Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €12 000 at a nominal rate of 4.8% per year, compounded quarterly. Find the value FV of the investment after 3 years.
You save €50 at the end of every month for 7 years. The account pays 7.2% per year, compounded monthly. How much have you saved at the end? Find FV.
A project costs €30 000 today and returns €8 000 at the end of each year for 6 years. The cost of capital is 10%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €40 000 and returns €14 500, €29 500, €16 000 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 3.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 3%. Find your holding period return HPR in percent.
A bond with 12 years to maturity (face value €1 000, annual coupon 6%) sells for €1 242. The issuer can call it in 6 years at a call price of €1 060. Find the yield to call YTC in percent.
A company has equity worth €5 000 000 and debt worth €3 000 000. Its cost of equity is 13%, its pre-tax cost of debt is 4% and the tax rate is 30%. Find the WACC.
A company has equity with a market value of €12 000 000 and a cost of equity of 14%. Its debt consists of 8 000 bonds (face value €1 000, annual coupon 6%, 6 years to maturity) that trade at €976 each. The tax rate is 25%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A product sells for €34 per unit with a variable cost of €30 per unit. Fixed costs are €6 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A product sells for €28 per unit, with a variable cost of €18 per unit and fixed costs of €25 000 per year. Budgeted sales are 2 880 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 13847 €
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 = 5440 €
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 €4 200 yourself; the rest is interest.
Answer: DPB = 4.934 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 (3.75 years).
Answer: IRR = 22.82 %
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 = 4.426 %
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 = 2.539 %
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 3.496%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 9.175 %
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 = 10.4 %
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 = 3000 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).
Answer: MS = 13.19 %
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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