Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You invest €12 000 at a nominal rate of 3.6% per year, compounded monthly. Find the value FV of the investment after 4 years.
A loan of €15 000 is repaid with equal monthly payments over 3 years. The interest rate is 7.2% per year, compounded monthly. Find the monthly payment A.
A bond with a face value of €1 000 pays a 2% annual coupon in two half-yearly payments and matures in 3 years. The yield to maturity is 9% per year. Find the price P of the bond.
A bond with 15 years to maturity (face value €1 000, annual coupon 9%) sells for €1 132. The issuer can call it in 5 years at a call price of €1 090. Find the yield to call YTC in percent.
A company has equity worth €6 000 000 and debt worth €3 000 000. Its cost of equity is 11%, its pre-tax cost of debt is 3% and the tax rate is 25.8%. Find the WACC.
A company has 5 000 000 shares at €17.00 each and 20 000 bonds trading at €980 each. The pre-tax cost of debt is 7% and the tax rate is 21%. The share’s beta is 1.3, the risk-free rate 2% and the market return 7.5%. Find the WACC.
A project costs €100 000 today. It generates €13 000 at the end of the first year, and the cash flow then grows by 1% per year forever. The cost of capital is 12%. Find the net present value NPV.
A project costs €40 000 and returns €24 500, €25 500, €17 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
Answer: FV = 13856 €
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 = 464.5 €
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: P = 819.5 €
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 above the coupon rate, so the bond sells at a discount.
Answer: YTC = 7.309 %
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 7.504%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 8.075 %
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 = 8.472 %
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 = 18182 €
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 positive, so the project earns more than the required return.
Answer: IRR = 33.53 %
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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