Training Financial Mathematics Present Value & Future Value
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Present Value & Future Value

23 min Financial Mathematics

Present Value & Future Value

Future Value (FV)

The value of an investment at a future date:

$$FV = PV\left(1 + \frac{r}{n}\right)^{nt}$$

Present Value (PV)

The current worth of a future sum, discounted back:

$$PV = \frac{FV}{\left(1 + \frac{r}{n}\right)^{nt}}$$

Discount Factor

$$d = \frac{1}{(1+i)^n}$$

Multiply by the future amount to get present value. $i$ = rate per period.

Example 1

How much must you invest today at $5\%$ compounded annually to have $\$10{,}000$ in 7 years?

$PV = 10000/(1.05)^7 \approx \$7{,}106.81$.

Example 2

FV of $\$3{,}000$ at $4\%$ compounded monthly for 10 years.

$FV = 3000(1 + 0.04/12)^{120} \approx \$4{,}475.47$.

Example 3

A bond pays $\$1{,}000$ in 5 years. Market rate is $6\%$. What is its present value?

$PV = 1000/(1.06)^5 \approx \$747.26$.

Practice Problems

1. PV of $\$20{,}000$ in 10 years at $4\%$ annually.
2. FV of $\$5{,}000$ at $3\%$ compounded monthly, 6 years.
3. How long until $\$8{,}000$ grows to $\$12{,}000$ at $5\%$?
4. PV of $\$50{,}000$ in 20 years at $7\%$.
5. What rate doubles money in 10 years (annual compounding)?
6. Discount factor for $i = 0.03$, $n = 15$.
7. FV of $\$1{,}000$ at $8\%$ compounded quarterly, 5 years.
8. PV of $\$100{,}000$ at $5\%$ continuously, 25 years.
9. Which is worth more today: $\$5{,}000$ in 3 years or $\$6{,}000$ in 5 years at $4\%$?
10. FV of $\$2{,}500$ at $6\%$ compounded semi-annually, 8 years.
11. How much to invest at $3\%$ monthly to have $\$15{,}000$ in 4 years?
12. If PV = $\$4{,}000$ and FV = $\$5{,}500$ after 6 years, find the annual rate.
Show Answer Key

1. $20000/(1.04)^{10} \approx \$13{,}511.47$

2. $5000(1.0025)^{72} \approx \$5{,}984.74$

3. $t = \ln(1.5)/\ln(1.05) \approx 8.31$ years

4. $50000/(1.07)^{20} \approx \$12{,}920.07$

5. $2 = (1+r)^{10}$; $r = 2^{0.1}-1 \approx 7.18\%$

6. $d = 1/(1.03)^{15} \approx 0.6419$

7. $1000(1.02)^{20} \approx \$1{,}485.95$

8. $100000 e^{-1.25} \approx \$28{,}650.48$

9. PV₁ = $5000/1.04^3 \approx \$4{,}444.98$; PV₂ = $6000/1.04^5 \approx \$4{,}931.53$; $\$6{,}000$ in 5 years

10. $2500(1.03)^{16} \approx \$4{,}011.73$

11. $P = 15000/(1+0.03/12)^{48} \approx \$13{,}312.42$

12. $5500/4000 = (1+r)^6$; $r = (1.375)^{1/6}-1 \approx 5.44\%$

Simple & Compound Interest Annuities