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 15% per year, compounded quarterly. Find the effective annual rate EAR in percent.
You save €250 at the end of every month for 19 years. The account pays 4.8% per year, compounded monthly. How much have you saved at the end? Find FV.
You buy a bond with 11 years to maturity (face value €1 000, annual coupon 7%) when its yield to maturity is 3.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 5%. Find your holding period return HPR in percent.
A bond with 15 years to maturity (face value €1 000, annual coupon 7%) sells for €1 208. The issuer can call it in 3 years at a call price of €1 070. Find the yield to call YTC in percent.
A company is financed with ordinary shares worth €8 000 000, preferred shares worth €500 000 and debt worth €2 000 000. The cost of equity is 12%, the cost of preferred stock 6% and the pre-tax cost of debt 5%. The tax rate is 25%. Find the WACC.
A company's shares have a beta of 1.1 at its current debt-to-equity ratio of 0.5. It plans to raise its debt-to-equity ratio to 1.25. The tax rate is 25%, the risk-free rate 3.5% and the market risk premium 6%. Find the cost of equity re after the change.
A machine costs €40 000 today, lasts 4 years and costs €3 000 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 €200 000 today. Its cash flow is €27 500 at the end of year 1 and grows by 3% per year until the last cash flow at the end of year 8. The cost of capital is 8%. Find the net present value NPV.
Answer: EAR = 15.87 %
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 = 92798 €
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 €57 000 yourself; the rest is interest.
Answer: HPR = −6.891 %
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: YTC = 2.064 %
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 4.997%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 10.14 %
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 = 12.8 %
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 = 14280 €
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 = −26418 €
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.
ExamProblems · examproblems.com/courses/investments-and-valuation