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

23 min Financial Mathematics
Present value (PV) and future value (FV) are two sides of the same coin: PV = FV / (1 + r)ⁿ discounts a future cash flow back to today's dollars, while FV = PV × (1 + r)ⁿ projects a current sum forward. The discount rate r captures the time value of money—the principle that a dollar today is worth more than a dollar tomorrow because it can be invested. These concepts are the bedrock of capital budgeting, where firms compare the PV of expected cash inflows against the cost of an investment. Net present value (NPV = Σ CFₜ / (1 + r)ᵗ − C₀) is positive when an investment creates value, and the internal rate of return (IRR) is the discount rate that makes NPV zero.

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?

  1. $PV = 10000/(1.05)^7 \approx \$7{,}106.81$.
Example 2

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

  1. $FV = 3000(1 + 0.04/12)^{120} \approx \$4{,}475.47$.
Example 3

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

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

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

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