Present Value

Overview

Engineering reference data for Present Value in economics.

Key Formulas

Present Value

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

Discount a future value to present.

Net Present Value

$NPV = \sum_{t=0}^{n} \frac{C_t}{(1+r)^t}$

Sum of discounted cash flows.

Compound Interest

$FV = PV(1+r)^n$

Future value with compound interest.

Variables

Symbol	Description	Unit
$PV$	Present value	$
$FV$	Future value	$
$r$	Interest/discount rate	—
$n$	Number of periods	years

Examples

Example 1: Variable Cash Flow Stream

The Net Present Worth (NPW) of investing **$1,000 today** and receiving savings of$250, $200,$300, $310, and$290 over the next 5 years, plus a final sale value of $310 in year 5, at a discount rate of 10%:

$P = \frac{-1000}{(1 + 0.1)^0} + \frac{250}{(1 + 0.1)^1} + \frac{200}{(1 + 0.1)^2} + \frac{300}{(1 + 0.1)^3} + \frac{310}{(1 + 0.1)^4} + \frac{(290 + 310)}{(1 + 0.1)^5} = 202$

Since the Net Present Worth is positive ($P > 0$), the investment is profitable.

*NPW = 202 | IRR = 16.5%

Example 2: Fixed Cash Flow with Growth Rate (Solar Power Investment)

An investment of $30,000 in a solar power system saves$1,000 on electricity bills the first year. The electricity price is assumed to rise 10% each year (growth rate). The alternative investment is a savings account at 3% interest rate (discount rate). The lifespan is 20 years.

Parameter	Value
Investment	$30,000
Annual cash flow (fixed)	$1,000
Growth rate	10% per year
Discount rate	3% per year
Lifespan	20 years

*NPW = 12,819 | IRR = 6.06%

The investment is profitable compared to the savings account (IRR of 6.06% > 3% discount rate).

Sensitivity Analysis

Using the same parameters but with a 15-year lifespan:

• *NPW = −3,581 (negative)
• IRR = 1.6% (below 3% discount rate)

The investment is not profitable compared to the savings account over 15 years.

Key Properties

Property	Description
Compound Amount Factor	$(1 + i)^n$ — the factor by which a present value grows to a future value
NPW Decision Rule	If $P > 0$, the investment is profitable; if $P < 0$, it is not
IRR	The discount rate at which $P = 0$; compare IRR to required rate of return
Sign Convention	Money out (investment) = negative; money in (returns) = positive

Interactive Charts

References

• Original Source