Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €1 000 at the end of each year for 14 years. The interest rate is 5% per year. Find the present value PV of the annuity.
You save €500 at the end of every month for 10 years. The account pays 3.6% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €60 000 gives these cash flows at the end of years 1 to 4: €20 000, €25 000, €28 000, €18 000. The cost of capital is 12%. Find the net present value NPV.
A project costs €80 000 today. Its cash flow is €12 000 at the end of year 1 and grows by 3% per year until the last cash flow at the end of year 8. The cost of capital is 8%. Find the net present value NPV.
A bond with a face value of €1 000 pays a 4% annual coupon in two half-yearly payments and matures in 6 years. The yield to maturity is 5% per year. Find the price P of the bond.
A bond with a face value of €1 000 pays a 4.5% annual coupon in two half-yearly payments and matures in 10 years. It sells for €762. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A share with a beta of 0.8 has a required return of 6.9%. The risk-free rate is 2.5%. What expected market return rm does CAPM imply?
A company's shares have a beta of 1.3 at its current debt-to-equity ratio of 0.4. It plans to raise its debt-to-equity ratio to 0.6. The tax rate is 20%, the risk-free rate 2.5% and the market risk premium 6.5%. Find the cost of equity re after the change.
A product sells for €45 per unit with a variable cost of €25 per unit. Fixed costs are €50 000 per month. How many units Q must be sold per month to make a profit of €2 000?
A company sells two products. Product A sells for €16 with a variable cost of €12; product B sells for €27 with a variable cost of €15. They sell in a fixed mix of 3 units of A for every 2 of B. Fixed costs are €14 400 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 9899 €
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 = 72093 €
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 €60 000 yourself; the rest is interest.
Answer: NPV = 9156 €
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 = −4255 €
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: P = 948.7 €
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 = 8.003 %
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 = 8 %
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 = 11.97 %
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 = 2600 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: Q = 2000 units
Break-even with a sales mix: when products sell in a fixed ratio, use the weighted average contribution margin: weight each product's p − v by its share of units. Break-even units = F/(weighted contribution margin), split again in the mix ratio.
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