Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You need €5 000 in 9 years. An account pays 4% per year, compounded monthly. How much must you deposit today? Find the present value PV.
A loan of €15 000 is repaid with equal monthly payments over 3 years. The interest rate is 5.4% per year, compounded monthly. Find the monthly payment A.
A bond with a face value of €1 000 matures in 4 years and pays annual coupons. It sells for €964 when the yield to maturity is 4%. Find the coupon rate c in percent.
A bond with 19 years to maturity (face value €1 000, annual coupon 5%) sells for €1 374. The issuer can call it in 4 years at a call price of €1 050. Find the yield to call YTC in percent.
A company has equity worth €8 000 000 and debt worth €6 000 000. Its cost of equity is 11%, its pre-tax cost of debt is 7% and the tax rate is 25%. Find the WACC.
A company has 5 000 000 shares trading at €44.00; the book value of its equity is €66 000 000. It just paid a dividend of €1.30 per share, and dividends grow by 5% per year. Its debt has a market value of €66 000 000 and a pre-tax cost of 6%. The tax rate is 28%. Find the WACC, using the dividend growth model for the cost of equity.
A machine costs €60 000 today. It generates €23 500 at the end of each year for 3 years and can then be sold for €6 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €10 000 and returns €4 500, €7 000, €6 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
Answer: PV = 3490 €
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: A = 452.3 €
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.
The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.
Answer: c = 3.008 %
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: 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 2.503%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 8.536 %
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 = 7.229 %
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 = 2949 €
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.
Answer: IRR = 32.24 %
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.
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