Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €5 000 at 5% per year with continuous compounding. Find the value FV after 20 years.
You want €20 000 in 19 years. You save the same amount at the end of every month in an account paying 3% per year, compounded monthly. Find the monthly deposit A.
An investment of €20 000 gives these cash flows at the end of years 1 to 4: €5 000, €5 000, €7 000, €8 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €10 000 and returns €5 500, €6 500 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 4% annual coupon in two half-yearly payments and matures in 7 years. The yield to maturity is 8% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 3% matures in 9 years and sells for €825. Find its yield to maturity YTM in percent.
A share with a beta of 0.9 has a required return of 7%. The risk-free rate is 2.5%. What expected market return rm does CAPM imply?
A company's shares have a beta of 1.5 at its current debt-to-equity ratio of 0.5. It plans to raise its debt-to-equity ratio to 0.6. The tax rate is 20%, the risk-free rate 3% and the market risk premium 6%. Find the cost of equity re after the change.
A product can be made in two ways. Method A has fixed costs of €10 000 per year and a variable cost of €8 per unit. Method B, more automated, has fixed costs of €30 000 per year and a variable cost of €4 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €42 per unit, with a variable cost of €30 per unit and fixed costs of €48 000 per year. Budgeted sales are 5 600 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 13591 €
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: A = 65.19 €
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: NPV = −599 €
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: IRR = 12.68 %
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 = 788.7 €
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 = 5.519 %
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: rm = 7.5 %
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 = 12.51 %
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 = 5000 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).
Below this volume method A is cheaper; above it the automated method B is cheaper.
Answer: MS = 28.57 %
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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