Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €50 000 in 9 years. An account pays 2% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
You deposit €2 500 at the start of every year for 16 years, the first deposit today. The account pays 5% per year. How much is in the account at the end of year 16? Find FV.
A project costs €40 000 today and returns €10 500, €10 000, €13 000, €12 000, €15 000, €17 000 at the end of years 1 to 6. The cost of capital is 8%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €20 000 and returns €13 000, €9 500, €12 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 matures in 5 years and pays annual coupons. It sells for €1 203 when the yield to maturity is 3.5%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 5% matures in 8 years and sells for €1 105. Find its yield to maturity YTM in percent.
A company has a debt-to-equity ratio (at market values) of 1.5. Its cost of equity is 14%, its pre-tax cost of debt is 7% and the tax rate is 21%. Find the WACC.
A company has 5 000 000 shares at €20.50 each and 20 000 bonds trading at €960 each. The pre-tax cost of debt is 5% and the tax rate is 30%. The share’s beta is 0.8, the risk-free rate 3% and the market return 8.5%. Find the WACC.
A company sells 3 000 units at €21 each. The variable cost is €6 per unit and fixed costs are €18 000. Find the degree of operating leverage DOL at this sales level.
A company sells 800 units at €17 each, with a variable cost of €15 per unit and fixed costs of €1 000. By what percentage %ΔΠ does operating profit change if sales volume rises by 5%?
Answer: PV = 41782 €
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 = 62101 €
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: DPB = 4.251 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 (3.542 years).
Answer: IRR = 33.92 %
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 = 7.996 %
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: YTM = 3.474 %
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 = 8.918 %
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 = 6.785 %
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: DOL = 1.667
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 = 13.33 %
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 = 840·2 − 1 000 = €680.00.
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