MRP Finance Internship Unit 10

Unit 10 · Practice solutions

Practice solutions

Cost of Capital · US → South Africa → Ghana

Worked answers to the optional practice questions. The required question is turned in to the TAs, and its answer is not published here.

Practice solutions

10.1

  1. Risk premium = 9.2% − 4.3% = 4.9 percentage points. The company's equity must require more than 9.2% because equity is a residual claim paid only after the bondholders are satisfied — the same operating cash flows are therefore more uncertain when viewed through the equity than through the debt, so the required return must be higher.
  2. A bond has contractual cash flows and an observable market price, so the discount rate that equates the two can be solved for directly. Equity has no promised cash flows — the holder receives whatever remains, which may be nothing — so there is no stream to discount back from the price. The premium must therefore be constructed from a model rather than read off a screen.
  3. Risk-free rate — the yield on a US Treasury security matching the cash-flow horizon, from the Treasury's published curve. Market risk premium — long-run historical excess returns of a broad index such as the S&P 500 over Treasuries, from a published long-run series. Beta — a data provider's regression estimate, checked against a second provider.
  4. "CAPM's theory is not the problem in Ghana — the difficulty is that the inputs it requires do not exist there. There is no reliable risk-free security, no long enough or deep enough index history to estimate a market premium, and no meaningful beta for a private company or one listed on a thin exchange."

10.2

  1. Diversifiable: (a) factory fire, (c) patent case, (e) chief executive's resignation. Systematic: (b) policy rates, (d) currency devaluation.
  2. From 1 to 10: (45.0 − 23.0)/45.0 = 48.9% of total risk eliminated. From 10 to 100: (23.0 − 19.4)/23.0 = 15.7%. The great majority of the available benefit comes from the first handful of holdings and the gains diminish sharply — which is why a portfolio of twenty or thirty names captures nearly all of what diversification can offer, and why going from thirty to three hundred is largely wasted effort.
  3. Because required return compensates only for risk that cannot be avoided, and firm-specific volatility can be avoided by holding other assets. An asset whose large swings are driven entirely by events unrelated to the economy contributes almost nothing to the risk of a diversified portfolio, since those swings cancel against other holdings. Its beta is therefore low even though its own standard deviation is high. Investors will accept a low expected return on it precisely because it does not add to the risk they actually bear.
  4. CAPM prices only systematic risk, on the assumption that the investor holds a diversified portfolio and has therefore eliminated everything else. This family has not diversified and cannot — nearly all their wealth sits in one firm, in one country, in one currency — so they bear the full firm-specific risk that the model assumes away. In valuing their business I would apply a discount for illiquidity and lack of marketability, or equivalently add several percentage points to the required return, and I would state the adjustment explicitly rather than hiding it in the beta.

10.3

  1. R_E = 4.2% + 0.85 × 5.5% = 4.2% + 4.675% = 8.875%.
  2. (a) Below 1.0 — electricity demand barely changes with the economy and tariffs are often regulated. (b) Above 1.0 — luxury purchases are deferred first in a downturn and surge in a boom. (c) Below 1.0 — people buy food in every economic condition. (d) Well above 1.0 — property development is among the most cyclical activities there is, and it is capital-intensive and sensitive to interest rates as well.
  3. At 1.05: 4 + 6.3 = 10.3%. At 1.38: 4 + 8.28 = 12.28%. I would not pick one — I would present the valuation at both, showing the range it produces, and investigate why they differ (different estimation windows, different index, different adjustment toward 1.0). If a reason emerged to prefer one, I would say so; if not, the two-point spread is a genuine feature of the estimate and belongs in the output rather than being resolved arbitrarily.
  4. Because it is the only model that connects risk to required return through a coherent theory, and the empirically-motivated alternatives are either no more accurate out of sample or considerably more arbitrary in what they include. It also forces the analyst to state each assumption separately — the risk-free rate, the market premium, and the relative riskiness of the business — which is valuable discipline even when the output is imprecise. And it is the market standard, so a valuation built any other way cannot easily be compared with the ones it will sit beside. The right use is as a way of organising a judgment, presented as a range rather than a point.

10.4

  1. After-tax cost = 8.4% × (1 − 0.27) = 6.132%. Tax shield = 50 × 8.4% × 27% = $1.134M per year.
  2. The 6.5% market yield. The WACC measures what it costs the firm to raise capital today, and today the market requires 6.5% to lend to this firm. The 3% coupon reflects conditions when the bond was issued and is irrelevant to any new financing decision or to valuing the firm going forward.
  3. The tax shield only has value if there is taxable profit for the interest deduction to reduce. A firm with large accumulated losses pays no tax, so an extra dollar of interest saves nothing in the current year and the deduction is merely carried forward. Since a deferred benefit is worth less than an immediate one, the effective after-tax cost of debt is closer to the full pre-tax rate — and for a firm with many years of losses, essentially equal to it.
  4. Because the 24% is computed on the contractual payments the market expects not to receive in full — a large part of that yield is compensation for expected default rather than a rate at which anyone would actually lend new money. Using it would overstate the firm's cost of debt and, through the WACC, distort the valuation. I would instead use the rate at which the firm could realistically raise new secured borrowing today, or the yield on bonds of similar-quality firms that are not distressed, and note the uncertainty explicitly.

