Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
A credit card charges a nominal interest rate of 21% per year, compounded daily. Find the effective annual rate EAR in percent.
You deposit €2 000 at the start of every year for 29 years, the first deposit today. The account pays 8% per year. How much is in the account at the end of year 29? Find FV.
An investment of €45 000 gives these cash flows at the end of years 1 to 3: €14 000, €10 000, €10 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €100 000 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 10. The cost of capital is 12%. Find the net present value NPV.
You buy a bond with 7 years to maturity (face value €1 000, annual coupon 6%) when its yield to maturity is 5%. 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 3% matures in 4 years and sells for €1 060. Find its yield to maturity YTM in percent.
A company is financed with ordinary shares worth €12 000 000, preferred shares worth €2 000 000 and debt worth €6 000 000. The cost of equity is 11%, the cost of preferred stock 6.5% and the pre-tax cost of debt 6%. The tax rate is 25%. Find the WACC.
A company has equity with a market value of €5 000 000 and a cost of equity of 11%. Its debt consists of 8 000 bonds (face value €1 000, annual coupon 7%, 10 years to maturity) that trade at €1 036 each. The tax rate is 21%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A company sells 6 000 units at €8 each. The variable cost is €6 per unit and fixed costs are €4 000. Find the degree of operating leverage DOL at this sales level.
A company sells two products. Product A sells for €17 with a variable cost of €12; product B sells for €30 with a variable cost of €10. They sell in a fixed mix of 3 units of A for every 2 of B. Fixed costs are €11 000 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: EAR = 23.36 %
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 = 224566 €
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.
Each deposit is invested one year longer.
Answer: NPV = −16495 €
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 = −43271 €
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: HPR = −2.086 %
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 = 1.445 %
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 = 8.6 %
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: WACC = 7.342 %
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: DOL = 1.5
Operating leverage: fixed costs make profit react more strongly than sales. The degree of operating leverage DOL = contribution margin/operating profit tells you that a 1% change in sales changes profit by DOL %. It is very high just above break-even and falls as sales grow.
Answer: Q = 1000 units
Break-even with a sales mix: when products sell in a fixed ratio, use the weighted average contribution margin: weight each product's p − v by its share of units. Break-even units = F/(weighted contribution margin), split again in the mix ratio.
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