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 19.5% per year, compounded semi-annually. Find the effective annual rate EAR in percent.
You save €100 at the end of every month for 13 years. The account pays 7.2% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €45 000 gives these cash flows at the end of years 1 to 4: €12 000, €18 000, €17 000, €15 000. The cost of capital is 10%. Find the net present value NPV.
An investment of €20 000 returns €6 000 at the end of each year for 4 years. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays an annual coupon of 4% and matures in 15 years. Its yield to maturity changes from 6% to 8%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 and an annual coupon of 5% matures in 7 years and sells for €890. Find its yield to maturity YTM in percent.
A company has equity worth €8 000 000 and debt worth €3 000 000. Its cost of equity is 14%, its pre-tax cost of debt is 4% and the tax rate is 20%. Find the WACC.
A company has 5 000 000 shares at €10.00 each and 10 000 bonds trading at €1 020 each. The pre-tax cost of debt is 4% and the tax rate is 25.8%. The share’s beta is 1.3, the risk-free rate 2.5% and the market return 9%. Find the WACC.
A product sells for €10 per unit with a variable cost of €6 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 company sells 2 000 units per month at €23 each, with a variable cost of €15 per unit and fixed costs of €6 400 per month. The variable cost rises by €3 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?
Answer: EAR = 20.45 %
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 = 25711 €
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 €15 600 yourself; the rest is interest.
Answer: NPV = 3803 €
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 = 7.714 %
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.
The annuity factor falls as the rate rises; bracket the rate and narrow down.
Answer: ΔP = −148.1 €
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 = 7.044 %
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 = 11.05 %
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 = 9.598 %
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 = 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: Q = 3200 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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