Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €15 000 in 9 years. An account pays 4% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
A loan of €8 000 is repaid with equal monthly payments over 3 years. The interest rate is 3.6% per year, compounded monthly. Find the monthly payment A.
A machine costs €60 000 today. It generates €13 000 at the end of each year for 5 years and can then be sold for €9 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €40 000 and returns €28 500, €15 000, €23 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 matures in 5 years and pays annual coupons. It sells for €922 when the yield to maturity is 9%. Find the coupon rate c in percent.
A bond with a face value of €1 000 pays an annual coupon of 6% and matures in 6 years. The yield to maturity is 7%. Find the Macaulay duration D in years.
A share with a beta of 0.8 has a required return of 6.7%. The risk-free rate is 3.5%. What expected market return rm does CAPM imply?
A company has 1 000 000 shares trading at €50.00; the book value of its equity is €15 000 000. It just paid a dividend of €2.50 per share, and dividends grow by 4% per year. Its debt has a market value of €50 000 000 and a pre-tax cost of 3%. The tax rate is 30%. Find the WACC, using the dividend growth model for the cost of equity.
A product sells for €14 with a variable cost of €12 per unit. Fixed costs are €10 000 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €4 000 after tax?
A company sells two products. Product A sells for €16 with a variable cost of €12; product B sells for €28 with a variable cost of €20. They sell in a fixed mix of 2 units of A for every 3 of B. Fixed costs are €19 200 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 10484 €
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 = 234.8 €
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: NPV = −5131 €
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: IRR = 32.39 %
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 falls as the rate rises, so the IRR lies between a rate with positive NPV and one with negative NPV.
Linear interpolation gives a first estimate; repeating it (or a calculator's IRR function) gives the exact rate.
Answer: c = 6.995 %
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 = 5.19 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: rm = 7.5 %
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 = 5.65 %
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 = 7500 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 = 3000 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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