Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €15 000 in 3 years. An account pays 4% per year, compounded monthly. How much must you deposit today? Find the present value PV.
A loan of €200 000 is repaid with equal monthly payments over 25 years. The interest rate is 4.2% per year, compounded monthly. Find the monthly payment A.
An investment of €75 000 gives these cash flows at the end of years 1 to 3: €23 000, €20 000, €23 000. The cost of capital is 6%. Find the net present value NPV.
A project costs €20 000 and returns €14 500, €7 500, €14 500 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 pays an annual coupon of 3% and matures in 4 years. Its yield to maturity changes from 8% to 6%. By how much ΔP does the bond price change?
A bond with 13 years to maturity (face value €1 000, annual coupon 7%) sells for €1 242. The issuer can call it in 6 years at a call price of €1 070. Find the yield to call YTC in percent.
A company has equity worth €3 000 000 and debt worth €1 000 000. Its cost of equity is 8%, its pre-tax cost of debt is 4.5% and the tax rate is 20%. Find the WACC.
A company's shares have a beta of 1.3 at its current debt-to-equity ratio of 0.25. It plans to raise its debt-to-equity ratio to 1.5. The tax rate is 25%, the risk-free rate 4% and the market risk premium 4%. Find the cost of equity re after the change.
A product sells for €12 per unit with a variable cost of €8 per unit. Fixed costs are €6 000 per month. How many units Q must be sold per month to make a profit of €5 000?
A product sells for €45 per unit, with a variable cost of €30 per unit and fixed costs of €18 000 per year. Budgeted sales are 1 920 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 13306 €
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 = 1078 €
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 = −16191 €
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 negative, so the project earns less than the required return.
Answer: IRR = 37.85 %
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: ΔP = 61.65 €
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: YTC = 3.523 %
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 to maturity is 4.501%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 6.9 %
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 = 13.31 %
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 = 2750 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: MS = 37.5 %
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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