Finance-Theory · Unit 21 · Video 4 · Interactive Practice

IRR as Yield to Maturity: Four Conditions, Loans and No IRR

IKey Formulas

Formula	Name	What you need
$I_0 = \dfrac{CF_1}{1+\text{IRR}} + \dfrac{CF_2}{(1+\text{IRR})^2} + \cdots + \dfrac{CF_t}{(1+\text{IRR})^t}$	IRR: the project's yield to maturity	The investment $I_0 = -CF_0$ and the later cash flows
$\text{accept if } \text{IRR} \ge \text{IRR}^*$	IRR rule (investment-type flows)	The hurdle rate $\text{IRR}^*$ , set equal to the cost of capital $R$
$\text{accept if } \text{IRR} < R$	Loan-type flows ( $+$ today, $-$ later)	The IRR is the rate you pay, so lower is better
$CF_0\,y^2 + CF_1\,y + CF_2 = 0,\quad y = 1 + \text{IRR}$	Two-period IRR equation	A real IRR needs $CF_1^2 - 4\,CF_0\,CF_2 \ge 0$

Key Insight: The IRR rule makes the same decisions as NPV when (1) the only cash outflow is at $t = 0$ , (2) only one project is under consideration, (3) the opportunity cost of capital is the same in every period, and (4) $\text{IRR}^*$ equals that cost of capital. Loans and missing IRRs both come from breaking condition (1).

IIMirror-Image Flows: One IRR, Opposite Answers

An investment and its mirror-image loan share one IRR, but the IRR rule can be right for only one of them.

Cost of capital R, % 10

IIIWhen No IRR Exists

With two periods the IRR equation is a quadratic in $y = 1 + \text{IRR}$ , and its discriminant decides whether any IRR exists.

CF₀ −105 CF₁ 250 CF₂ −150

💡 Flipping every sign multiplies the equation by $-1$ : the roots, and so the IRRs (or their absence), stay the same while the NPV verdict reverses.

IVQuiz Questions

Problem 1 · Solve for the IRR

Given: A project costs $140 today and pays $52 at the end of year 1 and $169 at the end of year 2. Find its IRR.

✅ Correct! At

30\%

the two receipts are worth

40 + 100 = 140

today, exactly the price paid:

30\%

is the project's yield to maturity.

❌ Not quite.

(52 + 169 - 140)/140/2 \approx 28.93\%

averages the total gain over two years and ignores when each dollar arrives. The IRR is the rate that discounts the two receipts back to

140

❌ Not quite.

(52 + 169)/140 - 1 \approx 57.86\%

is the undiscounted total return over two years, not a rate per period.

❌ Wrong root. The quadratic has a second root

y = -\tfrac{13}{14}

, but

y = 1 + \text{IRR}

must be positive: a rate below

-100\%

means nothing as an interest rate.

Show solution

Set the NPV to zero and write $y = 1 + \text{IRR}$ :

-140 + \frac{52}{y} + \frac{169}{y^2} = 0 \quad\Longrightarrow\quad -140y^2 + 52y + 169 = 0.

With $a = -140$ , $b = 52$ , $c = 169$ :

b^2 - 4ac = 2{,}704 + 94{,}640 = 97{,}344 = 312^2,\qquad y = \frac{-52 \pm 312}{-280}.

The roots are $y = 1.3$ and $y = -\tfrac{13}{14}$ . Only $y = 1.3$ is a usable rate, so $\text{IRR} = 30\%$ .

Check: $\dfrac{52}{1.3} + \dfrac{169}{1.69} = 40 + 100 = 140$ ✓. The project has one outflow, at $t = 0$ , so at a cost of capital of $6\%$ the IRR rule (accept, $30\% > 6\%$ ) agrees with NPV ( $\approx +59.47$ ).

Problem 2 · Supplier Financing

Given: A supplier offers financing: you receive $175 today and repay $189 in one year. Your opportunity cost of capital is $6\%$ . Decide whether to accept.

✅ Correct! The

8\%

IRR is the rate you would pay; borrowing at

8\%

when capital costs

6\%

destroys value.

❌ That is the investment rule applied to a loan. The flows are

+

then

-

, so condition (1) fails: the IRR is the interest you pay, and a loan is worth taking only when

\text{IRR} < R

❌ Check the signs. You receive

175

today (a

+

) and repay

189

later (a

-

). Writing the flows the other way round evaluates the lender's side of the deal.

❌ Not quite. The IRR exists:

175 = \tfrac{189}{1 + \text{IRR}}

gives

8\%

. Loans flip the direction of the IRR rule; they do not remove the IRR.

Show solution

The flows are $CF_0 = +175$ , $CF_1 = -189$ : the pattern of a loan.

