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 8 years.
A loan of €25 000 is repaid with equal monthly payments over 4 years at 4.2% per year, compounded monthly. How much interest I is paid in total over the life of the loan?
An investment of €75 000 gives these cash flows at the end of years 1 to 3: €18 000, €37 000, €17 000. The cost of capital is 8%. Find the net present value NPV.
Project A costs €30 000 and returns €16 500, €12 000, €6 500 at the end of years 1 to 3. Project B costs €35 000 and returns €9 500, €13 500, €18 000. Find the crossover rate r* at which both projects have the same NPV, in percent.
A bond with a face value of €1 000 pays an 8% annual coupon in two half-yearly payments and matures in 7 years. The yield to maturity is 2.5% per year. Find the price P of the bond.
A bond with a face value of €1 000 pays a 6% annual coupon in two half-yearly payments and matures in 4 years. It sells for €1 018. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company has equity worth €6 000 000 and debt worth €4 000 000. Its WACC is 7.18%, its pre-tax cost of debt is 6% and the tax rate is 25.8%. Find the cost of equity re.
A company has 2 500 000 shares at €40.50 each and 40 000 bonds trading at €1 020 each. The pre-tax cost of debt is 3.5% and the tax rate is 25%. The share’s beta is 1.3, the risk-free rate 1.5% and the market return 6.5%. Find the WACC.
A product sells for €32 per unit with a variable cost of €12 per unit. Fixed costs are €10 000 per month. How many units Q must be sold per month to make a profit of €10 000?
A company sells two products. Product A sells for €9 with a variable cost of €5; product B sells for €35 with a variable cost of €25. They sell in a fixed mix of 3 units of A for every 2 of B. Fixed costs are €9 600 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: FV = 4000 €
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: I = 2202 €
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 = −13117 €
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: r* = 3.536 %
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 3.536% both NPVs are €2 987.58.
Answer: P = 1351 €
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: YTM = 5.493 %
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: re = 8.999 %
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 = 6.456 %
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 = 1000 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 = 1500 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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