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 want €75 000 in 9 years. You save the same amount at the end of every month in an account paying 7.2% per year, compounded monthly. Find the monthly deposit A.
An investment of €45 000 gives these cash flows at the end of years 1 to 4: €22 000, €20 000, €16 000, €12 000. The cost of capital is 8%. Find the net present value NPV.
Project A costs €10 000 and returns €5 500, €4 000, €2 500 at the end of years 1 to 3. Project B costs €15 000 and returns €4 000, €6 000, €8 000. Find the crossover rate r* at which both projects have the same NPV, in percent.
You buy a bond with 3 years to maturity (face value €1 000, annual coupon 2%) when its yield to maturity is 8%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 7.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 4% matures in 10 years and sells for €855. Find its yield to maturity YTM in percent.
A company has equity worth €6 000 000 and debt worth €1 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 7% and the tax rate is 21%. Find the WACC.
A company has 2 000 000 shares at €23.50 each and 10 000 bonds trading at €1 000 each. The pre-tax cost of debt is 4% and the tax rate is 20%. The share’s beta is 0.6, the risk-free rate 1% and the market return 7.5%. Find the WACC.
A product sells for €24 per unit with a variable cost of €12 per unit. Fixed costs are €24 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A company sells two products. Product A sells for €16 with a variable cost of €10; product B sells for €30 with a variable cost of €15. They sell in a fixed mix of 2 units of A for every 3 of B. Fixed costs are €11 400 per month. Find the total number of units Q (A and B together) needed to break even.
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: A = 495.6 €
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: NPV = 14039 €
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: r* = 5.886 %
Crossover rate: the discount rate at which two projects have the same NPV. It is the IRR of the difference in their cash flows (B − A). Below it the project with the later, larger cash flows has the higher NPV; above it the other one wins.
Check: at 5.886% both NPVs are €867.70.
Answer: HPR = 8.975 %
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 = 5.967 %
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 = 9.361 %
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 = 4.602 %
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: Q = 1000 units
Break-even with a sales mix: when products sell in a fixed ratio, use the weighted average contribution margin: weight each product's p − v by its share of units. Break-even units = F/(weighted contribution margin), split again in the mix ratio.
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