Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €3 000 at a nominal rate of 4% per year, compounded quarterly. Find the value FV of the investment after 5 years.
You save €250 at the end of every month for 9 years. The account pays 3.6% 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: €18 000, €35 000, €31 000, €21 000. The cost of capital is 12%. Find the net present value NPV.
A project costs €50 000 today. Its cash flow is €10 000 at the end of year 1 and grows by 5% per year until the last cash flow at the end of year 5. The cost of capital is 12%. Find the net present value NPV.
A bond with a face value of €1 000 matures in 7 years and pays annual coupons. It sells for €1 061 when the yield to maturity is 3.5%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 3% matures in 6 years and sells for €875. Find its yield to maturity YTM in percent.
A company is financed with ordinary shares worth €10 000 000, preferred shares worth €500 000 and debt worth €6 000 000. The cost of equity is 10%, the cost of preferred stock 6.5% and the pre-tax cost of debt 6%. The tax rate is 21%. Find the WACC.
A company's shares have a beta of 1.4 at its current debt-to-equity ratio of 0.25. It plans to raise its debt-to-equity ratio to 1.5. The tax rate is 25.8%, the risk-free rate 3.5% and the market risk premium 4%. Find the cost of equity re after the change.
A product sells for €24 with a variable cost of €4 per unit. Fixed costs are €40 000 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €12 000 after tax?
A company sells 3 750 units at €24 each, with a variable cost of €18 per unit and fixed costs of €9 000. By what percentage %ΔΠ does operating profit change if sales volume rises by 15%?
Answer: FV = 3661 €
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 = 31831 €
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 €27 000 yourself; the rest is interest.
Answer: NPV = 4384 €
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 = −10599 €
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 = 4.498 %
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 = 5.502 %
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 = 7.981 %
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 = 13.48 %
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: Q = 2750 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 = 25 %
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 = 4 313·6 − 9 000 = €16 875.00.
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