Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You need €50 000 in 8 years. An account pays 4.8% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
A loan of €15 000 is repaid with equal monthly payments over 5 years at 3.6% 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 7% annual coupon in two half-yearly payments and matures in 8 years. The yield to maturity is 5% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 6% matures in 10 years and sells for €865. Find its yield to maturity YTM in percent.
A company has a debt-to-equity ratio (at market values) of 1. Its cost of equity is 8%, its pre-tax cost of debt is 5% and the tax rate is 20%. Find the WACC.
A company's business has an unlevered (asset) beta of 1.1. It plans a debt-to-equity ratio of 0.5. The tax rate is 21%, the risk-free rate is 3% and the market risk premium is 6.5%. Find the cost of equity re at the new capital structure.
A machine costs €60 000 today, lasts 4 years and costs €1 000 per year to run (paid at the end of each year). The cost of capital is 12%. Find its equivalent annual cost EAC.
A project costs €80 000 today. Its cash flow is €9 000 at the end of year 1 and grows by 3% per year until the last cash flow at the end of year 9. The cost of capital is 8%. Find the net present value NPV.
Answer: PV = 34135 €
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 = 1413 €
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 = 1131 €
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 = 8.013 %
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 = 6 %
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: re = 12.97 %
Levered beta (Hamada): debt makes equity riskier. βL = βU·(1 + (1 − T)·D/E), where βU is the beta of the business without debt. To move a beta to another capital structure, unlever it first and then relever it at the new D/E.
Answer: EAC = 20754 €
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: NPV = −17488 €
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 negative, so the project earns less than the required return.
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