Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €20 000 in 4 years. An account pays 3.6% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
You borrow €12 000 at 7.2% per year, compounded monthly, and repay €175 at the end of every month. How many months does it take until the loan is repaid (the last payment may be smaller)? Give the number of payments N.
A machine costs €40 000 today, lasts 4 years and costs €2 000 per year to run (paid at the end of each year). The cost of capital is 10%. Find its equivalent annual cost EAC.
A project costs €40 000 and returns €24 000, €25 500, €29 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 matures in 11 years and pays annual coupons. It sells for €907 when the yield to maturity is 3%. Find the coupon rate c in percent.
A bond with a face value of €1 000 pays a 2% annual coupon in two half-yearly payments and matures in 12 years. It sells for €493. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company is financed with ordinary shares worth €8 000 000, preferred shares worth €1 000 000 and debt worth €5 000 000. The cost of equity is 12%, the cost of preferred stock 6.5% and the pre-tax cost of debt 5%. The tax rate is 30%. Find the WACC.
A company has 4 000 000 shares at €50.50 each and 40 000 bonds trading at €980 each. The pre-tax cost of debt is 5% and the tax rate is 28%. The share’s beta is 1.8, the risk-free rate 4% and the market return 10%. Find the WACC.
A product can be made in two ways. Method A has fixed costs of €25 000 per year and a variable cost of €25 per unit. Method B, more automated, has fixed costs of €28 000 per year and a variable cost of €19 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €10 per unit, with a variable cost of €4 per unit and fixed costs of €15 000 per year. Budgeted sales are 4 000 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 17329 €
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: N = 89 payments
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.
88 full payments leave a small balance, which the last, smaller payment clears.
Answer: EAC = 14619 €
Equivalent annual cost: spread the present value of all costs over the life as an equal yearly amount (an annuity): EAC = PV·r/(1 − (1 + r)⁻ⁿ). It lets you compare machines with different lives.
Answer: IRR = 41.37 %
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 falls as the rate rises, so the IRR lies between a rate with positive NPV and one with negative NPV.
Linear interpolation gives a first estimate; repeating it (or a calculator's IRR function) gives the exact rate.
Answer: c = 1.995 %
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 = 8.993 %
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: WACC = 8.571 %
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 = 12.98 %
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 = 500 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 = 37.5 %
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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