Capital Investment Appraisal with an Inconsistent Discount Rate
Kivu Logistics Ltd runs 40 diesel trucks on the Nairobi–Mombasa corridor and is considering replacing 10 of them with electric units at a total cost of KES 50 million. The finance director's paper shows fuel and maintenance savings of KES 16 million a year for five years, stated in today's prices, with no residual value. The company has 400,000 ordinary shares trading at KES 1,000 each and a KES 200 million bank term loan at 14% interest. The risk-free rate is 10.5%, the market risk premium is 5.5% and the company's equity beta is 1.1. Corporation tax is 30%. The finance director discounted the savings at 12%, "the rate we used last year", and presented a positive NPV to the board. Inflation is expected to average 6% a year. One director wants to reject the project because a 12% return is "barely above the loan rate"; another argues that the savings should be grown at 6% a year in the appraisal.
1. Compute Kivu Logistics' weighted average cost of capital using the CAPM for the cost of equity. Show every step and state the assumptions you rely on.
Read the worked answer
Cost of equity (CAPM): Ke = Rf + β(Rm − Rf) = 10.5% + 1.1 × 5.5% = 16.55%. The beta measures the systematic risk of a logistics business, and the market risk premium compensates shareholders for that risk; the risk-free rate is the return on a default-free government security of matching maturity.
Cost of debt (after tax): Kd = 14% × (1 − 0.30) = 9.8%. Interest is tax-deductible, so the government effectively bears 30% of the cost, which is why the hurdle rate uses the after-tax figure. Dividends are paid from after-tax profit and get no such shield.
Weights: Equity E = 400,000 × KES 1,000 = KES 400 million. Debt D = KES 200 million. Total V = KES 600 million. Weights: 66.67% equity, 33.33% debt.
WACC = (400/600 × 16.55%) + (200/600 × 9.8%) = 11.033% + 3.267% = 14.30%.
Assumptions to state: the share price is a fair proxy for market value of equity; the book value of the loan approximates its market value; the bank facility is the company's marginal source of debt and no new equity is being raised, so the target weights are taken to equal existing weights; the project is in the same risk class as existing operations, so the same beta and WACC apply; there are no material issue costs; and the company is profitable enough to use the tax shield. If the project were riskier than the core business - for example a first move into an untested technology - a project-specific premium or a proxy-company beta would be more defensible than the corporate WACC.
2. Explain the inflation consistency problem in the finance director's appraisal, compute the NPV on a correct basis, and comment on the two directors' views.
Read the worked answer
The principle: cash flows and discount rates must be internally consistent. Either discount nominal (money) cash flows at a nominal rate, or real cash flows at a real rate. Discounting real cash flows at a nominal rate understates value; discounting nominal cash flows at a real rate overstates it. The 12% rate used is neither the company's WACC nor a properly derived real rate, so the paper is wrong twice over.
Real approach. Real rate = (1.1430 ÷ 1.06) − 1 = 7.83%. Five-year annuity factor at 7.83% = (1 − 1.0783⁻⁵) ÷ 0.0783 = 4.009. NPV = (16 × 4.009) − 50 = 64.14 − 50 = +KES 14.14 million.
Nominal approach (equivalent). Year 1 savings 16 × 1.06 = KES 16.96 million, growing 6% a year, discounted at 14.30% gives the same +KES 14.14 million. The two methods agree, which is the test that the pairing is right.
The 12% rate gives an annuity factor of 3.605 and an NPV of +KES 7.68 million - understated but still positive; discounting the real savings at 14.30% gives +KES 4.53 million. The correct figure is the largest of the three, so the error runs against the project.
On the directors' comments. The first director confuses the pre-tax cost of a single liability with the blended after-tax hurdle rate; the project earns roughly 18% in real terms (annuity factor 50 ÷ 16 = 3.125), far above the 7.83% real cost of capital, so the margin is wide, not "bare." The second director is right about the arithmetic but should understand it is a presentation choice, not an improvement - escalating the savings at 6% is only correct if the 16 million is also discounted at 16 × 1.06ⁿ equivalents. Neither inflation treatment changes the decision; what changes it is using the correct rate.
3. Compute the sensitivity of the NPV to the annual savings and advise the board on what else it should require before approving the project.
Read the worked answer
Sensitivity to annual savings. Present value of the savings is KES 64.14 million, so the NPV of KES 14.14 million can absorb a fall of 14.14 ÷ 64.14 = 22.0% before the project breaks even. That is, savings could fall from KES 16 million to KES 12.47 million a year - about KES 3.53 million a year - and still return the cost of capital. A 22% margin on an operating saving of this kind is reasonably robust; a sensitivity below about 10% would normally be treated as too thin to justify the capital commitment.
Sensitivity to the discount rate. The real IRR is about 18%, against a real cost of capital of 7.83% - the rate would have to more than double before the project failed. The appraisal is therefore driven by the cash flow assumption, not by the discount rate.
What the board should require before approving:
- Evidence for the KES 16 million. It should be broken down into fuel, servicing, downtime and driver costs, benchmarked against the diesel fleet's actual cost per kilometre, not a supplier's brochure.
- The excluded costs. Charging infrastructure, substation upgrades, driver and technician retraining, and battery replacement if the battery life is shorter than five years.
- Residual value. Assuming nil is conservative but should be confirmed against secondary-market evidence; battery degradation is the key variable.
- Tax and financing details. Capital allowances on the vehicles, import duty and VAT, and foreign-exchange exposure if the units are imported.
- Operational feasibility. Range versus the corridor's terrain and load, charging time versus turnaround, grid reliability and the risk of power outages, and the reliability of a new technology in a fleet whose revenue depends on uptime.
- A staged commitment. Approve a pilot of three vehicles, measure actual savings for six months, then exercise an option on the remaining seven.