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 €6 000 next year. The rent is expected to grow by 1.5% per year forever. The required return is 6.5% per year. Find the value PV of the property today.
You want €25 000 in 13 years. You save the same amount at the end of every month in an account paying 4.8% per year, compounded monthly. Find the monthly deposit A.
A machine costs €50 000 today, lasts 8 years and costs €4 000 per year to run (paid at the end of each year). The cost of capital is 8%. Find its equivalent annual cost EAC.
An investment of €10 000 returns €1 500 at the end of each year for 9 years. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays a 4.5% annual coupon in two half-yearly payments and matures in 3 years. The yield to maturity is 9% per year. Find the price P of the bond.
A bond with 12 years to maturity (face value €1 000, annual coupon 9%) sells for €1 159. The issuer can call it in 5 years at a call price of €1 090. Find the yield to call YTC in percent.
A share with a beta of 1.6 has a required return of 8.4%. The risk-free rate is 2%. What expected market return rm does CAPM imply?
A company has 5 000 000 shares at €39.50 each and 10 000 bonds trading at €1 050 each. The pre-tax cost of debt is 4% and the tax rate is 28%. The share’s beta is 1.1, the risk-free rate 1% and the market return 7%. Find the WACC.
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 €22 000 per year and a variable cost of €5 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €22 per unit, with a variable cost of €18 per unit and fixed costs of €8 000 per year. Budgeted sales are 3 600 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 120000 €
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: A = 115.7 €
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: EAC = 12701 €
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 = 6.463 %
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.
The annuity factor falls as the rate rises; bracket the rate and narrow down.
Answer: P = 883.9 €
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: YTC = 6.724 %
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 6.998%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: rm = 6 %
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 = 7.362 %
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 = 4000 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 = 44.44 %
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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