Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
A credit card charges a nominal interest rate of 22% per year, compounded monthly. Find the effective annual rate EAR in percent.
You deposit €5 000 at the start of every year for 5 years, the first deposit today. The account pays 7% per year. How much is in the account at the end of year 5? Find FV.
An investment of €75 000 gives these cash flows at the end of years 1 to 3: €16 000, €28 000, €15 000. The cost of capital is 9%. Find the net present value NPV.
A project costs €40 000 and returns €20 000, €22 500, €29 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
You buy a bond with 5 years to maturity (face value €1 000, annual coupon 2%) when its yield to maturity is 5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 4.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 3% matures in 7 years and sells for €1 135. Find its yield to maturity YTM in percent.
A company has equity worth €6 000 000 and debt worth €6 000 000. Its cost of equity is 13%, its pre-tax cost of debt is 3.5% and the tax rate is 30%. Find the WACC.
A company has 2 000 000 shares at €10.00 each and 5 000 bonds trading at €960 each. The pre-tax cost of debt is 3% and the tax rate is 30%. The share’s beta is 0.6, the risk-free rate 2% and the market return 8%. Find the WACC.
A product sells for €27 with a variable cost of €12 per unit. Fixed costs are €22 500 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 800 units at €47 each, with a variable cost of €45 per unit and fixed costs of €1 000. By what percentage %ΔΠ does operating profit change if sales volume rises by 10%?
Answer: EAR = 24.36 %
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 EAR is always above the nominal rate when interest compounds more than once a year.
Answer: FV = 30766 €
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.
Each deposit is invested one year longer.
Answer: NPV = −25171 €
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 = 33.14 %
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: HPR = 6.918 %
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: YTM = 0.994 %
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: WACC = 7.725 %
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 = 4.923 %
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 = 1900 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 = 26.67 %
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 = 880·2 − 1 000 = €760.00.
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