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 daily. Find the effective annual rate EAR in percent.
You want €50 000 in 19 years. You save the same amount at the end of every month in an account paying 4.2% per year, compounded monthly. Find the monthly deposit A.
You buy a bond with 9 years to maturity (face value €1 000, annual coupon 4.5%) when its yield to maturity is 7%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 6.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 6% matures in 10 years and sells for €1 160. Find its yield to maturity YTM in percent.
A company has equity worth €12 000 000 and debt worth €5 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 4% and the tax rate is 28%. Find the WACC.
A company has 4 000 000 shares at €43.00 each and 10 000 bonds trading at €960 each. The pre-tax cost of debt is 3% and the tax rate is 21%. The share’s beta is 0.6, the risk-free rate 4% and the market return 9%. Find the WACC.
A project costs €50 000 today and returns €10 500, €16 000, €21 000, €16 000 at the end of years 1 to 4. The cost of capital is 6%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €150 000 today. Its cash flow is €23 500 at the end of year 1 and grows by 4% per year until the last cash flow at the end of year 7. The cost of capital is 12%. Find the net present value NPV.
Answer: EAR = 24.6 %
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: A = 143.7 €
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.
Answer: HPR = 10.29 %
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 = 4.025 %
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 = 7.906 %
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 = 6.755 %
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: DPB = 3.649 years
Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.
The discounted payback is always longer than the simple payback (3.156 years).
Answer: NPV = −31108 €
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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