Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €50 000 in 5 years. An account pays 4% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
You want €50 000 in 12 years. You save the same amount at the end of every month in an account paying 5.4% per year, compounded monthly. Find the monthly deposit A.
A machine costs €30 000 today. It generates €10 000 at the end of each year for 3 years and can then be sold for €4 500. The cost of capital is 10%. Find the net present value NPV.
A project costs €25 000 and returns €16 000, €12 000 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
You buy a bond with 6 years to maturity (face value €1 000, annual coupon 8%) when its yield to maturity is 7%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 8.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 pays an 8% annual coupon in two half-yearly payments and matures in 7 years. It sells for €1 144. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company has equity worth €10 000 000 and debt worth €3 000 000. Its cost of equity is 12%, its pre-tax cost of debt is 3.5% and the tax rate is 21%. Find the WACC.
A company has 4 000 000 shares trading at €16.00; the book value of its equity is €25 600 000. It just paid a dividend of €2.00 per share, and dividends grow by 2% per year. Its debt has a market value of €38 400 000 and a pre-tax cost of 6%. The tax rate is 25.8%. Find the WACC, using the dividend growth model for the cost of equity.
A product sells for €50 per unit with a variable cost of €45 per unit. Fixed costs are €25 000 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 €9 with a variable cost of €5; product B sells for €35 with a variable cost of €15. They sell in a fixed mix of 4 units of A for every 1 of B. Fixed costs are €10 800 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 40977 €
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 = 247.5 €
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 = −1751 €
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 = 8.315 %
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: HPR = 1.206 %
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 = 5.494 %
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 = 9.869 %
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 = 10.89 %
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 = 5400 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 = 1500 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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