Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €3 000 at the end of each year for 10 years. The interest rate is 3% per year. Find the present value PV of the annuity.
You deposit €2 500 at the start of every year for 10 years, the first deposit today. The account pays 6% per year. How much is in the account at the end of year 10? Find FV.
An investment of €30 000 gives these cash flows at the end of years 1 to 4: €13 000, €11 000, €9 000, €8 000. The cost of capital is 6%. Find the net present value NPV.
A project costs €10 000 and returns €7 500, €5 000 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
You buy a bond with 7 years to maturity (face value €1 000, annual coupon 3%) when its yield to maturity is 6%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 5.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 4% matures in 6 years and sells for €880. Find its yield to maturity YTM in percent.
A company is financed with ordinary shares worth €4 000 000, preferred shares worth €2 000 000 and debt worth €5 000 000. The cost of equity is 12%, the cost of preferred stock 6.5% and the pre-tax cost of debt 5%. The tax rate is 25%. Find the WACC.
A company has equity with a market value of €10 000 000 and a cost of equity of 12%. Its debt consists of 2 000 bonds (face value €1 000, annual coupon 4%, 8 years to maturity) that trade at €967 each. The tax rate is 25.8%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A product sells for €16 per unit with a variable cost of €4 per unit. Fixed costs are €30 000 per month. How many units Q must be sold per month to make a profit of €12 000?
A company sells 3 000 units per month at €30 each, with a variable cost of €20 per unit and fixed costs of €12 000 per month. The variable cost rises by €2 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 = 25591 €
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 = 34929 €
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 = 5947 €
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: IRR = 17.54 %
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 = 8.718 %
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 = 6.477 %
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 = 7.25 %
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 = 10.6 %
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 = 3500 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 = 3750 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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