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 €10 000 next year. The rent is expected to grow by 2.5% per year forever. The required return is 7.5% per year. Find the value PV of the property today.
You save €300 at the end of every month for 9 years. The account pays 2.4% per year, compounded monthly. How much have you saved at the end? Find FV.
A project costs €40 000 today and returns €11 000 at the end of each year for 5 years. The cost of capital is 10%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €20 000 and returns €9 500, €12 000 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 9 years and pays annual coupons. It sells for €1 040 when the yield to maturity is 2.5%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 6% matures in 4 years and sells for €920. Find its yield to maturity YTM in percent.
A company has equity worth €4 000 000 and debt worth €3 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 4% and the tax rate is 30%. Find the WACC.
A company's shares have a beta of 1.1 at its current debt-to-equity ratio of 0.4. It plans to raise its debt-to-equity ratio to 0.75. The tax rate is 20%, the risk-free rate 4% and the market risk premium 4%. Find the cost of equity re after the change.
A company sells 1 250 units at €23 each. The variable cost is €8 per unit and fixed costs are €15 000. Find the degree of operating leverage DOL at this sales level.
A company sells 3 200 units at €31 each, with a variable cost of €25 per unit and fixed costs of €12 000. By what percentage %ΔΠ does operating profit change if sales volume rises by 15%?
Answer: PV = 200000 €
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 = 36125 €
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 €32 400 yourself; the rest is interest.
Answer: DPB = 4.751 years
Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.
The discounted payback is always longer than the simple payback (3.636 years).
Answer: IRR = 4.769 %
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 = 3.002 %
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 = 8.439 %
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 = 6.914 %
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: re = 9.333 %
Levered beta (Hamada): debt makes equity riskier. βL = βU·(1 + (1 − T)·D/E), where βU is the beta of the business without debt. To move a beta to another capital structure, unlever it first and then relever it at the new D/E.
Answer: DOL = 5
Operating leverage: fixed costs make profit react more strongly than sales. The degree of operating leverage DOL = contribution margin/operating profit tells you that a 1% change in sales changes profit by DOL %. It is very high just above break-even and falls as sales grow.
Answer: %ΔΠ = 40 %
Operating leverage: fixed costs make profit react more strongly than sales. The degree of operating leverage DOL = contribution margin/operating profit tells you that a 1% change in sales changes profit by DOL %. It is very high just above break-even and falls as sales grow.
Check: new profit = 3 680·6 − 12 000 = €10 080.00.
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