Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
An annuity pays €1 000 at the end of each year for 20 years. The interest rate is 10% per year. Find the present value PV of the annuity.
A loan of €20 000 is repaid with equal monthly payments over 3 years at 5.4% per year, compounded monthly. How much interest I is paid in total over the life of the loan?
A bond with a face value of €1 000 pays a 4.5% annual coupon in two half-yearly payments and matures in 9 years. The yield to maturity is 5% per year. Find the price P of the bond.
A bond with 13 years to maturity (face value €1 000, annual coupon 5%) sells for €1 275. The issuer can call it in 4 years at a call price of €1 050. Find the yield to call YTC in percent.
A share with a beta of 2 has a required return of 18%. The risk-free rate is 4%. What expected market return rm does CAPM imply?
A company has 1 000 000 shares trading at €25.50; the book value of its equity is €7 700 000. It just paid a dividend of €0.50 per share, and dividends grow by 4% per year. Its debt has a market value of €7 700 000 and a pre-tax cost of 6%. The tax rate is 21%. Find the WACC, using the dividend growth model for the cost of equity.
A machine costs €60 000 today, lasts 7 years and costs €1 500 per year to run (paid at the end of each year). The cost of capital is 5%. Find its equivalent annual cost EAC.
A project costs €50 000 and returns €25 500, €33 500 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
Answer: PV = 8514 €
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: I = 1709 €
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: P = 964.1 €
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 above the coupon rate, so the bond sells at a discount.
Answer: YTC = 50 %
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 to maturity is 2.497%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: rm = 11 %
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 = 5.738 %
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 = 11869 €
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 = 11.23 %
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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