Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You invest €20 000 at 3% per year with continuous compounding. Find the value FV after 9 years.
You save €300 at the end of every month for 21 years. The account pays 7.2% per year, compounded monthly. How much have you saved at the end? Find FV.
A bond with a face value of €1 000 matures in 9 years and pays annual coupons. It sells for €700 when the yield to maturity is 9%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 8% matures in 6 years and sells for €915. Find its yield to maturity YTM in percent.
A company has equity worth €5 000 000 and debt worth €3 000 000. Its cost of equity is 14%, its pre-tax cost of debt is 7% and the tax rate is 30%. Find the WACC.
A company has 1 000 000 shares trading at €12.00; the book value of its equity is €7 200 000. It just paid a dividend of €1.10 per share, and dividends grow by 3% per year. Its debt has a market value of €3 600 000 and a pre-tax cost of 4%. The tax rate is 25%. Find the WACC, using the dividend growth model for the cost of equity.
A project costs €400 000 today. It generates €22 000 at the end of the first year, and the cash flow then grows by 2% per year forever. The cost of capital is 8%. Find the net present value NPV.
An investment of €25 000 returns €5 500 at the end of each year for 5 years. Find its internal rate of return IRR in percent.
Answer: FV = 26199 €
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 = 175767 €
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 €75 600 yourself; the rest is interest.
Answer: c = 3.996 %
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 = 9.949 %
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 = 10.59 %
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 = 10.26 %
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: NPV = −33333 €
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 = 3.263 %
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.
The annuity factor falls as the rate rises; bracket the rate and narrow down.
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