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 17.5% per year, compounded daily. Find the effective annual rate EAR in percent.
A loan of €25 000 is repaid with equal monthly payments over 5 years. The interest rate is 3% per year, compounded monthly. Find the monthly payment A.
An investment of €75 000 gives these cash flows at the end of years 1 to 4: €23 000, €30 000, €15 000, €26 000. The cost of capital is 6%. Find the net present value NPV.
A project costs €20 000 and returns €13 500, €12 000, €12 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 matures in 5 years and pays annual coupons. It sells for €1 116 when the yield to maturity is 2.5%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 8% matures in 5 years and sells for €925. Find its yield to maturity YTM in percent.
A share with a beta of 1.2 has a required return of 6.8%. The risk-free rate is 2%. What expected market return rm does CAPM imply?
A company has 5 000 000 shares at €31.00 each and 5 000 bonds trading at €980 each. The pre-tax cost of debt is 6% and the tax rate is 21%. The share’s beta is 0.9, the risk-free rate 2% and the market return 7%. Find the WACC.
A company sells 1 800 units at €22 each. The variable cost is €18 per unit and fixed costs are €4 800. Find the degree of operating leverage DOL at this sales level.
A company sells two products. Product A sells for €18 with a variable cost of €12; product B sells for €28 with a variable cost of €20. They sell in a fixed mix of 2 units of A for every 1 of B. Fixed costs are €12 000 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: EAR = 19.12 %
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 = 449.2 €
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 present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.
Answer: NPV = 6587 €
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 positive, so the project earns more than the required return.
Answer: IRR = 40.56 %
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 falls as the rate rises, so the IRR lies between a rate with positive NPV and one with negative NPV.
Linear interpolation gives a first estimate; repeating it (or a calculator's IRR function) gives the exact rate.
Answer: c = 4.997 %
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 = 9.977 %
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: rm = 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 = 6.446 %
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 = 3
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 = 1800 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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