = 29 months; i = 1% per month

Our equation tells us that the PV factor is 0.750. Since the rate for discounting is 1% per month, we look at the PV of 1 Table in the 1% column and find the amount closest to 0.750; this amount would be 0.749. We see it is located in the row where n = 29. This tells us that the missing component, length of time, is approximately 29 months (about 2.4 years).





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