Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €40 000 in 3 years. An account pays 4.8% per year, compounded monthly. How much must you deposit today? Find the present value PV.
A loan of €25 000 is repaid with equal monthly payments over 6 years at 2.4% per year, compounded monthly. How much interest I is paid in total over the life of the loan?
A project costs €20 000 today and returns €5 500 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 €50 000 and returns €33 500, €20 000, €30 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
You buy a bond with 11 years to maturity (face value €1 000, annual coupon 4.5%) when its yield to maturity is 5.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 5% matures in 4 years and sells for €950. Find its yield to maturity YTM in percent.
A company has equity worth €8 000 000 and debt worth €5 000 000. Its WACC is 8%, its pre-tax cost of debt is 4% and the tax rate is 20%. Find the cost of equity re.
A company has equity with a market value of €8 000 000 and a cost of equity of 11%. Its debt consists of 8 000 bonds (face value €1 000, annual coupon 4%, 7 years to maturity) that trade at €942 each. The tax rate is 25.8%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A product sells for €8 per unit with a variable cost of €4 per unit. Fixed costs are €16 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A product sells for €19 per unit, with a variable cost of €15 per unit and fixed costs of €20 000 per year. Budgeted sales are 7 000 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 34645 €
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: I = 1868 €
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 = 4.751 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 (3.636 years).
Answer: IRR = 31.85 %
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 = 9.501 %
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: YTM = 6.458 %
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: re = 11 %
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 = 7.465 %
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 = 5500 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 = 28.57 %
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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