Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €2 000 at a nominal rate of 4% per year, compounded quarterly. Find the value FV of the investment after 7 years.
You save €50 at the end of every month for 22 years. The account pays 2.4% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €75 000 gives these cash flows at the end of years 1 to 4: €27 000, €18 000, €20 000, €31 000. The cost of capital is 5%. Find the net present value NPV.
A project costs €50 000 today. Its cash flow is €4 500 at the end of year 1 and grows by 4% per year until the last cash flow at the end of year 10. The cost of capital is 8%. Find the net present value NPV.
A bond with a face value of €1 000 matures in 6 years and pays annual coupons. It sells for €1 138 when the yield to maturity is 2.5%. Find the coupon rate c in percent.
A bond with 10 years to maturity (face value €1 000, annual coupon 8%) sells for €1 147. The issuer can call it in 4 years at a call price of €1 080. Find the yield to call YTC in percent.
A company has equity worth €5 000 000 and debt worth €4 000 000. Its cost of equity is 12%, its pre-tax cost of debt is 4% and the tax rate is 28%. Find the WACC.
A company's business has an unlevered (asset) beta of 1.1. It plans a debt-to-equity ratio of 0.4. The tax rate is 25%, the risk-free rate is 3% and the market risk premium is 7%. Find the cost of equity re at the new capital structure.
A company sells 4 000 units at €26 each. The variable cost is €6 per unit and fixed costs are €40 000. Find the degree of operating leverage DOL at this sales level.
A company sells 2 400 units at €33 each, with a variable cost of €30 per unit and fixed costs of €4 500. By what percentage %ΔΠ does operating profit change if sales volume rises by 15%?
Answer: FV = 2643 €
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 = 17366 €
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.
You paid in €13 200 yourself; the rest is interest.
Answer: NPV = 9821 €
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: NPV = −14634 €
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: c = 5.005 %
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 = 5.632 %
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 6.003%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 7.947 %
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 = 13.01 %
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
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: %ΔΠ = 40 %
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 = 2 760·3 − 4 500 = €3 780.00.
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