Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €25 000 in 10 years. An account pays 3% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
A loan of €25 000 is repaid with equal monthly payments over 6 years. The interest rate is 4.8% per year, compounded monthly. Find the monthly payment A.
A machine costs €60 000 today, lasts 4 years and costs €5 000 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 €40 000 and returns €18 000, €22 000, €17 500 at the end of years 1 to 3. Cash flows can be reinvested at the cost of capital of 6%. Find the modified internal rate of return MIRR in percent.
A bond with a face value of €1 000 pays a 3% annual coupon in two half-yearly payments and matures in 4 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 5% matures in 3 years and sells for €935. Find its yield to maturity YTM in percent.
A company has equity worth €4 000 000 and debt worth €5 000 000. Its WACC is 5.75%, its pre-tax cost of debt is 5% and the tax rate is 21%. Find the cost of equity re.
A company has equity with a market value of €5 000 000 and a cost of equity of 9%. Its debt consists of 5 000 bonds (face value €1 000, annual coupon 5%, 9 years to maturity) that trade at €965 each. The tax rate is 28%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A product sells for €32 per unit with a variable cost of €12 per unit. Fixed costs are €100 000 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 €14 with a variable cost of €8; product B sells for €38 with a variable cost of €30. They sell in a fixed mix of 4 units of A for every 1 of B. Fixed costs are €32 000 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 18541 €
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 = 400.3 €
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.
The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.
Answer: EAC = 24754 €
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 = 15.13 %
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 = 831.7 €
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 = 7.499 %
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: re = 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: WACC = 6.526 %
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 = 5100 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 = 5000 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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