Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
A credit card charges a nominal interest rate of 20% per year, compounded quarterly. Find the effective annual rate EAR in percent.
You want €10 000 in 9 years. You save the same amount at the end of every month in an account paying 4.8% per year, compounded monthly. Find the monthly deposit A.
An investment of €30 000 gives these cash flows at the end of years 1 to 3: €10 000, €12 000, €10 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €10 000 and returns €5 500, €6 500, €7 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
You buy a bond with 3 years to maturity (face value €1 000, annual coupon 6%) when its yield to maturity is 9%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 10%. Find your holding period return HPR in percent.
A bond with 19 years to maturity (face value €1 000, annual coupon 8%) sells for €1 161. The issuer can call it in 6 years at a call price of €1 080. Find the yield to call YTC in percent.
A company has equity worth €3 000 000 and debt worth €2 000 000. Its cost of equity is 14%, its pre-tax cost of debt is 4.5% and 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 13%. Its debt consists of 5 000 bonds (face value €1 000, annual coupon 7%, 6 years to maturity) that trade at €977 each. The tax rate is 21%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A product can be made in two ways. Method A has fixed costs of €20 000 per year and a variable cost of €10 per unit. Method B, more automated, has fixed costs of €23 000 per year and a variable cost of €7 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €32 per unit, with a variable cost of €30 per unit and fixed costs of €10 000 per year. Budgeted sales are 8 000 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: EAR = 21.55 %
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 EAR is always above the nominal rate when interest compounds more than once a year.
Answer: A = 74.21 €
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: NPV = −3479 €
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 = 38.46 %
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.
Answer: HPR = 7.198 %
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 = 5.888 %
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 6.5%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 9.66 %
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 = 9.5 %
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: Q = 1000 units
Cost-volume-profit: each unit sold earns a contribution margin, price minus variable cost per unit. Profit = Q·(p − v) − F, where F is the fixed cost. At the break-even point profit is zero, so Q = F/(p − v).
Below this volume method A is cheaper; above it the automated method B is cheaper.
Answer: MS = 37.5 %
Cost-volume-profit: each unit sold earns a contribution margin, price minus variable cost per unit. Profit = Q·(p − v) − F, where F is the fixed cost. At the break-even point profit is zero, so Q = F/(p − v).
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