Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €10 000 at 5% per year with continuous compounding. Find the value FV after 7 years.
You borrow €8 000 at 4.8% per year, compounded monthly, and repay €250 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 €75 000 gives these cash flows at the end of years 1 to 4: €34 000, €33 000, €30 000, €23 000. The cost of capital is 10%. Find the net present value NPV.
Project A costs €40 000 and returns €23 000, €16 000, €8 500 at the end of years 1 to 3. Project B costs €50 000 and returns €12 500, €21 000, €26 500. Find the crossover rate r* at which both projects have the same NPV, in percent.
A bond with a face value of €1 000 matures in 9 years and pays annual coupons. It sells for €1 234 when the yield to maturity is 3%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 5% matures in 6 years and sells for €925. 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 5% and the tax rate is 21%. Find the WACC.
A company has 5 000 000 shares at €9.00 each and 10 000 bonds trading at €1 050 each. The pre-tax cost of debt is 6% and the tax rate is 28%. The share’s beta is 0.9, the risk-free rate 1% and the market return 6.5%. Find the WACC.
A product sells for €19 per unit with a variable cost of €15 per unit. Fixed costs are €8 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A company sells 1 600 units per month at €45 each, with a variable cost of €30 per unit and fixed costs of €12 000 per month. The variable cost rises by €5 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?
Answer: FV = 14191 €
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 = 35 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.
34 full payments leave a small balance, which the last, smaller payment clears.
Answer: NPV = 21431 €
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* = 5.194 %
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 5.194% both NPVs are €3 625.29.
Answer: c = 6.005 %
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 = 6.552 %
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 = 8.475 %
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 = 5.642 %
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 = 3500 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 = 2400 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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