0.3 The Time Value of Money and Interest (Solutions and Explanations)
The $1 in the question is referred to as “Present Value (PV).” The amount of money we will have in the future is referred to as “Future Value (FV).” Remember, we are, so far, assuming that the interest payments are made at the end of each relevant payment period.
| Question | Calculation | What we learn from this |
| 1 | $1 (1.05) = $1.05 | $1 times one (representing 100% of your principal) plus the interest rate |
| 2 | $1 (1.10) = $1.10 | As interest rates increase, so too do FVs. |
| 3 | $1 (1.10) (1.10) =
$1 (1.10)2 = $1.21 |
As time increases, so too do FVs. |
| 4 | $1 (1.05) (1.05) =
$1 (1.05)2 = $1.1025 |
Rates are always quoted in annualized terms, unless otherwise indicated. You must make the necessary adjustments. Here we see that as the number of compounding periods per year increases, so too does the FV. |
| 5 | $1 (1.05)4 = $1.2155 | We have two things going on here |
| 6 | See “summary” below | |
| 7.1 | $1 ÷ 1.05 = $0.9524 | Whereas solving for FV involves multiplication (i.e., compounding), solving for PV involves division or discounting. |
| 7.2 | $1 ÷ 1.10 = $0.9091 | As interest rates increase, PVs decrease |
| 7.3 | $1 ÷ (1.10)2 = $0.8264 | As time increases, PVs decrease |
| 7.4 | $1 ÷ (1.05)2 = $0.9070 | As discounting frequency increases, PV decreases |
| 7.5 | $1 ÷ (1.05)4 = $0.8227 | |
| 8. | Yes! | This is how bonds and many financial things are figured. Can you provide examples? |
Definition: A “reciprocal” is the “opposite” of a number, which is arrived at by turning the number on its head – by dividing! So, 1/2 or 0.5 is the reciprocal of 2. The reciprocal of 5 is 20% (i.e., 1/5).
Summary: As interest rates, the number of compounding periods per year, and time increase, the Future Value increases and the Present Value decreases.