Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €25 000 in 2 years. An account pays 5% 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 9 years. The account pays 3.6% per year, compounded monthly. How much have you saved at the end? Find FV.
A machine costs €30 000 today, lasts 4 years and costs €4 000 per year to run (paid at the end of each year). The cost of capital is 12%. Find its equivalent annual cost EAC.
A project costs €20 000 and returns €7 500, €7 500, €8 000, €6 000 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 4.5% annual coupon in two half-yearly payments and matures in 4 years. The yield to maturity is 3.5% per year. Find the price P of the bond.
A bond with a face value of €1 000 pays a 7% annual coupon in two half-yearly payments and matures in 4 years. It sells for €1 072. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company has equity worth €6 000 000 and debt worth €1 000 000. Its cost of equity is 11%, its pre-tax cost of debt is 4.5% and the tax rate is 28%. Find the WACC.
A company has 2 000 000 shares at €8.00 each and 5 000 bonds trading at €1 050 each. The pre-tax cost of debt is 6% and the tax rate is 25.8%. The share’s beta is 0.6, the risk-free rate 3.5% and the market return 10%. Find the WACC.
A product can be made in two ways. Method A has fixed costs of €20 000 per year and a variable cost of €25 per unit. Method B, more automated, has fixed costs of €30 000 per year and a variable cost of €23 per unit. At what yearly volume Q do both methods cost the same?
A company sells two products. Product A sells for €13 with a variable cost of €8; product B sells for €18 with a variable cost of €10. They sell in a fixed mix of 3 units of A for every 2 of B. Fixed costs are €12 400 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 22635 €
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 = 19099 €
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 €16 200 yourself; the rest is interest.
Answer: EAC = 13877 €
Equivalent annual cost: spread the present value of all costs over the life as an equal yearly amount (an annuity): EAC = PV·r/(1 − (1 + r)⁻ⁿ). It lets you compare machines with different lives.
Answer: MIRR = 14.07 %
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 = 1037 €
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 below the coupon rate, so the bond sells at a premium.
Answer: YTM = 4.992 %
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 = 9.891 %
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: WACC = 6.672 %
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: 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: 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.
ExamProblems · examproblems.com/courses/corporate-finance