Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €50 000 in 2 years. An account pays 4% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
A mortgage of €250 000 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 15? Find the outstanding balance B.
A machine costs €80 000 today, lasts 10 years and costs €1 500 per year to run (paid at the end of each year). The cost of capital is 6%. Find its equivalent annual cost EAC.
A project costs €20 000 and returns €9 000, €13 000 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 matures in 12 years and pays annual coupons. It sells for €832 when the yield to maturity is 6%. Find the coupon rate c in percent.
A bond with 10 years to maturity (face value €1 000, annual coupon 9%) sells for €1 140. The issuer can call it in 4 years at a call price of €1 090. Find the yield to call YTC in percent.
A company has a debt-to-equity ratio (at market values) of 1. Its cost of equity is 9%, its pre-tax cost of debt is 3.5% and the tax rate is 25.8%. Find the WACC.
A company has 1 000 000 shares at €20.00 each and 5 000 bonds trading at €1 020 each. The pre-tax cost of debt is 3.5% and the tax rate is 30%. The share’s beta is 0.8, the risk-free rate 1.5% and the market return 8.5%. 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 €20 per unit. Method B, more automated, has fixed costs of €32 500 per year and a variable cost of €17 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €25 per unit, with a variable cost of €20 per unit and fixed costs of €20 000 per year. Budgeted sales are 6 400 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 46174 €
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 = 147238 €
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: EAC = 12369 €
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 = 6.203 %
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 = 3.996 %
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: YTC = 6.906 %
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 to maturity is 7.006%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 5.798 %
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 = 6.155 %
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 = 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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