175 = \frac{189}{1 + \text{IRR}} \quad\Longrightarrow\quad \text{IRR} = \frac{189}{175} - 1 = 8\%.

\text{NPV at } 6\% = 175 - \frac{189}{1.06} \approx 175 - 178.30 = -3.30.

NPV is negative, so reject. The loan rule says the same: accept only if $\text{IRR} < R$ , and $8\% > 6\%$ . The plain IRR rule ( $8\% > 6\%$ , accept) gets this backwards.

Problem 3 · Is There an IRR?

Given: A project's cash flows are $CF_0 = -48$ , $CF_1 = 130$ , $CF_2 = -90$ ($ thousands), and the cost of capital is $6\%$ .

What is the discriminant $b^2 - 4ac$ of the IRR quadratic in $y = 1 + \text{IRR}$ ?

What should the firm do?

✅ Correct! The discriminant is negative, so both roots are complex and no IRR exists; NPV still decides, and it is below zero at every rate.

❌ Sign slip.

4ac = 4(-48)(-90) = +17{,}280

: two negatives make a positive, and it is then subtracted.

❌ Check the order. The discriminant is

b^2 - 4ac = 16{,}900 - 17{,}280

, not the other way round.

❌ Not quite.

16{,}900

is only

b^2

; the

4ac

term still has to be subtracted.

❌ Check the discriminant. Use

a = CF_0

b = CF_1

c = CF_2

❌ That is not a rate. The roots are

y \approx 1.354 \pm 0.203\,i

; dropping the imaginary part does not turn a complex number into an interest rate.

❌ NPV still answers. With no real root the NPV never crosses zero, so its sign is the same at every discount rate.

❌ Not quite. The loan rule needs an IRR to compare with

R

. Here there is none; read the sign of the NPV instead.

❌ Not quite. When no IRR exists, the NPV keeps one sign at every rate.

Show solution

Multiply $-48 + \dfrac{130}{y} - \dfrac{90}{y^2} = 0$ by $y^2$ :

-48y^2 + 130y - 90 = 0,\qquad a = -48,\ b = 130,\ c = -90.

b^2 - 4ac = 16{,}900 - 4(-48)(-90) = 16{,}900 - 17{,}280 = -380 < 0.

The roots $y = \dfrac{65 \pm \sqrt{95}\,i}{48} \approx 1.354 \pm 0.203\,i$ are complex: no IRR.

The NPV never reaches zero, so it keeps the sign of $CF_0$ at every rate. At $6\%$ :

\text{NPV} = -48 + \frac{130}{1.06} - \frac{90}{1.06^2} \approx -48 + 122.64 - 80.10 = -5.46.

Even at its best rate (about $38.46\%$ ) the NPV is only about $-1.06$ . Reject.

Problem 4 · A Hurdle with a Safety Margin

Given: A firm's opportunity cost of capital is $6\%$ in every year. Its CFO sets the IRR hurdle at $\text{IRR}^* = 7.5\%$ "as a safety margin". The only project under review costs $210 today and pays $80 at the end of each of the next three years; its IRR is about $6.99\%$ (at that rate the three payments are worth about $210$ today).

What is the right decision?

Which of the four conditions fails?

✅ Correct! Conditions (1) to (3) hold; the padded hurdle breaks condition (4) and turns away a project that creates value.

❌ That is what the padded rule does. Comparing with

7.5\%

rejects the project, but the project earns more than the

6\%

that capital actually costs.

❌ Wrong discount rate. NPV discounts at the opportunity cost of capital,

6\%

; the

7.5\%

hurdle is an arbitrary threshold.

❌ Not always. They agree only under all four conditions, and here the hurdle differs from the cost of capital.

❌ Not quite. Discount the three payments at the opportunity cost of capital.

❌ That condition holds. The only outflow is at

t = 0

, there is one project, and the cost of capital is

6\%

every year. Look at how

\text{IRR}^*

was set.

Show solution

\text{NPV at } 6\% = -210 + \frac{80}{1.06} + \frac{80}{1.06^2} + \frac{80}{1.06^3} \approx -210 + 213.84 = +3.84 > 0 \quad\Longrightarrow\quad \text{accept}.

Check the conditions: (1) one outflow, at $t = 0$ ✓; (2) one project ✓; (3) $6\%$ in every year ✓; (4) $\text{IRR}^* = 7.5\% \ne R = 6\%$ ✗.

With the hurdle set at $\text{IRR}^* = 6\%$ the IRR rule would read $6.99\% > 6\%$ and accept, in agreement with NPV. The safety margin is what makes the rule reject a positive-NPV project.

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