Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
An annuity pays €8 000 at the end of each year for 20 years. The interest rate is 6% per year. Find the present value PV of the annuity.
You deposit €3 000 at the start of every year for 14 years, the first deposit today. The account pays 5% per year. How much is in the account at the end of year 14? Find FV.
A bond with a face value of €1 000 pays a 5% annual coupon in two half-yearly payments and matures in 9 years. The yield to maturity is 3.5% 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 9 years and sells for €925. Find its yield to maturity YTM in percent.
A share with a beta of 1.5 has a required return of 13.5%. The risk-free rate is 3%. What expected market return rm does CAPM imply?
A company has equity with a market value of €15 000 000 and a cost of equity of 13%. Its debt consists of 4 000 bonds (face value €1 000, annual coupon 6%, 5 years to maturity) that trade at €1 043 each. The tax rate is 25.8%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A machine costs €20 000 today, lasts 7 years and costs €1 000 per year to run (paid at the end of each year). The cost of capital is 8%. Find its equivalent annual cost EAC.
A project costs €25 000 and returns €18 500, €14 500 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
Answer: PV = 91759 €
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 = 61736 €
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: P = 1115 €
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 = 4.009 %
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: rm = 10 %
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 = 10.98 %
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: EAC = 4841 €
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: IRR = 21.67 %
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.
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