Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €3 000 at a nominal rate of 3.6% per year, compounded monthly. Find the value FV of the investment after 5 years.
You borrow €20 000 at 2.4% per year, compounded monthly, and repay €375 at the end of every month. How many months does it take until the loan is repaid (the last payment may be smaller)? Give the number of payments N.
An investment of €45 000 gives these cash flows at the end of years 1 to 4: €11 000, €19 000, €19 000, €18 000. The cost of capital is 7.000000000000001%. Find the net present value NPV.
A project costs €10 000 and returns €5 000, €5 500, €7 000 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 a 6% annual coupon in two half-yearly payments and matures in 12 years. The yield to maturity is 4% per year. Find the price P of the bond.
A bond with 14 years to maturity (face value €1 000, annual coupon 9%) sells for €1 225. The issuer can call it in 3 years at a call price of €1 090. Find the yield to call YTC in percent.
A share with a beta of 1.25 has a required return of 8.5%. The risk-free rate is 1%. What expected market return rm does CAPM imply?
A company has 2 000 000 shares at €35.50 each and 40 000 bonds trading at €1 000 each. The pre-tax cost of debt is 6% and the tax rate is 28%. The share’s beta is 1.5, the risk-free rate 2% and the market return 7%. Find the WACC.
A product sells for €21 with a variable cost of €15 per unit. Fixed costs are €9 000 per month and profit is taxed at 25%. How many units Q must be sold per month to earn €4 500 after tax?
A company sells 4 000 units at €32 each, with a variable cost of €30 per unit and fixed costs of €4 000. By what percentage %ΔΠ does operating profit change if sales volume falls by 12%?
Answer: FV = 3591 €
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: N = 57 payments
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.
56 full payments leave a small balance, which the last, smaller payment clears.
Answer: NPV = 11117 €
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 = 31.92 %
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: P = 1189 €
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: YTC = 3.81 %
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 6.503%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: rm = 7 %
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: WACC = 7.633 %
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: Q = 2500 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 = 3 520·2 − 4 000 = €3 040.00.
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