Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €15 000 in 10 years. An account pays 5% per year, compounded monthly. How much must you deposit today? Find the present value PV.
You save €300 at the end of every month for 12 years. The account pays 4.8% per year, compounded monthly. How much have you saved at the end? Find FV.
A project costs €30 000 today and returns €10 000 at the end of each year for 6 years. The cost of capital is 8%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €40 000 and returns €16 000, €26 500 at the end of years 1 to 2. 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 €1 075 when the yield to maturity is 5.5%. Find the coupon rate c in percent.
A bond with 11 years to maturity (face value €1 000, annual coupon 8%) sells for €1 202. The issuer can call it in 3 years at a call price of €1 080. Find the yield to call YTC in percent.
A company has equity worth €10 000 000 and debt worth €5 000 000. Its cost of equity is 9%, its pre-tax cost of debt is 4.5% and the tax rate is 28%. Find the WACC.
A company's shares have a beta of 0.9 at its current debt-to-equity ratio of 0.4. It plans to raise its debt-to-equity ratio to 1.25. The tax rate is 28%, the risk-free rate 3% and the market risk premium 4.5%. Find the cost of equity re after the change.
A product sells for €45 per unit with a variable cost of €25 per unit. Fixed costs are €16 000 per month. How many units Q must be sold per month to make a profit of €5 000?
A company sells 3 750 units per month at €65 each, with a variable cost of €45 per unit and fixed costs of €50 000 per month. The variable cost rises by €5 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 = 9107 €
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 = 58265 €
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.
You paid in €43 200 yourself; the rest is interest.
Answer: DPB = 3.575 years
Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.
The discounted payback is always longer than the simple payback (3 years).
Answer: IRR = 3.815 %
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: c = 7.001 %
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 = 3.384 %
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 5.503%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 7.08 %
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 = 8.974 %
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 = 1050 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 = 5000 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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