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Net present value and internal rate of return | net present value actuarial | Lemma

Lemma/CM1 Foundation/Equation of Value and Applications

CM1 Foundation · Equation of Value and Applications

Net present value and internal rate of return

Value a project's cashflows with NPV and locate the internal rate of return as the root of the equation of value.

Skills in this lesson

1. Net present value of a project0/2 verified
2. Internal rate of returnLocked

Prerequisites

No prior lessons required — this is your entry point to CM1 Foundation.

Teach

A project generates a stream of cashflows $c_t$ at times $t$ . The net present value (NPV) discounts every cashflow back to time $0$ at a chosen rate of interest $i$ and adds them up. The internal rate of return (IRR) is the rate that makes the NPV zero — it is the yield the project earns on the capital tied up in it.

Net present value

With discount factor ,

v = (1+i)^{-1}

v = (1 + i)^{- 1}

\text{NPV}(i) = \sum_t c_t\, v^t = \sum_t \frac{c_t}{(1+i)^t}.

Inflows are positive, outflows negative. A project is profitable at rate $i$ when $\text{NPV}(i) > 0$ .

Worked example: NPV of a two-year project

Problem. Outlay Rs. 1,000 now returns Rs. 600 at $t=1$ and Rs. 600 at $t=2$ . Find the NPV at $i = 8\%$ .

Solution.

\text{NPV} = -1000 + 600(1.08)^{-1} + 600(1.08)^{-2} \approx -1000 + 555.6 + 514.4 = \text{Rs. } 70.

The project adds value at 8%.

Internal rate of return

The IRR is the solution $i^{*}$ of the equation of value

\text{NPV}(i^{*}) = \sum_t c_t\,(1+i^{*})^{-t} = 0.

For non-conventional cashflows (several sign changes) there may be no IRR or several, so the NPV rule is the more reliable decision tool.

Decision rule

Accept a project if $\text{NPV} > 0$ at the firm's cost of capital, equivalently if $\text{IRR} >$ cost of capital. The two rules agree for conventional projects.

Practice

Practice 1

$c_t$

$\sum_t c_t (1+i)^{t}$$\sum_t c_t (1+i)^{-t}$$\sum_t c_t / t$$\sum_t c_t$

Hint

Each cashflow is discounted back to time 0.

Practice 2

$t=1$

Hint

$-1000 + 600(1.08)^{-1} + 600(1.08)^{-2}$.

Practice 3

$t=1$

Hint

$-500 + 550/1.10$.

Practice 4

$i$

the NPV is maximisedthe NPV equals zerothe payback period is zeroall cashflows are equal

Hint

It is the root of the equation of value.

Practice 5

$Invest Rs. 1,000 now and receive Rs. 1,100 in one year. Give the IRR as a percentage.$

Hint

$1100/1000 - 1$.

Practice 6

$For a conventional project (outflow then inflows), the IRR is unique because the NPV is:$

increasing in $i$a monotonically decreasing function of $i$constant in $i$always negative

Mastery checks

Pass each skill by answering correctly on 2 of 3 checks (fewer if fewer are available). Practice questions above are rehearsal — only mastery checks count toward progress. Retry wrong answers until you succeed.

Net present value of a project

0/2 verified

Check 1

$i=10\%$

Hint

$1000\,a_{\overline{3}|} - 2000$ with $a_{\overline{3}|}\approx 2.4869$.

Check 2

$A project should be accepted on the NPV rule when:$

NPV < 0 at the cost of capitalNPV = 0 at any rateNPV > 0 at the cost of capitalthe payback exceeds the term

Check 3

$t=1$

Hint

With zero interest, discount factors are 1.

Internal rate of return

Locked — complete prior skills first

Check 1(Complete prior skills first)

$Invest Rs. 1,000 now and receive Rs. 1,210 in two years. Give the IRR as a percentage.$

Hint

$(1210/1000)^{1/2} - 1$.

Check 2(Complete prior skills first)

$Non-conventional cashflows (several sign changes) may have several IRRs. The preferred decision tool is then:$

the smallest IRRthe payback periodthe NPV at the cost of capitalthe largest IRR

Check 3(Complete prior skills first)

$On the IRR rule, a conventional project is accepted when its IRR is:$

below the cost of capitalabove the cost of capitalequal to zeronegative