Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €1 500 at the end of each year for 13 years. The interest rate is 10% per year. Find the present value PV of the annuity.
You save €200 at the end of every month for 28 years. The account pays 5.4% per year, compounded monthly. How much have you saved at the end? Find FV.
A project costs €20 000 today and returns €6 500, €7 000, €4 500, €7 500, €7 500 at the end of years 1 to 5. The cost of capital is 5%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €50 000 and returns €20 000, €35 000 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 pays a 5% annual coupon in two half-yearly payments and matures in 8 years. The yield to maturity is 9% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 7% matures in 9 years and sells for €910. Find its yield to maturity YTM in percent.
A company has equity worth €3 000 000 and debt worth €6 000 000. Its cost of equity is 9%, its pre-tax cost of debt is 4% and the tax rate is 21%. Find the WACC.
A company's business has an unlevered (asset) beta of 0.6. It plans a debt-to-equity ratio of 0.6. The tax rate is 30%, the risk-free rate is 1.5% and the market risk premium is 4.5%. Find the cost of equity re at the new capital structure.
A company sells 1 600 units at €8 each. The variable cost is €6 per unit and fixed costs are €1 600. Find the degree of operating leverage DOL at this sales level.
A product sells for €38 per unit, with a variable cost of €30 per unit and fixed costs of €8 000 per year. Budgeted sales are 1 800 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 10655 €
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 = 156464 €
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 €67 200 yourself; the rest is interest.
Answer: DPB = 3.579 years
Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.
The discounted payback is always longer than the simple payback (3.267 years).
Answer: IRR = 6.023 %
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: P = 775.3 €
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 is above the coupon rate, so the bond sells at a discount.
Answer: YTM = 8.469 %
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 = 5.107 %
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 = 5.334 %
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: MS = 44.44 %
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).
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