Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
A credit card charges a nominal interest rate of 22% per year, compounded monthly. Find the effective annual rate EAR in percent.
You save €50 at the end of every month for 25 years. The account pays 4.8% per year, compounded monthly. How much have you saved at the end? Find FV.
A bond with a face value of €1 000 matures in 15 years and pays annual coupons. It sells for €772 when the yield to maturity is 7%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 8% matures in 7 years and sells for €925. Find its yield to maturity YTM in percent.
A company has equity worth €5 000 000 and debt worth €1 000 000. Its cost of equity is 10%, 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 0.25. The tax rate is 21%, the risk-free rate is 3% and the market risk premium is 5.5%. Find the cost of equity re at the new capital structure.
An investment of €45 000 gives these cash flows at the end of years 1 to 3: €10 000, €14 000, €17 000. The cost of capital is 5%. Find the net present value NPV.
A project costs €80 000 today. Its cash flow is €20 000 at the end of year 1 and grows by 2% per year until the last cash flow at the end of year 4. The cost of capital is 8%. Find the net present value NPV.
Answer: EAR = 24.36 %
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 EAR is always above the nominal rate when interest compounds more than once a year.
Answer: FV = 28902 €
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 €15 000 yourself; the rest is interest.
Answer: c = 4.497 %
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 = 9.516 %
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 %
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 = 10.24 %
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: NPV = −8093 €
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.
Answer: NPV = −11873 €
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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