Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €2 000 at a nominal rate of 3% per year, compounded quarterly. Find the value FV of the investment after 2 years.
You want €20 000 in 14 years. You save the same amount at the end of every month in an account paying 3% per year, compounded monthly. Find the monthly deposit A.
A project costs €50 000 today and returns €19 000, €11 000, €18 500, €18 500 at the end of years 1 to 4. The cost of capital is 5%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €150 000 today. Its cash flow is €27 000 at the end of year 1 and grows by 5% per year until the last cash flow at the end of year 5. The cost of capital is 10%. Find the net present value NPV.
A bond with a face value of €1 000 pays a 6% annual coupon in two half-yearly payments and matures in 5 years. The yield to maturity is 3.5% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 6% matures in 6 years and sells for €950. Find its yield to maturity YTM in percent.
A company has equity worth €8 000 000 and debt worth €1 000 000. Its cost of equity is 8%, its pre-tax cost of debt is 3.5% and the tax rate is 30%. Find the WACC.
A company has 2 000 000 shares trading at €20.00; the book value of its equity is €16 000 000. It just paid a dividend of €1.20 per share, and dividends grow by 5% per year. Its debt has a market value of €20 000 000 and a pre-tax cost of 4.5%. The tax rate is 30%. Find the WACC, using the dividend growth model for the cost of equity.
A product sells for €35 per unit with a variable cost of €25 per unit. Fixed costs are €40 000 per month. How many units Q must be sold per month to make a profit of €2 000?
A product sells for €51 per unit, with a variable cost of €45 per unit and fixed costs of €4 800 per year. Budgeted sales are 1 040 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 2123 €
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 = 95.94 €
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: DPB = 3.391 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 (3.081 years).
Answer: NPV = −37934 €
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: P = 1114 €
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 below the coupon rate, so the bond sells at a premium.
Answer: YTM = 7.051 %
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: WACC = 7.383 %
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 = 8.583 %
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 = 4200 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 = 23.08 %
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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