Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You need €15 000 in 3 years. An account pays 3% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
A mortgage of €150 000 is repaid with equal monthly payments over 30 years at 4.2% per year, compounded monthly. How much is still owed just after the payment at the end of year 22? Find the outstanding balance B.
A bond with a face value of €1 000 matures in 15 years and pays annual coupons. It sells for €1 222 when the yield to maturity is 4%. Find the coupon rate c in percent.
A bond with 15 years to maturity (face value €1 000, annual coupon 8%) sells for €1 091. The issuer can call it in 4 years at a call price of €1 080. Find the yield to call YTC in percent.
A company is financed with ordinary shares worth €6 000 000, preferred shares worth €1 000 000 and debt worth €5 000 000. The cost of equity is 13%, the cost of preferred stock 6% and the pre-tax cost of debt 4%. The tax rate is 20%. Find the WACC.
A company has 4 000 000 shares at €52.50 each and 10 000 bonds trading at €980 each. The pre-tax cost of debt is 6% and the tax rate is 21%. The share’s beta is 1.2, the risk-free rate 2.5% and the market return 8%. Find the WACC.
A project costs €50 000 today and returns €16 500 at the end of each year for 6 years. The cost of capital is 8%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €40 000 and returns €25 000, €27 000, €21 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
Answer: PV = 13714 €
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: B = 59721 €
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.
Whatever is still owed must be exactly paid off by the remaining payments.
Answer: c = 5.997 %
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 = 7.106 %
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 7.001%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 8.333 %
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 = 8.906 %
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: DPB = 3.617 years
Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.
The discounted payback is always longer than the simple payback (3.03 years).
Answer: IRR = 38.56 %
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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