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 timeline depicts the information we know, along with the unknown component (i):
| PV= |
$7,300 |
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FV= |
$10,000 |
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| 0 |
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| n = 4 years; i = ?? per year |
8a. Calculation Using a Present Value of 1 Table
As the timeline 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:
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PV = |
FV x [ PV factor for n = 4 years; i = ??% per year ] |
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$7,300 = |
$10,000 x [ PV of factor for n = 4 years; i = ??% per year ] |
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$7,300 / $10,000 = |
PV factor for n = 4 years; i = ??% per year |
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0.730 = |
PV factor for n = 4 years; i = ??% per year |
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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 the 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 timeline depicts the information we know, along with the unknown component (i):
| PV= |
$67,000 |
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FV= |
$100,000 |
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..... |
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← 3 months → |
← 3 months → |
← 3 months → |
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← 3 months → |
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3 |
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19 |
20 |
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| n = 20 (5 years X 4 quarters each year); i = ?? per quarter |
9a. Calculation Using a PV of 1 Table
As the timeline 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 ] |
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$67,000 / $100,000 = |
PV factor for n = 20 quarters; i = ??% per quarter |
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0.670 = |
PV factor for n = 20 quarters; i = ??% per quarter |
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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 timeline depicts the information we know, along with the unknown component (i):
| PV= |
$790 |
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FV= |
$1,000 |
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..... |
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← 1 month → |
← 1 month → |
← 1 month → |
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← 1 month → |
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2 |
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23 |
24 |
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| n = 24 (2 years X 12 months per year); i = ?? per month |
10a. Calculation Using a PV of 1 Table
As the timeline 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:
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PV = |
FV x [ PV factor for n = 24 months; i = ??% per month ] |
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$790 = |
$1,000 x [ PV factor for n = 24 months; i = ??% per month ] |
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$790 / $1,000 = |
PV factor for n = 24 months; i = ??% per month |
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0.790 = |
PV factor for n = 24 months; i = ??% per month |
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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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