Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €8 000 at a nominal rate of 5% per year, compounded monthly. Find the value FV of the investment after 2 years.
A loan of €15 000 is repaid with equal monthly payments over 6 years. The interest rate is 3.6% 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 3: €10 000, €11 000, €8 000. The cost of capital is 6%. Find the net present value NPV.
A project costs €50 000 today. Its cash flow is €7 500 at the end of year 1 and grows by 2% per year until the last cash flow at the end of year 8. The cost of capital is 12%. Find the net present value NPV.
You buy a bond with 12 years to maturity (face value €1 000, annual coupon 4%) when its yield to maturity is 5.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 7%. Find your holding period return HPR in percent.
A bond with 16 years to maturity (face value €1 000, annual coupon 9%) sells for €1 089. The issuer can call it in 3 years at a call price of €1 090. Find the yield to call YTC in percent.
A company has equity worth €4 000 000 and debt worth €6 000 000. Its cost of equity is 12%, its pre-tax cost of debt is 5% 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 1. The tax rate is 30%, the risk-free rate is 1.5% and the market risk premium is 6%. Find the cost of equity re at the new capital structure.
A product sells for €23 per unit with a variable cost of €18 per unit. Fixed costs are €6 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A company sells 2 250 units at €38 each, with a variable cost of €30 per unit and fixed costs of €12 000. By what percentage %ΔΠ does operating profit change if sales volume rises by 8%?
Answer: FV = 8840 €
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 = 232 €
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 = −4059 €
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: NPV = −10491 €
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: HPR = −6.395 %
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 = 8.293 %
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 7.995%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 7.026 %
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 = 13.74 %
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 = 2400 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: %ΔΠ = 24 %
Operating leverage: fixed costs make profit react more strongly than sales. The degree of operating leverage DOL = contribution margin/operating profit tells you that a 1% change in sales changes profit by DOL %. It is very high just above break-even and falls as sales grow.
Check: new profit = 2 430·8 − 12 000 = €7 440.00.
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