Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
A property is expected to earn net rent of €20 000 next year. The rent is expected to grow by 1.5% per year forever. The required return is 3.5% per year. Find the value PV of the property today.
You deposit €6 000 at the start of every year for 27 years, the first deposit today. The account pays 4% per year. How much is in the account at the end of year 27? Find FV.
An investment of €75 000 gives these cash flows at the end of years 1 to 4: €20 000, €28 000, €23 000, €26 000. The cost of capital is 15%. Find the net present value NPV.
A project costs €25 000 and returns €9 500, €14 500, €11 000, €8 000 at the end of years 1 to 4. Cash flows can be reinvested at the cost of capital of 12%. Find the modified internal rate of return MIRR in percent.
You buy a bond with 11 years to maturity (face value €1 000, annual coupon 4%) when its yield to maturity is 9%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 9.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 pays an annual coupon of 5% and matures in 6 years. The yield to maturity is 9%. Find the Macaulay duration D in years.
A company has equity worth €10 000 000 and debt worth €4 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 4% and the tax rate is 30%. Find the WACC.
A company's business has an unlevered (asset) beta of 1.2. It plans a debt-to-equity ratio of 0.5. The tax rate is 25.8%, the risk-free rate is 4% and the market risk premium is 6%. Find the cost of equity re at the new capital structure.
A product can be made in two ways. Method A has fixed costs of €10 000 per year and a variable cost of €8 per unit. Method B, more automated, has fixed costs of €14 500 per year and a variable cost of €5 per unit. At what yearly volume Q do both methods cost the same?
A company sells two products. Product A sells for €19 with a variable cost of €15; product B sells for €25 with a variable cost of €10. They sell in a fixed mix of 3 units of A for every 2 of B. Fixed costs are €25 200 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 1 × 10⁶ €
Time value of money: money today is worth more than the same amount later, because it can earn interest. With interest rate i per period, an amount grows as FV = PV·(1 + i)n. A series of equal payments (an annuity) is valued by adding up each payment, which gives the annuity formulas.
Answer: FV = 293805 €
Time value of money: money today is worth more than the same amount later, because it can earn interest. With interest rate i per period, an amount grows as FV = PV·(1 + i)n. A series of equal payments (an annuity) is valued by adding up each payment, which gives the annuity formulas.
Each deposit is invested one year longer.
Answer: NPV = −6448 €
Net present value: discount every cash flow back to today with the required rate of return r, then add them up: NPV = Σ CFₜ/(1 + r)ᵗ. A project with NPV > 0 earns more than r and adds value. The internal rate of return (IRR) is the rate at which NPV = 0.
NPV is negative, so the project earns less than the required return.
Answer: MIRR = 20.01 %
Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.
Answer: HPR = 5.294 %
Bond pricing: a bond’s price is the present value of its cash flows, discounted at the yield to maturity (YTM): the coupons form an annuity and the face value is paid back at maturity. P = C·(1 − (1 + y)⁻ⁿ)/y + F/(1 + y)ⁿ. When the yield is above the coupon rate the bond trades below face value, and the other way round.
Answer: D = 5.248 years
Duration: the Macaulay duration is the weighted average time at which a bond pays its cash flows, with the present values of the cash flows as weights: D = Σ t·PVₜ/P. The modified duration D/(1 + y) estimates the percentage price change for a small change in yield: %ΔP ≈ −Dmod·Δy.
Answer: WACC = 7.943 %
Cost of capital: CAPM gives the return shareholders require: re = rf + β·(rm − rf). The weighted average cost of capital mixes equity and debt by their market values: WACC = E/V·re + D/V·rd·(1 − T), with V = E + D. Interest is tax-deductible, hence the (1 − T).
Answer: re = 13.87 %
Levered beta (Hamada): debt makes equity riskier. βL = βU·(1 + (1 − T)·D/E), where βU is the beta of the business without debt. To move a beta to another capital structure, unlever it first and then relever it at the new D/E.
Answer: Q = 1500 units
Cost-volume-profit: each unit sold earns a contribution margin, price minus variable cost per unit. Profit = Q·(p − v) − F, where F is the fixed cost. At the break-even point profit is zero, so Q = F/(p − v).
Below this volume method A is cheaper; above it the automated method B is cheaper.
Answer: Q = 3000 units
Break-even with a sales mix: when products sell in a fixed ratio, use the weighted average contribution margin: weight each product's p − v by its share of units. Break-even units = F/(weighted contribution margin), split again in the mix ratio.
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