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 deposit €3 000 at the start of every year for 25 years, the first deposit today. The account pays 3% per year. How much is in the account at the end of year 25? Find FV.
A machine costs €100 000 today. It generates €18 500 at the end of each year for 8 years and can then be sold for €10 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €10 000 and returns €6 000, €5 000 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 pays an annual coupon of 4% and matures in 13 years. Its yield to maturity changes from 2.5% to 3.5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 and an annual coupon of 7% matures in 4 years and sells for €920. Find its yield to maturity YTM in percent.
A company has equity worth €10 000 000 and debt worth €2 000 000. Its cost of equity is 11%, its pre-tax cost of debt is 6% and the tax rate is 30%. Find the WACC.
A company has 4 000 000 shares at €47.50 each and 40 000 bonds trading at €960 each. The pre-tax cost of debt is 3.5% and the tax rate is 25.8%. The share’s beta is 1.5, the risk-free rate 3% and the market return 10%. Find the WACC.
A product sells for €25 with a variable cost of €20 per unit. Fixed costs are €4 000 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €1 000 after tax?
A company sells 4 000 units per month at €55 each, with a variable cost of €45 per unit and fixed costs of €20 000 per month. The variable cost rises by €5 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?
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 = 112659 €
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.
Each deposit is invested one year longer.
Answer: NPV = 3361 €
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.
Answer: IRR = 6.811 %
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: ΔP = −113.2 €
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 = 9.496 %
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 = 9.867 %
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 = 11.67 %
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 = 1050 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: Q = 8000 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).
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