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 quarterly. Find the effective annual rate EAR in percent.
You deposit €1 000 at the start of every year for 13 years, the first deposit today. The account pays 8% per year. How much is in the account at the end of year 13? Find FV.
A machine costs €80 000 today, lasts 9 years and costs €2 000 per year to run (paid at the end of each year). The cost of capital is 6%. Find its equivalent annual cost EAC.
A project costs €20 000 and returns €6 500, €11 000, €8 500, €12 000 at the end of years 1 to 4. Cash flows can be reinvested at the cost of capital of 8%. Find the modified internal rate of return MIRR in percent.
A bond with a face value of €1 000 pays an annual coupon of 3% and matures in 11 years. Its yield to maturity changes from 9% to 9.5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 and an annual coupon of 6% matures in 10 years and sells for €1 075. Find its yield to maturity YTM in percent.
A share with a beta of 1.4 has a required return of 12.4%. The risk-free rate is 4%. What expected market return rm does CAPM imply?
A company has 4 000 000 shares at €12.50 each and 20 000 bonds trading at €1 000 each. The pre-tax cost of debt is 4% and the tax rate is 25%. The share’s beta is 1.8, the risk-free rate 4% and the market return 11%. Find the WACC.
A product sells for €18 with a variable cost of €8 per unit. Fixed costs are €5 000 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €1 000 after tax?
A company sells 8 000 units at €55 each, with a variable cost of €45 per unit and fixed costs of €50 000. By what percentage %ΔΠ does operating profit change if sales volume falls by 8%?
Answer: EAR = 18.68 %
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 = 23215 €
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: EAC = 13762 €
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: MIRR = 20.52 %
Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.
Answer: ΔP = −23.76 €
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 = 5.027 %
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 = 10 %
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 = 12.71 %
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: Q = 625 units
Cost-volume-profit: each unit sold earns a contribution margin, price minus variable cost per unit. Profit = Q·(p − v) − F, where F is the fixed cost. At the break-even point profit is zero, so Q = F/(p − v).
Answer: %ΔΠ = −21.33 %
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.
Check: new profit = 7 360·10 − 50 000 = €23 600.00.
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