Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €12 000 at a nominal rate of 6% per year, compounded quarterly. Find the value FV of the investment after 9 years.
You save €100 at the end of every month for 5 years. The account pays 5.4% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €60 000 gives these cash flows at the end of years 1 to 3: €18 000, €23 000, €16 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €40 000 and returns €30 000, €22 500, €25 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays a 7% annual coupon in two half-yearly payments and matures in 12 years. The yield to maturity is 3% per year. Find the price P of the bond.
A bond with 20 years to maturity (face value €1 000, annual coupon 6%) sells for €1 355. The issuer can call it in 4 years at a call price of €1 060. Find the yield to call YTC in percent.
A company has a debt-to-equity ratio (at market values) of 0.6. Its cost of equity is 9%, its pre-tax cost of debt is 3% and the tax rate is 20%. Find the WACC.
A company has 5 000 000 shares at €43.50 each and 40 000 bonds trading at €1 050 each. The pre-tax cost of debt is 4% and the tax rate is 20%. The share’s beta is 1.1, the risk-free rate 1% and the market return 5%. Find the WACC.
A product sells for €35 per unit with a variable cost of €15 per unit. Fixed costs are €16 000 per month. How many units Q must be sold per month to make a profit of €5 000?
A company sells 12 500 units per month at €35 each, with a variable cost of €30 per unit and fixed costs of €25 000 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 = 20510 €
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: FV = 6870 €
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 €6 000 yourself; the rest is interest.
Answer: NPV = −12607 €
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: IRR = 44.48 %
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 falls as the rate rises, so the IRR lies between a rate with positive NPV and one with negative NPV.
Linear interpolation gives a first estimate; repeating it (or a calculator's IRR function) gives the exact rate.
Answer: P = 1401 €
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: YTC = 50 %
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 to maturity is 3.502%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 6.525 %
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 = 5.044 %
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 = 1050 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 = 15625 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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