Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You need €40 000 in 6 years. An account pays 3% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
You save €100 at the end of every month for 20 years. The account pays 4.8% 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 8 years and pays annual coupons. It sells for €1 251 when the yield to maturity is 2.5%. Find the coupon rate c in percent.
A bond with 12 years to maturity (face value €1 000, annual coupon 7%) sells for €1 177. The issuer can call it in 3 years at a call price of €1 070. Find the yield to call YTC in percent.
A company is financed with ordinary shares worth €12 000 000, preferred shares worth €500 000 and debt worth €4 000 000. The cost of equity is 10%, the cost of preferred stock 6% and the pre-tax cost of debt 4%. The tax rate is 25.8%. Find the WACC.
A company has 4 000 000 shares at €42.50 each and 10 000 bonds trading at €1 020 each. The pre-tax cost of debt is 6% and the tax rate is 21%. The share’s beta is 0.6, the risk-free rate 4% and the market return 8.5%. Find the WACC.
An investment of €30 000 gives these cash flows at the end of years 1 to 4: €15 000, €8 000, €14 000, €15 000. The cost of capital is 6%. Find the net present value NPV.
A project costs €25 000 and returns €12 000, €13 500, €12 500 at the end of years 1 to 3. Cash flows can be reinvested at the cost of capital of 10%. Find the modified internal rate of return MIRR in percent.
Answer: PV = 33433 €
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 = 40168 €
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 €24 000 yourself; the rest is interest.
Answer: c = 6.001 %
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 = 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.
The yield to maturity is 5.003%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 8.174 %
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 = 6.589 %
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 = 14907 €
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: MIRR = 18.76 %
Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.
ExamProblems · examproblems.com/courses/investments-and-valuation