Present value and net present value

This is a backward calculation using the compound interest formula.

If the amount receivable is $\$133.10$, what was the amount invested if the compound interest rate is $10\%$ over $3$ years?

Present value formula

Use this formula when you know the future value and want to find the amount invested today.

$$\text{Present value}=\frac{\text{Future value}}{(1+r{)}^n}$$

Example 1

Solution

Please note that the examiner will test you thoroughly on these computations.

Investment appraisal using discounted cash flow

Discounting cash flows is useful when deciding whether a project is viable.

We focus on two appraisal methods:

• Net Present Value (NPV)
• Internal Rate of Return (IRR)

Net Present Value (NPV)

Previously, we discounted single cash flows using compound interest. NPV extends this idea by discounting multiple cash inflows and outflows and combining them into one figure.

NPV is calculated as:

$$\text{NPV}=\text{Present value of cash inflows}-\text{Present value of cash outflows}$$

The discount factor used is also known as the rate of return or cost of capital.

Key decision rules

• A project is viable if NPV is positive
• Projects with higher NPV should be preferred
• An NPV of zero means the project breaks even

Discount factors may be obtained from tables (e.g., CIMA Present Value Tables) or calculated using:

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

For example, at $8.5\%$ for one year:

$$\frac{1}{1.085}=0.922$$

Example 3

Kuzet Tutoring Academy is considering an investment costing $\$10,000$ initially. The discount rate is $10\%$.

Year	Cash flow	Discount factor	Present value
0	(10,000)	1.000	(10,000)
1	2,000	0.909	1,818
2	5,000	0.826	4,130
3	6,000	0.751	4,506
4	3,000	0.683	2,049
5	2,000	0.621	1,242

$$\text{NPV}=3,745$$

Since NPV is positive, the project is viable.

Below is a structured illustration comparing two projects, including visible calculations.

Project details

Project	Initial investment	Annual cash inflow	Duration (years)
A	$8,000$	$2,400$	$4$
B	$10,000$	$3,000$	$4$

NPV calculation steps

Assumptions

Discount Rate = $10\%$ per year
Cash inflows occur at the end of each year

Formula

$$\begin{array}{rl}\text{NPV}& =\frac{{\text{Cash flow}}_1}{(1+r{)}^1}+\frac{{\text{Cash flow}}_2}{(1+r{)}^2}+\cdots +\frac{{\text{Cash flow}}_n}{(1+r{)}^n}-\text{Initial investment}\end{array}$$

Calculations for Project A

Year	Cash flow	Discount factor	Present value
$1$	$2,400$	$0.909$	$2,182$
$2$	$2,400$	$0.826$	$1,982$
$3$	$2,400$	$0.751$	$1,802$
$4$	$2,400$	$0.683$	$1,639$

$$\begin{array}{rl}\text{Total present value of cash inflows}& =\$2,182+\$1,982+\$1,802+\$1,639=\$7,605\\ \text{NPV}& =\$7,605-\$8,000=-\$395\end{array}$$

Calculations for Project B

Year	Cash inflow	Discount factor	Present value
$1$	$3,000$	$0.909$	$2,727$
$2$	$3,000$	$0.826$	$2,478$
$3$	$3,000$	$0.751$	$2,253$
$4$	$3,000$	$0.683$	$2,049$

$$\begin{array}{rl}\text{Total present value of cash inflows}& =\$2,727+\$2,478+\$2,253+\$2,049=\$9,507\\ \text{NPV}& =\$9,507-\$10,000=-\$493\end{array}$$

Advantages of NPV

• Considers time value of money
• Measures absolute profitability
• Considers entire project lifespan
• Maximizes shareholder wealth

Disadvantages of NPV

• Difficult to understand conceptually
• Based on estimates
• Assumes cash flows occur at period boundaries
• Discount rate may change over time