Finance-Theory · Unit 20 · Video 5 · Interactive Practice

Synergies, Firm-Wide Cost of Capital and Moving Discount Rates

IKey Formulas

Formula	Name	Use
$\text{Value to the firm} = \text{NPV}_{\text{standalone}} + \text{PV}(\text{synergies})$	Project plus its interactions	Cross-selling and shared people add value the standalone NPV misses
$\text{PV} = \dfrac{S}{r}$	Perpetuity of $S$ a year, first at $t = 1$	A level synergy that lasts indefinitely
$V = \dfrac{C}{r}, \quad \text{multiple} = \dfrac{V}{C} = \dfrac{1}{r}$	Value of a level perpetual cash flow $C$ ; value per dollar of it	A higher rate charged on the existing business lowers its multiple, and the lower multiple applies to all of its cash flows
$\text{NPV} = CF_0 + \displaystyle\sum_{t=1}^{T} \frac{CF_t}{(1 + r_t)^t}, \quad r_t = R_{f,t} + \text{premium}$	NPV with date-specific rates	The risk-free base moves over time and with the horizon

Key Insight: A project is worth its standalone NPV plus whatever it does to the rest of the firm. Synergies add to it; a higher firm-wide cost of capital should cost nothing for a small division that investors can value on its own, but at scale a repricing of the whole firm is real value destroyed. And because $R_f$ moves, the same cash flows can be worth taking in one year and not in the next.

IIVisualization 1 — Synergies on Top of the Standalone NPV

A project that fails on its own can still add value once its synergies with the firm are counted.

Standalone NPV Synergy per year Synergy rate

IIIVisualization 2 — Does the Division Reprice the Rest of the Firm?

Charge the existing business a higher rate and it loses value. Whether the project should bear that loss depends on its size. (Our illustration: the rest of the firm earns $90M a year forever and its own risk calls for 9%; the division's own rate is 13%.)

Rate investors charge the rest of the firm The new division is

IVVisualization 3 — The Same Project as the Risk-Free Rate Moves

With a constant 5% premium, a rise in $R_f$ from 2% to 4.5% can turn the same cash flows from accept to reject.

Risk-free rate

R_f

VQuiz Questions

Problem 1 · Counting the Synergy

Given: a new division has a standalone NPV of −$2.5 million. Through cross-selling it adds $0.36 million a year to the existing business, starting in one year and lasting indefinitely; those extra cash flows are discounted at $9\%$ . Find the project's value to the firm.

✅ Correct! The synergies are a perpetuity worth $0.36 million ÷ 0.09 = $4.0 million, which more than covers the −$2.5 million standalone NPV.

❌ That counts one year of synergy, undiscounted. The $0.36 million arrives every year forever, so its value is a perpetuity,

S / r

❌ That is the synergies alone. The division's own cash flows still lose $2.5 million in present value; add the two.

❌ That is the standalone view. The synergies never show up in the division's own cash flows, but they are incremental cash flows of the firm with the project versus without it.

Show solution

The synergy is a level perpetuity starting at $t = 1$ , worth $S / r$ :

\text{PV}(\text{synergies}) = \frac{0.36}{0.09} = 4.0 \text{ (millions)}

Value to the firm = standalone NPV + PV(synergies):

-2.5 + 4.0 = 1.5 \text{ (millions)}

The project adds $1.5 million and should be taken, even though it would be rejected as a standalone subsidiary.

Problem 2 · Which Rate for the Division?

Given: a firm's existing business has a cost of capital of $9\%$ . It is considering a small, riskier division whose own (pure-play) cost of capital is $13\%$ . The division costs $10 million today and is expected to produce after-tax cash flows of $1.17 million a year forever, starting in one year. Ignore synergies. Find the division's NPV and the decision.

✅ Correct! At its own

13\%

the division is worth $1.17 million ÷ 0.13 = $9.0 million, less than the $10 million it costs.

