Present Value of a
Single Amount

Now we will show how to find the interest rate (i) for discounting the future amount in a present value (PV) calculation. To do this, we need to know the three other components in the PV calculation: present value amount (PV), future amount (FV), and the length of time before the future amount is received (n).


8. Exercise #8. What is the annual interest rate necessary to discount a future amount of $10,000 to a present value of $7,300 over a four-year period, assuming the interest is compounded annually?

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

PV=	$7,300			FV=	$10,000
0	1	2	3	4
n = 4 years;  i = ?? per year

8a. Calculation Using a Present Value of 1 Table
As the time line indicates, we know the future value is $10,000 and the present value is $7,300. The number of years (n) is four. The unknown component is the annual interest rate (i), which is compounded annually. In equation form, Exercise #8 looks like this:

	PV   =	FV x [ PV factor for n = 4 years; i = ??% per year ]
	$7,300   =	$10,000 x [ PV of factor for n = 4 years; i = ??% per year ]
	$7,300 / $10,000   =	PV factor for n = 4 years; i = ??% per year
	0.730   =	PV factor for n = 4 years; i = ??% per year
	0.730   =	PV factor for n = 4 years; i = 8% per year

Our equation tells us that the PV factor is 0.730. Since the number of periods is four, we look at the PV of 1 Table in the row where n = 4 and find the amount closest to 0.730; this would be 0.735. We see it is located in the column where i = 8%. This tells us that missing component, the annual interest rate, is approximately 8% compounded annually.




9. Exercise #9. What is the annual interest rate necessary to discount a future amount of $100,000 to a present value of $67,000 over a five-year period, assuming the interest is compounded quarterly?

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

PV=	$67,000					FV=	$100,000
					.....
	← 3 months →	← 3 months →	← 3 months →
0	1	2	3			20
n = 20 (5 years X 4 quarters each year);   i = ?? per quarter

9a. Calculation Using a PV of 1 Table
As the time line indicates, we know the future value is $100,000 and the present value is $67,000. The number of periods (n) is 20 quarters (5 years x 4 quarters per year). The unknown component is the interest rate (i), which will be expressed as a quarterly rate. In equation form, Exercise #9 looks like this:

	PV   =	FV x [ PV factor for n = 20 quarters; i = ??% per quarter ]
	$67,000   =	$100,000 x [ PV factor for n = 20 quarters; i = ??% per quarter ]
	$67,000 / $100,000   =	PV factor for n = 20 quarters; i = ??% per quarter
	0.670   =	PV factor for n = 20 quarters; i = ??% per quarter
	0.670   =	PV factor for n = 20 quarters; i = 2% per quarter

Our equation tells us that the PV factor is 0.670. Since the number of periods is 20 quarters, we look at the PV of 1 Table in the row where n = 20 and we find the amount closest to 0.670; this would be 0.673. We see it is located in the column where i = 2%. This tells us that the missing component, the interest rate is approximately 2% per quarter. However, the exercise asked for the annual interest rate, compounded quarterly. The annual interest rate is approximately 8% (the quarterly interest rate x four quarters).



10. Exercise #10. A future amount of $1,000 will be due in exactly two years. The obligation can be settled today for $790. What is the annual interest rate involved in this arrangement if interest is compounded monthly?

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

PV=	$790					FV=	$1,000
					.....
	← 1 month →	← 1 month →	← 1 month →
0	1	2	3			24
n = 24 (2 years X 12 months per year);   i = ?? per month

10a. Calculation Using a PV of 1 Table
As the time line indicates, we know the future value is $1,000 and the present value is $790. Since the interest is compounded monthly, the number of time periods (n) is 24 (2 years x 12 months per year). The unknown component is the monthly interest rate (i).

In equation form, Exercise #10 looks like this:

	PV   =	FV x [ PV factor for n = 24 months; i = ??% per month ]
	$790   =	$1,000 x [ PV factor for n = 24 months; i = ??% per month ]
	$790 / $1,000   =	PV factor for n = 24 months; i = ??% per month
	0.790   =	PV factor for n = 24 months; i = ??% per month
	0.790   =	PV factor for n = 24 months; i = 1% per month

Our equation tells us that the PV factor is 0.790. Since the number of periods is 24 months, we look at the PV of 1 Table in the row where n = 24 and find the amount closest to 0.790; this would be 0.788. We see it is located in the column where i = 1%.

This tells us that the missing component, the interest rate (i), is approximately 1% per month. However, the exercise asked for the annual interest rate, compounded monthly. The annual interest rate is approximately 12% (the approximate monthly interest rate x 12 months).





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