Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €25 000 in 6 years. An account pays 2% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
You deposit €2 000 at the start of every year for 17 years, the first deposit today. The account pays 4% per year. How much is in the account at the end of year 17? Find FV.
A machine costs €60 000 today. It generates €16 000 at the end of each year for 4 years and can then be sold for €15 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €10 000 and returns €3 500, €7 000, €6 500 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 9 years and pays annual coupons. It sells for €1 117 when the yield to maturity is 3%. Find the coupon rate c in percent.
A bond with 15 years to maturity (face value €1 000, annual coupon 8%) sells for €1 194. The issuer can call it in 5 years at a call price of €1 080. Find the yield to call YTC in percent.
A company is financed with ordinary shares worth €10 000 000, preferred shares worth €2 000 000 and debt worth €5 000 000. The cost of equity is 12%, the cost of preferred stock 6% and the pre-tax cost of debt 6%. The tax rate is 30%. Find the WACC.
A company's shares have a beta of 1.2 at its current debt-to-equity ratio of 0.4. It plans to raise its debt-to-equity ratio to 0.75. The tax rate is 25.8%, the risk-free rate 4% and the market risk premium 7%. Find the cost of equity re after the change.
A company sells 800 units at €26 each. The variable cost is €6 per unit and fixed costs are €10 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €21 per unit, with a variable cost of €18 per unit and fixed costs of €15 000 per year. Budgeted sales are 5 750 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 22180 €
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: FV = 49291 €
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 = 963 €
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.
Answer: IRR = 28.67 %
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 = 4.503 %
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 = 4.971 %
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 6.002%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 9 %
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 = 14.08 %
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: DOL = 2.667
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: MS = 13.04 %
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).
ExamProblems · examproblems.com/courses/corporate-finance