Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €15 000 in 6 years. An account pays 3% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
You save €150 at the end of every month for 29 years. The account pays 6% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €20 000 gives these cash flows at the end of years 1 to 4: €8 000, €5 000, €8 000, €5 000. The cost of capital is 9%. Find the net present value NPV.
A project costs €25 000 and returns €18 000, €17 000 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 matures in 3 years and pays annual coupons. It sells for €1 071 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 6 years and sells for €895. Find its yield to maturity YTM in percent.
A company has equity worth €2 000 000 and debt worth €4 000 000. Its cost of equity is 9%, its pre-tax cost of debt is 6% and the tax rate is 21%. Find the WACC.
A company's business has an unlevered (asset) beta of 0.7. It plans a debt-to-equity ratio of 0.6. The tax rate is 30%, the risk-free rate is 2% and the market risk premium is 7%. Find the cost of equity re at the new capital structure.
A product sells for €18 per unit with a variable cost of €8 per unit. Fixed costs are €40 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A company sells 6 000 units at €33 each, with a variable cost of €30 per unit and fixed costs of €12 000. By what percentage %ΔΠ does operating profit change if sales volume falls by 20%?
Answer: PV = 12537 €
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 = 140181 €
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 €52 200 yourself; the rest is interest.
Answer: NPV = 1267 €
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 = 25.98 %
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 = 4.986 %
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 = 10.44 %
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 = 6.16 %
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 = 8.958 %
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 = 4600 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 = −60 %
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 = 4 800·3 − 12 000 = €2 400.00.
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