Finance-Theory · Unit 20 · Video 4 · Interactive Practice

Discount the Project, Not the Company: Pricing Bloomberg Press with a Pure-Play Beta

IKey Formulas

Formula	Name	What you need
$\bar{R}_{\text{project}} = R_f + \beta_{\text{project}}\left(\bar{R}_m - R_f\right)$	Project cost of capital (CAPM)	Risk-free rate, the project's beta, the market risk premium
$\beta_{\text{project}} \approx \beta_{\text{pure play}}$	Pure-play estimate	A traded firm doing only the project's business
$\text{NPV} = CF_0 + \dfrac{CF_1}{1 + \bar{R}_{\text{project}}} + \cdots + \dfrac{CF_T}{(1 + \bar{R}_{\text{project}})^T}$	NPV at the project's rate	The project's cash flows and its own discount rate

Key Insight: The beta belongs to the project, not the company. A single company-wide rate is a horizontal line that sets the hurdle too high for low-beta projects and too low for high-beta ones.

IIProject Hurdle vs Company-Wide Rate

A parent with $\beta = 1.6$ sets one flat $13.9\%$ rate; each project's own hurdle rises along the SML.

Project β 0.95 Expected return, % 11.0

IIIA Range from Comparables

No comparable matches exactly: each comparable's $\beta$ gives one estimate, and judgment picks a rate inside the range.

A · closest pure play, β 0.85 B · large pure play, β 0.95 C · small pure play, β 1.05 D · conglomerate, β 1.40

💡 A conglomerate's $\beta$ is a value-weighted blend of all its businesses, so it says little about publishing alone; Problem 4 takes one apart.

IVQuiz Questions

Problem 1 · The Pure-Play Rate

Given: A technology firm with $\beta = 1.6$ is considering a publishing imprint. The closest pure-play publisher has $\beta = 0.85$ ; $R_f = 3.5\%$ and $\bar{R}_m - R_f = 6.5\%$ . Find the imprint's cost of capital.

✅ Correct!

3.5\% + 0.85 \times 6.5\% = 9.025\%

: the imprint is discounted at a publisher's rate, not the parent's.

❌ That is only the risk premium.

0.85 \times 6.5\% = 5.525\%

is what the project earns above the risk-free rate; add

R_f = 3.5\%

❌ That is the parent's rate.

3.5\% + 1.6 \times 6.5\% = 13.9\%

prices the technology business, not publishing. Use the pure play's

\beta

❌ Not quite. Multiplying

\beta

by the market's total return,

0.85 \times 10\%

, misplaces

R_f

: only the premium

\bar{R}_m - R_f

is scaled by

\beta

❌ Not quite. Use

\bar{R}_{\text{project}} = R_f + \beta_{\text{project}}(\bar{R}_m - R_f)

with the pure play's

\beta

Show solution

The project's beta is estimated by the pure play's, $\beta_{\text{project}} \approx 0.85$ , not the parent's $1.6$ :

\bar{R}_{\text{project}} = 0.035 + 0.85 \times 0.065 = 0.035 + 0.05525 = 0.09025 = 9.025\%

The parent's own rate, $0.035 + 1.6 \times 0.065 = 13.9\%$ , would overstate the imprint's hurdle by $4.875$ percentage points.

Problem 2 · The Company-Wide Trap

Given: The imprint in Problem 1 costs $2,000,000 today and is expected to generate $200,000 at the end of every year forever. Find its NPV at the correct discount rate and the decision.

✅ Correct! At the imprint's own

9.025\%

the perpetuity is worth about $2,216,066, more than its $2,000,000 cost.

❌ That uses the parent's 13.9%. This is exactly the low-beta project a company-wide rate wrongly rejects: it clears its own hurdle.

❌ That discounts at the market's 10%. The imprint's

\beta

0.85

, below

1

, so its rate is below the market's.

❌ That uses a conglomerate's 12.6%. A

\beta

1.40

blends other businesses' risks; the pure play's

0.85

measures publishing.

❌ Not quite. Value the perpetuity as

C/r

at the project's rate, then subtract the cost.

