Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €20 000 at 2% per year with continuous compounding. Find the value FV after 17 years.
A loan of €8 000 is repaid with equal monthly payments over 4 years. The interest rate is 4.2% per year, compounded monthly. Find the monthly payment A.
An investment of €30 000 gives these cash flows at the end of years 1 to 4: €13 000, €9 000, €13 000, €10 000. The cost of capital is 5%. Find the net present value NPV.
An investment of €50 000 returns €8 500 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 pays a 7% annual coupon in two half-yearly payments and matures in 9 years. The yield to maturity is 3.5% per year. Find the price P of the bond.
A bond trades at €1 020 with a yield to maturity of 5% (annual compounding) and a Macaulay duration of 9.2 years. Use the modified duration to estimate the new price P if the yield falls by 150 basis points.
A company has equity worth €6 000 000 and debt worth €1 000 000. Its cost of equity is 9%, its pre-tax cost of debt is 3% and the tax rate is 25.8%. Find the WACC.
A company's business has an unlevered (asset) beta of 1.2. It plans a debt-to-equity ratio of 0.6. The tax rate is 25.8%, the risk-free rate is 1% and the market risk premium is 6%. Find the cost of equity re at the new capital structure.
A product sells for €40 per unit with a variable cost of €30 per unit. Fixed costs are €10 000 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 €40 with a variable cost of €20. They sell in a fixed mix of 3 units of A for every 2 of B. Fixed costs are €31 200 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: FV = 28099 €
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 = 181.3 €
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 = 10001 €
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 = 7.389 %
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: P = 1268 €
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.
The yield is below the coupon rate, so the bond sells at a premium.
Answer: P = 1154 €
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 = 8.032 %
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 = 11.41 %
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 = 1200 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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