Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €10 000 at 2% per year with continuous compounding. Find the value FV after 11 years.
You borrow €5 000 at 7.2% per year, compounded monthly, and repay €175 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 €45 000 gives these cash flows at the end of years 1 to 4: €14 000, €21 000, €19 000, €19 000. The cost of capital is 5%. Find the net present value NPV.
A project costs €40 000 and returns €21 000, €23 000, €23 500 at the end of years 1 to 3. Cash flows can be reinvested at the cost of capital of 10%. Find the modified internal rate of return MIRR in percent.
A bond with a face value of €1 000 pays an annual coupon of 8% and matures in 12 years. Its yield to maturity changes from 9% to 10.5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 pays an annual coupon of 4.5% and matures in 5 years. The yield to maturity is 3%. Find the Macaulay duration D in years.
A company has equity worth €8 000 000 and debt worth €4 000 000. Its WACC is 10.08%, its pre-tax cost of debt is 3% and the tax rate is 25%. Find the cost of equity re.
A company has 4 000 000 shares at €50.00 each and 40 000 bonds trading at €1 020 each. The pre-tax cost of debt is 4.5% and the tax rate is 25.8%. The share’s beta is 1.2, the risk-free rate 2.5% and the market return 7%. Find the WACC.
A product can be made in two ways. Method A has fixed costs of €20 000 per year and a variable cost of €15 per unit. Method B, more automated, has fixed costs of €32 000 per year and a variable cost of €12 per unit. At what yearly volume Q do both methods cost the same?
A company sells 2 000 units per month at €31 each, with a variable cost of €25 per unit and fixed costs of €6 000 per month. The variable cost rises by €1 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 = 12461 €
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 = 32 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.
31 full payments leave a small balance, which the last, smaller payment clears.
Answer: NPV = 19425 €
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: MIRR = 22.88 %
Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.
Answer: ΔP = −94.64 €
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: D = 4.603 years
Duration: the Macaulay duration is the weighted average time at which a bond pays its cash flows, with the present values of the cash flows as weights: D = Σ t·PVₜ/P. The modified duration D/(1 + y) estimates the percentage price change for a small change in yield: %ΔP ≈ −Dmod·Δy.
Answer: re = 14 %
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 = 7.127 %
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 = 4000 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).
Below this volume method A is cheaper; above it the automated method B is cheaper.
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).
ExamProblems · examproblems.com/courses/corporate-finance