Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €1 000 at the end of each year for 10 years. The interest rate is 4% per year. Find the present value PV of the annuity.
You save €500 at the end of every month for 10 years. The account pays 3.6% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €75 000 gives these cash flows at the end of years 1 to 3: €30 000, €34 000, €17 000. The cost of capital is 15%. Find the net present value NPV.
Project A costs €40 000 and returns €22 500, €15 000, €9 000 at the end of years 1 to 3. Project B costs €45 000 and returns €11 500, €19 000, €23 500. Find the crossover rate r* at which both projects have the same NPV, in percent.
A bond with a face value of €1 000 matures in 9 years and pays annual coupons. It sells for €962 when the yield to maturity is 3.5%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 6% matures in 9 years and sells for €905. Find its yield to maturity YTM in percent.
A company has equity worth €3 000 000 and debt worth €3 000 000. Its WACC is 8.19%, its pre-tax cost of debt is 3% and the tax rate is 21%. Find the cost of equity re.
A company has 1 000 000 shares trading at €55.00; the book value of its equity is €22 000 000. It just paid a dividend of €1.40 per share, and dividends grow by 5% per year. Its debt has a market value of €44 000 000 and a pre-tax cost of 4%. The tax rate is 28%. Find the WACC, using the dividend growth model for the cost of equity.
A product can be made in two ways. Method A has fixed costs of €20 000 per year and a variable cost of €12 per unit. Method B, more automated, has fixed costs of €35 000 per year and a variable cost of €6 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €34 per unit, with a variable cost of €30 per unit and fixed costs of €20 000 per year. Budgeted sales are 6 000 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 8111 €
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 = 72093 €
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.
You paid in €60 000 yourself; the rest is interest.
Answer: NPV = −12026 €
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* = 7.166 %
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 7.166% both NPVs are €1 369.32.
Answer: c = 3.001 %
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: YTM = 7.489 %
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: re = 14.01 %
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 = 5.543 %
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 = 2500 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: MS = 16.67 %
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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