Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €20 000 in 2 years. An account pays 3.6% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
You deposit €5 000 at the start of every year for 19 years, the first deposit today. The account pays 5% per year. How much is in the account at the end of year 19? Find FV.
An investment of €20 000 gives these cash flows at the end of years 1 to 3: €7 000, €10 000, €9 000. The cost of capital is 7.000000000000001%. Find the net present value NPV.
A project costs €50 000 today. Its cash flow is €5 500 at the end of year 1 and grows by 5% per year until the last cash flow at the end of year 9. The cost of capital is 10%. Find the net present value NPV.
A bond with a face value of €1 000 matures in 13 years and pays annual coupons. It sells for €763 when the yield to maturity is 8%. Find the coupon rate c in percent.
A bond with a face value of €1 000 pays an annual coupon of 5% and matures in 4 years. The yield to maturity is 2.5%. Find the Macaulay duration D in years.
A company has equity worth €4 000 000 and debt worth €5 000 000. Its cost of equity is 12%, its pre-tax cost of debt is 7% and the tax rate is 28%. Find the WACC.
A company has 5 000 000 shares at €29.00 each and 5 000 bonds trading at €1 020 each. The pre-tax cost of debt is 3.5% and the tax rate is 21%. The share’s beta is 0.8, the risk-free rate 1% and the market return 5%. Find the WACC.
A product sells for €35 with a variable cost of €30 per unit. Fixed costs are €5 000 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €2 500 after tax?
A company sells 1 600 units per month at €16 each, with a variable cost of €8 per unit and fixed costs of €6 400 per month. The variable cost rises by €3 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?
Answer: PV = 18617 €
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 = 160330 €
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 = 2623 €
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 = −12371 €
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: c = 5.001 %
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: D = 3.737 years
Duration: the Macaulay duration is the weighted average time at which a bond pays its cash flows, with the present values of the cash flows as weights: D = Σ t·PVₜ/P. The modified duration D/(1 + y) estimates the percentage price change for a small change in yield: %ΔP ≈ −Dmod·Δy.
Answer: WACC = 8.133 %
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 = 4.151 %
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 = 1625 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 = 2560 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).
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