4.4 Present Value
For many investment decisions, it is necessary to find the principal, or present value, that corresponds to a given future value.
Example 4.4.1
Consider a note that will pay $10,000 to whoever owns it three years from now. If an investor wants to earn 10% compounded annually, what is the most he or she should pay for the note?
You have:
- i=10%=0.10 per year
- n=3 years
- FV= $10,000
Thus, using the compound-interest formula:
[latex]\begin{align*} &PV(1+i)^n=FV\\ &PV(1.10)^3=$10,000\\ &PV=\frac{$10,000}{1.1^3} =$7,513.15\\ \end{align*}[/latex]
This result can be checked by accumulating the money in an account, as shown in the next box.
| Time | Interest | Balance |
|---|---|---|
| 0 | $7,513.15 | |
| 1 | $751.32 | 8,264.46 |
| 2 | 826.45 | 9,090.91 |
| 3 | 909.09 | 10,000.00 |
Formula for Present Value
The compound-interest formula can be rewritten to give the present value directly. Divide both sides of the equation by [latex](1+i)^n[/latex]
[latex]\frac{PV(1+i)^n}{(1+i)^n}=\frac{FV}{(1+i)^n}[/latex]
Cancel and rewrite:
[latex]PV = \frac{FV}{(1+i)^n}[/latex]
Knowledge Check 4.3
Find the present values by using the formula for PV given above.
- The present value of $5,849.29 due in two years if interest is at 8% compounded semi-annually.
- The present value of $8,998.91 due in nine months if interest is at 16% compounded quarterly.
- The principal of a loan that would amount to $50,000 in six years at 8.5% compounded annually.
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