Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €10 000 in 4 years. An account pays 4.8% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
You deposit €2 000 at the start of every year for 14 years, the first deposit today. The account pays 3% per year. How much is in the account at the end of year 14? Find FV.
An investment of €60 000 gives these cash flows at the end of years 1 to 4: €15 000, €23 000, €23 000, €24 000. The cost of capital is 9%. Find the net present value NPV.
An investment of €50 000 returns €13 500 at the end of each year for 8 years. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 matures in 6 years and pays annual coupons. It sells for €1 157 when the yield to maturity is 4%. Find the coupon rate c in percent.
A bond trades at €1 045 with a yield to maturity of 6% (annual compounding) and a Macaulay duration of 9.4 years. Use the modified duration to estimate the new price P if the yield falls by 75 basis points.
A company has equity worth €4 000 000 and debt worth €3 000 000. Its cost of equity is 12%, its pre-tax cost of debt is 7% and the tax rate is 20%. Find the WACC.
A company's business has an unlevered (asset) beta of 0.9. It plans a debt-to-equity ratio of 0.75. The tax rate is 28%, the risk-free rate is 3% and the market risk premium is 6.5%. Find the cost of equity re at the new capital structure.
A product sells for €20 per unit with a variable cost of €12 per unit. Fixed costs are €9 600 per month. How many units Q must be sold per month to make a profit of €6 000?
A company sells two products. Product A sells for €10 with a variable cost of €5; product B sells for €50 with a variable cost of €30. They sell in a fixed mix of 5 units of A for every 2 of B. Fixed costs are €65 000 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 8263 €
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 = 35198 €
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 = 7883 €
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 = 21.2 %
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.
The annuity factor falls as the rate rises; bracket the rate and narrow down.
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: P = 1115 €
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.
Duration is a linear estimate; for large yield changes convexity makes the true price a little higher.
Answer: WACC = 9.257 %
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: re = 12.01 %
Levered beta (Hamada): debt makes equity riskier. βL = βU·(1 + (1 − T)·D/E), where βU is the beta of the business without debt. To move a beta to another capital structure, unlever it first and then relever it at the new D/E.
Answer: Q = 1950 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 = 7000 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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