❌ That discounts at the firm's $9\%$ . $1.17 million ÷ 0.09 = $13 million, but the existing business's rate reflects the existing business's risk. A project is discounted at the rate matching its own systematic risk,

13\%

here.

❌ The NPV is right; the reasoning is not. Investors who supply the capital for this division demand

13\%

for its risk, whatever the rest of the firm pays. A negative NPV at the right rate means reject.

❌ Check the sign. The cash flows are worth $9.0 million and the cost is $10 million, so NPV

= 9.0 - 10 = -1.0

(millions).

Show solution

The division is valued standalone at the rate for its own risk, $13\%$ , not at the firm's $9\%$ :

\text{NPV} = -10 + \frac{1.17}{0.13} = -10 + 9.0 = -1.0 \text{ (millions)}

At the firm's $9\%$ it would look like $-10 + 1.17/0.09 = +3.0$ million, a $4 million error that comes entirely from the wrong discount rate. Reject the division unless synergies worth more than $1.0 million in present value can be shown.

Problem 3 · A Rate for Each Date

Given: today the one-year risk-free rate is $2\%$ and the two-year risk-free rate is $4.5\%$ a year; the project's risk premium is $5\%$ at both horizons, so $r_1 = 7\%$ and $r_2 = 9.5\%$ . The project costs $16 million and pays $7 million in one year and $11 million in two years. Find its NPV.

✅ Correct! Discounting the year-2 cash flow at its own

9.5\%

rate leaves the project about $284,000 short, so it is rejected.

❌ That uses the one-year rate for both dates. The year-2 cash flow has a two-year horizon and gets

r_2 = 9.5\%

11/1.095^2

, not

11/1.07^2

❌ That uses $9.5\%$ for both dates. The year-1 cash flow is discounted at the one-year rate,

r_1 = 7\%

❌ That discounts at the risk-free rates alone. The project's cash flows are risky; each rate is

R_{f,t}

plus the

5\%

premium.

Show solution

Each cash flow is discounted at the rate for its own date:

\text{NPV} = -16 + \frac{7}{1.07} + \frac{11}{1.095^2}

\approx -16 + 6.5421 + 9.1741 \approx -0.2838 \text{ (millions)}

The NPV is about −$284,000: reject. With the one-year rate used for both dates the year-2 term would be $11/1.07^2 \approx 9.6078$ and the NPV about +$150,000, the wrong decision.

Problem 4 · A Division Big Enough to Reprice the Firm

Given: a firm's existing business produces $45 million a year forever and is priced at its own $9\%$ . A large trading division, priced at its own $13\%$ , has a standalone NPV of +$25 million. Taking it would make the division so large a part of the firm that investors would price all of the existing business's cash flows at $13\%$ as well. Find the change in the firm's value from taking the division.

✅ Correct! The existing business falls from $500 million to about $346.15 million, a loss of about $153.85 million that swamps the division's +$25 million.

❌ That is the standalone NPV only. It would be the answer for a small division that investors value on its own. Here the division reprices the rest of the firm, and that loss belongs to the project.

❌ That is the repricing loss alone. The division still brings its own +$25 million; net the two.

❌ The repricing is a loss, not a gain. A higher rate on the same $45 million a year lowers its value:

45/0.13 \lt 45/0.09

Show solution

Value the existing business at each rate (perpetuity, millions):

\frac{45}{0.09} = 500 \qquad \frac{45}{0.13} \approx 346.15

The loss elsewhere is about $500 - 346.15 = 153.85$ . Equivalently, the multiple on each dollar of annual cash flow drops from $1/0.09 \approx 11.11$ to $1/0.13 \approx 7.69$ , and the lower multiple applies to all of it.

\Delta \text{Value} \approx 25 - 153.85 = -128.85 \text{ (millions)}

The division destroys about $128.85 million of value. This is the prop-trading case from the video in our own numbers: once investors see the firm as the risky business, the lower multiple applies to all of its cash flows. The same division, small enough not to change how investors see the firm, would simply add its $25 million.

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