3.7 Calculating Present Value
Key Takeaways
Whenever interest is paid for the use of money, the value of the original principal will increase in relation to time. This concept is known as the time value of money.
Quite often, you need to calculate the Principal (when the future or maturity value, the rate and the time are known). You call this principal the present value. It can be found by rewriting the relationship as follows:
[latex]FV = P(1+rt)[/latex]
Therefore, dividing both sides by , you get:
[latex]P = \frac{FV}{1+rt}[/latex]
Example 3.7.1
Calculate the present value of a 6-month loan which requires a repayment of $6,500 including interest calculated at 8.25% pa simple interest.
[latex]FV=$6,500;\; r=8.25%=0.0825;\; t=\frac{6}{12} year=0.5[/latex]
Therefore,
[latex]P=\frac{$6,500}{(1+0.0825\times0.5)}=$6,242.50[/latex]
Example 3.7.2
How much should be invested on April 6, 2001 to amount to $9,200 (FV or maturity value) on September 19, 2001 at 8.5% simple interest?
[latex]t =\frac{262 -96}{365}=\frac{166}{365}\; years[/latex] (from table 3.1)
OR
Using the Calculator:
DT1 = 4.0601 [ENTER]
↓DT2 = 9.1901 [ENTER]
↓[CPT] DBD = 166
[latex]FV=$9,200;\; r=8.5%=0.085;\; t=1\frac{166}{365}\: years[/latex]
Therefore,
[latex]P=\frac{$9,200}{1+0.085\times\frac{166}{365}}=$8,857.59[/latex]
Key Takeaways
Key Takeaways
Example 3.7.3
How long will it take to earn $50 interest if $1,000 is deposited at 6%?
[latex]I=$50;\; r=6%=0.06;\; P=$1,000[/latex]
[latex]I=Prt[/latex]
[latex]t=\frac{I}{Pr}=\frac{$50}{$1000\times 0.06}=0.833\; years[/latex]
Note that $t$ will be in “years” since the interest rate is understood to be “per year.”
[latex]Actual\; time = 0.833 \;years \times\frac{365\; days}{1 \;year} = 304.2\; or \;305\; days.[/latex]
Now look at the same type of problem in a slightly different way:
Example 3.7.4
How long will it take $2,000 to accumulate to $2,100 if the simple interest rate is 6%?
[latex]FV=P(1+rt)[/latex]
So:
[latex]1+rt=\frac{FV}{P}[/latex]
Then:
[latex]rt=\frac{FV}{P}-1[/latex]
And (given that [latex]r\neq 0[/latex]):
[latex]t=\frac{\frac{FV}{P}-1}{r}=\frac{\frac{$2,100}{$2,000}-1}{0.06}=0.833\; years=305\; days[/latex]
Knowledge Check 3.3
- What rate of simple interest is used if a deposit of $2,000 amounts to $2,210 over 1.5 years?
- What deposit will amount to $1,871.25 over a period of 33 months if interest is calculated at 9% simple?
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