3.7 Calculating Present Value

Key Takeaways

“Moving” money backwards in time.

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.

Timeline showing Present Value (PV) and Future Value (FV=$6,500)

[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?

Timeline showing Present Value (PV) and Future Value (FV=$9,200)

[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

When you use your calculator, be careful when you apply the ORDER OF OPERATIONS. Also, DO NOT ROUND OFF any intermediate values; instead use appropriate keystroke sequences or the calculator memory.
[latex]\frac{$9,200}{1+0.085\times\frac{166}{365}}[/latex]
The following is a suggested sequence of keystrokes for the BAII Plus Calculator:
[166][÷][365][×][0.085][=][+][1][=][1/x][×][9200][=]

 

Key Takeaways

The two formulae, FV = P(1+rt) and I = Prt, interlink five variables, P, r, t, I  and FV. Depending on the context any unknown of the five variables may be calculated if you know three of the variables. A useful way to deal with problems in this area is to write down the “known,” and look for a formula which includes the “known” variables and the “unknown” variable.

 

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%?

 

Timeline showing Present Value (PV) and Future Value (FV)

[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

  1. What rate of simple interest is used if a deposit of $2,000 amounts to $2,210 over 1.5 years?
  2. What deposit will amount to $1,871.25 over a period of 33 months if interest is calculated at 9% simple?

Solutions at the end of the chapter

 

 

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