Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €3 000 at the end of each year for 10 years. The interest rate is 8% per year. Find the present value PV of the annuity.
A mortgage of €300 000 is repaid with equal monthly payments over 30 years at 7.2% per year, compounded monthly. How much is still owed just after the payment at the end of year 22? Find the outstanding balance B.
A project costs €60 000 today and returns €21 500 at the end of each year for 6 years. The cost of capital is 12%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €10 000 and returns €4 500, €3 500, €5 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
You buy a bond with 10 years to maturity (face value €1 000, annual coupon 4%) when its yield to maturity is 8%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 8.5%. Find your holding period return HPR in percent.
A bond trades at €1 040 with a yield to maturity of 3% (annual compounding) and a Macaulay duration of 4.2 years. Use the modified duration to estimate the new price P if the yield rises by 50 basis points.
A company has equity worth €3 000 000 and debt worth €5 000 000. Its cost of equity is 14%, its pre-tax cost of debt is 7% and the tax rate is 25.8%. Find the WACC.
A company has equity with a market value of €12 000 000 and a cost of equity of 12%. Its debt consists of 5 000 bonds (face value €1 000, annual coupon 5%, 7 years to maturity) that trade at €972 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 sells for €47 per unit with a variable cost of €45 per unit. Fixed costs are €5 000 per month. How many units Q must be sold per month to make a profit of €2 000?
A company sells two products. Product A sells for €21 with a variable cost of €15; product B sells for €35 with a variable cost of €15. They sell in a fixed mix of 2 units of A for every 3 of B. Fixed costs are €57 600 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 20130 €
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: B = 148278 €
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.
Whatever is still owed must be exactly paid off by the remaining payments.
Answer: DPB = 3.612 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 (2.791 years).
Answer: IRR = 14.09 %
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 = 4.517 %
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: P = 1019 €
Duration: the Macaulay duration is the weighted average time at which a bond pays its cash flows, with the present values of the cash flows as weights: D = Σ t·PVₜ/P. The modified duration D/(1 + y) estimates the percentage price change for a small change in yield: %ΔP ≈ −Dmod·Δy.
Duration is a linear estimate; for large yield changes convexity makes the true price a little higher.
Answer: WACC = 8.496 %
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 = 9.792 %
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 = 3500 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: Q = 4000 units
Break-even with a sales mix: when products sell in a fixed ratio, use the weighted average contribution margin: weight each product's p − v by its share of units. Break-even units = F/(weighted contribution margin), split again in the mix ratio.
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