Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €5 000 at 6% per year with continuous compounding. Find the value FV after 12 years.
You borrow €12 000 at 5.4% per year, compounded monthly, and repay €375 at the end of every month. How many months does it take until the loan is repaid (the last payment may be smaller)? Give the number of payments N.
An investment of €60 000 gives these cash flows at the end of years 1 to 4: €14 000, €26 000, €17 000, €24 000. The cost of capital is 7.000000000000001%. Find the net present value NPV.
A project costs €10 000 and returns €7 000, €6 500 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
You buy a bond with 3 years to maturity (face value €1 000, annual coupon 4%) when its yield to maturity is 2.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 4%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 6% matures in 5 years and sells for €920. Find its yield to maturity YTM in percent.
A company has equity worth €12 000 000 and debt worth €3 000 000. Its cost of equity is 8%, its pre-tax cost of debt is 5% and the tax rate is 25.8%. Find the WACC.
A company has 1 000 000 shares at €59.50 each and 40 000 bonds trading at €1 050 each. The pre-tax cost of debt is 5% and the tax rate is 20%. The share’s beta is 1.5, the risk-free rate 3.5% and the market return 8%. Find the WACC.
A product sells for €26 per unit with a variable cost of €6 per unit. Fixed costs are €30 000 per month. How many units Q must be sold per month to make a profit of €10 000?
A product sells for €35 per unit, with a variable cost of €30 per unit and fixed costs of €2 500 per year. Budgeted sales are 580 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 10272 €
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: N = 35 payments
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.
34 full payments leave a small balance, which the last, smaller payment clears.
Answer: NPV = 7980 €
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 positive, so the project earns more than the required return.
Answer: IRR = 22.89 %
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 = −0.2724 %
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 = 8.004 %
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 = 7.142 %
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 = 7.664 %
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 = 2000 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 = 13.79 %
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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