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 €20 000 next year. The rent is expected to grow by 2.5% per year forever. The required return is 8.5% per year. Find the value PV of the property today.
You save €250 at the end of every month for 18 years. The account pays 4.8% per year, compounded monthly. How much have you saved at the end? Find FV.
A machine costs €40 000 today, lasts 6 years and costs €5 000 per year to run (paid at the end of each year). The cost of capital is 5%. Find its equivalent annual cost EAC.
A project costs €100 000 today. Its cash flow is €9 000 at the end of year 1 and grows by 2% per year until the last cash flow at the end of year 10. The cost of capital is 8%. Find the net present value NPV.
A bond with a face value of €1 000 pays an 8% annual coupon in two half-yearly payments and matures in 8 years. The yield to maturity is 5% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 3% matures in 5 years and sells for €895. Find its yield to maturity YTM in percent.
A company has equity worth €12 000 000 and debt worth €4 000 000. 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 4 000 000 shares at €24.00 each and 10 000 bonds trading at €1 000 each. The pre-tax cost of debt is 7% and the tax rate is 25%. The share’s beta is 0.9, the risk-free rate 2.5% and the market return 8.5%. Find the WACC.
A product sells for €27 with a variable cost of €25 per unit. Fixed costs are €3 000 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €1 000 after tax?
A product sells for €48 per unit, with a variable cost of €45 per unit and fixed costs of €3 600 per year. Budgeted sales are 1 680 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 333333 €
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 = 85534 €
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 €54 000 yourself; the rest is interest.
Answer: EAC = 12881 €
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: NPV = −34695 €
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: P = 1196 €
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 below the coupon rate, so the bond sells at a premium.
Answer: YTM = 5.456 %
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 = 8.344 %
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 = 7.65 %
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 = 2125 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).
Answer: MS = 28.57 %
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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