10.5

  1. V = 800 + 300 = $1,100M. E/V = 0.7273, D/V = 0.2727. After-tax cost of debt = 7% × 0.75 = 5.25%. WACC = 12.5% × 0.7273 + 5.25% × 0.2727 = 9.091% + 1.432% = 10.52%.
  2. Book V = 250 + 320 = $570M. E/V = 0.4386, D/V = 0.5614. WACC = 12.5% × 0.4386 + 5.25% × 0.5614 = 5.482% + 2.947% = 8.43%. The error understates the WACC by 209 basis points.
  3. At 10.52%: 60 / (0.1052 − 0.025) = $748M. At 8.43%: 60 / (0.0843 − 0.025) = $1,012M. Overstated by 35.3%.
  4. Because the weights and the rates are not independent of each other. Adding debt places a larger fixed claim ahead of the equity, which makes the residual cash flows more volatile, raises the equity beta, and therefore raises the cost of equity. It simultaneously makes the debt itself riskier, so lenders require a higher yield. Beyond a point, real costs of financial distress appear — lost customers, tightened supplier terms, management distraction — so the WACC stops falling and begins to rise.
  5. For revising: the WACC formula uses market values, the market value of equity has changed, so mechanically the computed WACC has changed and any valuation using the old figure is inconsistent with observable prices. Against: the cost of capital is meant to reflect the long-run riskiness of the business, and nothing about the business changed — recomputing it every time the share price moves produces a rate that chases sentiment and makes valuations incomparable across dates. The sensible practice is to review the capital structure weights periodically using an average or a target rather than a spot figure, and to revisit the rate when something real changes. Note also the circularity: if the share fell because the market now requires a higher return, then the higher WACC is partly the market telling you something, and distinguishing that from noise is the actual judgment.

10.6

  1. R_E(USD) = 4.2% + 1.05 × 5.8% + 4.5% = 4.2% + 6.09% + 4.5% = 14.79%.
  2. 1 + r_KES = 1.1479 × 1.06 / 1.025 = 1.1871, so 18.71%.
  3. First, most of the 25% is compensation for roughly 20% inflation rather than a real return, so it is a high number rather than a high yield. Second and more fundamentally, it is not risk-free: Ghana restructured its domestic debt in 2022–23 and holders did not receive what they were promised, so the security carries genuine default risk. CAPM's logic requires an asset whose return is certain, against which the market premium is measured. Using a defaultable, inflation-laden yield as the risk-free rate corrupts every term that follows.
  4. Correct cedi rate ≈ 34.7%, so value = 50 × 1.20 / (0.347 − 0.20) = 60/0.147 = GH₵408M. At the incorrect 14.5% the denominator is 0.145 − 0.20 = negative, so the formula breaks down entirely and returns a meaningless negative value — the growth rate exceeds the discount rate. The error is not "small"; it is fatal, and the model's failure to produce any answer at all is the clearest possible signal that the currency bases have been mixed.
  5. Real growth = 1.03/1.22 − 1 = −15.6%. The analyst has almost certainly imported a terminal growth rate from a low-inflation textbook or a previous engagement in a developed market, without noticing that 3% there is a real rate and 3% here is nominal. I would use a nominal rate somewhere around 23–24%, implying real growth of roughly 1% to 2% — and I would state the real growth rate explicitly in the write-up so the assumption is visible to anyone reviewing it.

10.7

  1. Dollar cost of equity = 4.0 + 0.90 × 6.0 + 6.0 = 15.4%. In cedis: 1.154 × 1.20 ÷ 1.02 − 1 = 35.76%. Against an approved rate of 20%, the gap is 15.8 percentage points — the approved rate is not slightly low, it is a little over half of what the risk requires.
  2. After-tax cost of debt = 28% × 0.75 = 21.0%. WACC = 0.70 × 35.76% + 0.30 × 21.0% = 31.34%. Note that blending in the cheaper debt brings the rate down by about four and a half points, which is real but nothing like enough to reach 20%.
  3. At 20%: 12 / (0.20 − 0.18) = GH₵600M. At 31.34%: 12 / (0.3134 − 0.18) = GH₵90M, a value 85% lower. The gap is enormous because the approved rate leaves a denominator of only two percentage points: when r and g are close the value is governed by their difference rather than by the cash flow, so a change in r that looks moderate divides the answer by nearly seven. Any valuation in which r − g is under about five points should be treated as unstable and presented as a range rather than a figure.
  4. Because it is the analyst, not the client, who is representing that the valuation is defensible, and a discount rate is an assumption like any other — it does not become correct by being imposed. A client's rate is useful evidence about what the client believes and may well be right, but it still has to be tested against something observable, such as the sovereign yield and what comparable borrowers actually pay. The right response is not to refuse it. Run the valuation at both the imposed rate and a built one, present both, and state plainly what the difference rests on — which converts a disagreement about a number into a discussion about reasoning, where it can actually be settled.
  5. The reversals are evidence that the method works, not that it is fragile. A discounted cash flow is a machine for turning assumptions into a value, and it reversed because two of the assumptions were wrong — which is exactly what a good model should do, since a method that returned the same answer regardless of the discount rate would be telling you nothing at all. What the two examples actually show is that sensitivity analysis has to include the rate, and that the rate deserves at least as much scrutiny as the operating assumptions it sits beside. The conclusion is to present valuations as ranges over the rate rather than as point estimates, not to abandon the method.