Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €2 000 at a nominal rate of 3.6% per year, compounded quarterly. Find the value FV of the investment after 8 years.
You want €20 000 in 8 years. You save the same amount at the end of every month in an account paying 4.8% per year, compounded monthly. Find the monthly deposit A.
A machine costs €80 000 today. It generates €15 000 at the end of each year for 6 years and can then be sold for €8 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €150 000 today. Its cash flow is €19 500 at the end of year 1 and grows by 3% per year until the last cash flow at the end of year 7. The cost of capital is 12%. Find the net present value NPV.
A bond with a face value of €1 000 pays an annual coupon of 2% and matures in 7 years. Its yield to maturity changes from 9% to 8.5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 and an annual coupon of 6% matures in 6 years and sells for €950. Find its yield to maturity YTM in percent.
A company has equity worth €3 000 000 and debt worth €4 000 000. Its cost of equity is 9%, its pre-tax cost of debt is 3% and the tax rate is 20%. Find the WACC.
A company has 5 000 000 shares trading at €41.00; the book value of its equity is €102 500 000. It just paid a dividend of €1.80 per share, and dividends grow by 5% per year. Its debt has a market value of €164 000 000 and a pre-tax cost of 5%. The tax rate is 25%. Find the WACC, using the dividend growth model for the cost of equity.
A product can be made in two ways. Method A has fixed costs of €20 000 per year and a variable cost of €8 per unit. Method B, more automated, has fixed costs of €32 000 per year and a variable cost of €2 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €9 per unit, with a variable cost of €6 per unit and fixed costs of €1 500 per year. Budgeted sales are 800 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 2664 €
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 = 171.3 €
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.
Answer: NPV = −10155 €
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.
Answer: NPV = −53872 €
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: ΔP = 19.6 €
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: YTM = 7.051 %
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 = 5.229 %
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 = 7.005 %
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 = 2000 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 = 37.5 %
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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