Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You invest €5 000 at 7% per year with continuous compounding. Find the value FV after 6 years.
A loan of €8 000 is repaid with equal monthly payments over 4 years. The interest rate is 5.4% per year, compounded monthly. Find the monthly payment A.
A bond with a face value of €1 000 pays a 4.5% annual coupon in two half-yearly payments and matures in 3 years. The yield to maturity is 4% per year. Find the price P of the bond.
A bond with 17 years to maturity (face value €1 000, annual coupon 7%) sells for €1 225. The issuer can call it in 5 years at a call price of €1 070. Find the yield to call YTC in percent.
A share with a beta of 1.25 has a required return of 8.38%. The risk-free rate is 1.5%. What expected market return rm does CAPM imply?
A company has 5 000 000 shares trading at €46.50; the book value of its equity is €116 300 000. It just paid a dividend of €1.60 per share, and dividends grow by 4% per year. Its debt has a market value of €116 300 000 and a pre-tax cost of 6%. The tax rate is 20%. Find the WACC, using the dividend growth model for the cost of equity.
A machine costs €50 000 today, lasts 6 years and costs €5 000 per year to run (paid at the end of each year). The cost of capital is 10%. Find its equivalent annual cost EAC.
An investment of €40 000 returns €14 000 at the end of each year for 3 years. Find its internal rate of return IRR in percent.
Answer: FV = 7610 €
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: A = 185.7 €
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.
The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.
Answer: P = 1014 €
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: YTC = 3.348 %
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 5.004%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: rm = 7.004 %
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 = 6.652 %
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 = 16480 €
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 = 2.48 %
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.
The annuity factor falls as the rate rises; bracket the rate and narrow down.
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