Show solution

A level perpetuity starting in one year is worth $C / r$ . In dollars, at $r = 9.025\%$ :

\text{PV} = \frac{200{,}000}{0.09025} \approx 2{,}216{,}066, \qquad \text{NPV} \approx 2{,}216{,}066 - 2{,}000{,}000 = 216{,}066

NPV ≈ +$216,066 > 0, so accept. At the parent's $13.9\%$ the same cash flows are worth $200{,}000 / 0.139 \approx 1{,}438{,}849$ , an NPV of about −$561,151, and a value-adding project would be turned down.

Problem 3 · Where Judgment Matters

Given: Two more pure-play publishers have $\beta = 0.95$ (large) and $\beta = 1.05$ (small), so the pure-play estimates span $\beta = 0.85$ to $1.05$ . Same $R_f$ , premium and imprint as above. Find the cost of capital at the top of this range and the imprint's NPV there.

Cost of capital at $\beta = 1.05$

Imprint's NPV at that rate

✅ Correct! The NPV is positive at

\beta = 0.85

and

0.95

but negative at

1.05

: the sign flips inside the range, so where judgment places the rate decides the project.

❌ That is the middle of the range.

3.5\% + 0.95 \times 6.5\% = 9.675\%

uses the large publisher's

\beta

; the top of the range is

\beta = 1.05

❌ That is only the risk premium.

1.05 \times 6.5\% = 6.825\%

; add

R_f = 3.5\%

❌ That is the conglomerate's rate.

3.5\% + 1.40 \times 6.5\% = 12.6\%

lies outside the pure-play range, which stops at

\beta = 1.05

❌ Wrong rate. That is the NPV at

9.675\%

(

\beta = 0.95

), the middle of the range, not its top.

❌ Wrong rate. That is the NPV at

9.025\%

(

\beta = 0.85

), the bottom of the range.

❌ Wrong rate. That is the NPV at the conglomerate's

12.6\%

; use

10.325\%

❌ Check the rate.

R_f + 1.05 \times 6.5\%

: the premium is

6.825\%

, and

R_f = 3.5\%

must be added to it.

❌ Check the NPV. Divide $200,000 by

0.10325

, then subtract the $2,000,000 cost.

Show solution

\bar{R} = 0.035 + 1.05 \times 0.065 = 0.035 + 0.06825 = 0.10325 = 10.325\%

In dollars:

\text{PV} = \frac{200{,}000}{0.10325} \approx 1{,}937{,}046, \qquad \text{NPV} \approx 1{,}937{,}046 - 2{,}000{,}000 = -62{,}954

Across the range: at $9.025\%$ the NPV is about +$216,066, at $9.675\%$ about +$67,183, at $10.325\%$ about −$62,954. The NPV is zero at exactly $10\%$ , the rate for $\beta = 1$ . The decision rests on judging how risky this imprint is relative to the comparables.

Problem 4 · Inside a Conglomerate's Beta

Given: The conglomerate comparable has $\beta = 1.40$ . Publishing makes up $60\%$ of the conglomerate's total market value and has the pure-play $\beta = 0.85$ ; the other $40\%$ of its value is a ratings-and-data business. A firm's $\beta$ is the value-weighted average of its businesses' betas. Find the $\beta$ of the ratings-and-data business.

✅ Correct! Its required return is

3.5\% + 2.225 \times 6.5\% = 17.9625\%

. The conglomerate's

12.6\%

averages a

9.025\%

business with this one, which is why it misprices publishing.

❌ Not quite.

1.40 - 0.85

ignores the weights; solve

1.40 = 0.6 \times 0.85 + 0.4\,\beta_{\text{other}}

❌ Not quite.

1.40 / 0.4

forgets to remove publishing's contribution,

0.6 \times 0.85 = 0.51

, first.

❌ The weights are swapped. Publishing carries the

60\%

weight and the other business the

40\%

❌ Not quite. Write the conglomerate's

\beta

as a weighted average and solve for the unknown.

Show solution

1.40 = 0.6 \times 0.85 + 0.4\,\beta_{\text{other}} \;\Rightarrow\; \beta_{\text{other}} = \frac{1.40 - 0.51}{0.4} = \frac{0.89}{0.4} = 2.225

Its cost of capital is $0.035 + 2.225 \times 0.065 = 0.179625 \approx 18.0\%$ . Check: $0.6 \times 9.025\% + 0.4 \times 17.9625\% = 5.415\% + 7.185\% = 12.6\%$ , the conglomerate's own rate. Reading publishing's risk off $\beta = 1.40$ would mix in a business whose $\beta$ is about $2.6$ times publishing's.

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