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 |
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FV= |
$5,000 |
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..... |
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| 0 |
1 |
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n=?? |
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| 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:
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PV = |
FV x [ PV factor for n = ?? years; i = 8% per year ] |
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$1,000 = |
$5,000 x [ PV factor for n = ?? years; i = 8% per year ] |
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$1,000 / $5,000 = |
PV factor for n = ?? years; i = 8% per year |
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0.20000 = |
PV factor for n = ?? years; i = 8% per year |
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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 |
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FV= |
$10,000 |
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..... |
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← 6 months → |
← 6 months → |
← 6 months → |
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| 0 |
1 |
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n |
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| 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:
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PV = |
FV x [ PV factor for n = ?? semiannual periods; i = 5% per semiannual period ] |
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$3,000 = |
$10,000 x [ PV factor for n = ?? semiannual periods; i = 5% per semiannual period ] |
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$3,000 / $10,000 = |
PV factor for n = ?? semiannual periods; i = 5% per semiannual period |
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0.30000 = |
PV factor for n = ?? semiannual periods; i = 5% per semiannual period |
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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 |
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FV= |
$100,000 |
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..... |
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← 1 month → |
← 1 month → |
← 1 month → |
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| 0 |
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n |
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| 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):
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PV = |
FV x [ PV factor for n = ?? months ; i = 1% per month ] |
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$75,000 = |
$100,000 x [ PV factor for n = ?? months; i = 1% per month ] |
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$75,000 / $100,000 = |
PV factor for n = ?? months; i = 1% per month |
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0.75000 = |
PV factor for n = ?? months; i = 1% per month |
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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).
Part 1
Part 2
Part 3
Part 4
Part 5
Part 6
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