Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You invest €5 000 at 7% per year with continuous compounding. Find the value FV after 2 years.
You save €100 at the end of every month for 8 years. The account pays 4.2% 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 3 years and pays annual coupons. It sells for €1 014 when the yield to maturity is 2.5%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 5% matures in 10 years and sells for €1 170. Find its yield to maturity YTM in percent.
A company is financed with ordinary shares worth €5 000 000, preferred shares worth €1 500 000 and debt worth €2 000 000. The cost of equity is 13%, the cost of preferred stock 6% and the pre-tax cost of debt 4%. The tax rate is 30%. Find the WACC.
A company has equity with a market value of €5 000 000 and a cost of equity of 11%. Its debt consists of 8 000 bonds (face value €1 000, annual coupon 7%, 6 years to maturity) that trade at €977 each. The tax rate is 30%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
An investment of €75 000 gives these cash flows at the end of years 1 to 3: €24 000, €27 000, €15 000. The cost of capital is 9%. Find the net present value NPV.
A project costs €25 000 and returns €18 000, €9 500, €10 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
Answer: FV = 5751 €
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 = 11386 €
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 €9 600 yourself; the rest is interest.
Answer: c = 2.99 %
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 = 3.006 %
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 = 9.365 %
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 = 7.489 %
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 = −18674 €
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 = 27.59 %
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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