Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €2 000 at the end of each year for 11 years. The interest rate is 8% per year. Find the present value PV of the annuity.
You save €50 at the end of every month for 16 years. The account pays 4.8% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €45 000 gives these cash flows at the end of years 1 to 4: €12 000, €20 000, €21 000, €13 000. The cost of capital is 6%. Find the net present value NPV.
A project costs €40 000 and returns €26 000, €27 000, €18 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
You buy a bond with 9 years to maturity (face value €1 000, annual coupon 7%) when its yield to maturity is 4%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 3%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 pays a 4% annual coupon in two half-yearly payments and matures in 10 years. It sells for €851. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company is financed with ordinary shares worth €6 000 000, preferred shares worth €1 500 000 and debt worth €2 000 000. The cost of equity is 12%, the cost of preferred stock 6% and the pre-tax cost of debt 5%. The tax rate is 28%. Find the WACC.
A company has equity with a market value of €12 000 000 and a cost of equity of 9%. Its debt consists of 5 000 bonds (face value €1 000, annual coupon 7%, 4 years to maturity) that trade at €1 035 each. The tax rate is 28%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A company sells 3 000 units at €31 each. The variable cost is €25 per unit and fixed costs are €6 000. Find the degree of operating leverage DOL at this sales level.
A company sells 1 600 units at €42 each, with a variable cost of €30 per unit and fixed costs of €12 000. By what percentage %ΔΠ does operating profit change if sales volume rises by 20%?
Answer: PV = 14278 €
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 = 14402 €
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 €9 600 yourself; the rest is interest.
Answer: NPV = 12050 €
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 positive, so the project earns more than the required return.
Answer: IRR = 38.12 %
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: HPR = 10.44 %
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 = 6.003 %
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 = 9.284 %
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 = 7.588 %
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 = 1.5
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 = 53.33 %
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 = 1 920·12 − 12 000 = €11 040.00.
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