Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €5 000 at 4% per year with continuous compounding. Find the value FV after 7 years.
You deposit €3 000 at the start of every year for 19 years, the first deposit today. The account pays 4% per year. How much is in the account at the end of year 19? Find FV.
An investment of €75 000 gives these cash flows at the end of years 1 to 3: €19 000, €28 000, €26 000. The cost of capital is 9%. Find the net present value NPV.
A project costs €25 000 and returns €9 500, €13 500, €12 500, €10 500 at the end of years 1 to 4. Cash flows can be reinvested at the cost of capital of 10%. Find the modified internal rate of return MIRR in percent.
A bond with a face value of €1 000 pays a 7% annual coupon in two half-yearly payments and matures in 9 years. The yield to maturity is 8% per year. Find the price P of the bond.
A bond with 14 years to maturity (face value €1 000, annual coupon 8%) sells for €1 186. The issuer can call it in 6 years at a call price of €1 080. Find the yield to call YTC in percent.
A company has a debt-to-equity ratio (at market values) of 0.5. Its cost of equity is 8%, its pre-tax cost of debt is 4.5% and the tax rate is 25.8%. Find the WACC.
A company's shares have a beta of 1 at its current debt-to-equity ratio of 0.25. It plans to raise its debt-to-equity ratio to 0.75. The tax rate is 20%, the risk-free rate 3% and the market risk premium 7%. Find the cost of equity re after the change.
A product sells for €34 per unit with a variable cost of €30 per unit. Fixed costs are €6 000 per month. How many units Q must be sold per month to make a profit of €5 000?
A product sells for €16 per unit, with a variable cost of €4 per unit and fixed costs of €30 000 per year. Budgeted sales are 4 500 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 6616 €
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 = 86334 €
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.
Each deposit is invested one year longer.
Answer: NPV = −13925 €
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: MIRR = 20.8 %
Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.
Answer: P = 936.7 €
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: YTC = 5.446 %
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 5.999%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 6.446 %
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 = 12.33 %
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: 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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