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 €12 000 next year. The rent is expected to grow by 2.5% per year forever. The required return is 5.5% per year. Find the value PV of the property today.
A mortgage of €200 000 is repaid with equal monthly payments over 30 years at 2.4% per year, compounded monthly. How much is still owed just after the payment at the end of year 10? Find the outstanding balance B.
A project costs €200 000 today. It generates €8 000 at the end of the first year, and the cash flow then grows by 4% per year forever. The cost of capital is 9%. Find the net present value NPV.
Project A costs €20 000 and returns €11 500, €8 000, €4 000 at the end of years 1 to 3. Project B costs €25 000 and returns €7 000, €10 000, €13 500. Find the crossover rate r* at which both projects have the same NPV, in percent.
A bond with a face value of €1 000 pays a 2% annual coupon in two half-yearly payments and matures in 11 years. The yield to maturity is 8% per year. Find the price P of the bond.
A bond trades at €905 with a yield to maturity of 4% (annual compounding) and a Macaulay duration of 11 years. Use the modified duration to estimate the new price P if the yield falls by 75 basis points.
A company has equity worth €6 000 000 and debt worth €2 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 4.5% and the tax rate is 25%. Find the WACC.
A company has equity with a market value of €15 000 000 and a cost of equity of 10%. Its debt consists of 4 000 bonds (face value €1 000, annual coupon 7%, 8 years to maturity) that trade at €915 each. The tax rate is 30%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A product can be made in two ways. Method A has fixed costs of €20 000 per year and a variable cost of €20 per unit. Method B, more automated, has fixed costs of €35 000 per year and a variable cost of €17 per unit. At what yearly volume Q do both methods cost the same?
A company sells 10 000 units per month at €24 each, with a variable cost of €4 per unit and fixed costs of €100 000 per month. The variable cost rises by €4 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?
Answer: PV = 400000 €
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: B = 148536 €
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.
Whatever is still owed must be exactly paid off by the remaining payments.
Answer: NPV = −40000 €
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: r* = 8.445 %
Crossover rate: the discount rate at which two projects have the same NPV. It is the IRR of the difference in their cash flows (B − A). Below it the project with the later, larger cash flows has the higher NPV; above it the other one wins.
Check: at 8.445% both NPVs are €543.37.
Answer: P = 566.5 €
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.
The yield is above the coupon rate, so the bond sells at a discount.
Answer: P = 976.8 €
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.
Duration is a linear estimate; for large yield changes convexity makes the true price a little higher.
Answer: WACC = 8.344 %
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: WACC = 9.207 %
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: Q = 5000 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 = 12500 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).
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