Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €2 000 at 8% per year with continuous compounding. Find the value FV after 3 years.
You save €100 at the end of every month for 24 years. The account pays 6% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €30 000 gives these cash flows at the end of years 1 to 4: €8 000, €6 000, €9 000, €8 000. The cost of capital is 7.000000000000001%. Find the net present value NPV.
A project costs €50 000 and returns €36 000, €32 000 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays an annual coupon of 4% and matures in 5 years. Its yield to maturity changes from 2.5% to 4.5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 and an annual coupon of 4% matures in 8 years and sells for €1 070. Find its yield to maturity YTM in percent.
A company has equity worth €8 000 000 and debt worth €3 000 000. Its cost of equity is 8%, its pre-tax cost of debt is 6% and the tax rate is 25.8%. Find the WACC.
A company has 2 000 000 shares at €8.00 each and 10 000 bonds trading at €1 000 each. The pre-tax cost of debt is 4.5% and the tax rate is 25.8%. The share’s beta is 0.8, the risk-free rate 1.5% and the market return 7%. Find the WACC.
A product sells for €16 per unit with a variable cost of €6 per unit. Fixed costs are €25 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A product sells for €14 per unit, with a variable cost of €6 per unit and fixed costs of €40 000 per year. Budgeted sales are 8 000 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 2542 €
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 = 64112 €
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 €28 800 yourself; the rest is interest.
Answer: NPV = −3833 €
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 = 23.73 %
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: ΔP = −91.64 €
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: YTM = 3.003 %
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 = 7.032 %
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 = 4.915 %
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 = 3100 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: MS = 37.5 %
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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