Corporate Finance: cram pack

Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.

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1Formulas and methods

Time value of money

Future value: FV = PV·(1 + i)ⁿ
Present value: PV = FV/(1 + i)ⁿ
Several periods per year: i = r/m, N = m·n
Annuity, present value: PV = PMT·[1 − (1 + i)⁻ⁿ]/i
Annuity, future value: FV = PMT·[(1 + i)ⁿ − 1]/i

Method

Draw a timeline with every cash flow and its timing.
Match the rate to the period: a monthly rate for monthly payments.
Use FV = PV·(1 + i)ⁿ for single amounts and the annuity formulas for equal payments.
Check the sign and the order of magnitude of the answer.

NPV & IRR

Net present value: NPV = −C₀ + Σ CF_t/(1 + r)^t
Equal cash flows: NPV = −C₀ + CF·[1 − (1 + r)⁻ⁿ]/r
Internal rate of return: the r at which NPV = 0
Decision: accept when NPV > 0 (or IRR > required return)

Method

List all cash flows with their year; the investment at year 0 is negative.
Discount each cash flow: CF_t/(1 + r)^t.
Add them up to get the NPV. Positive means the project earns more than r.
For the IRR, find the rate that makes NPV zero by trial or interpolation.

Bond pricing & yield

Bond price: P = C·(1 − (1 + y)⁻ⁿ)/y + F/(1 + y)ⁿ
Semi-annual: C/2, y/2 and 2n periods
Current yield: C/P
Price and yield: yield up → price down

Method

Find the coupon per period and the yield per period.
Price = present value of the coupons (an annuity) + present value of the face value.
For semi-annual coupons use half the coupon, half the yield and twice the periods.
Check: if the yield is above the coupon rate, the price is below face value.

CAPM & WACC

CAPM: r_e = r_f + β·(r_m − r_f)
WACC: E/V·r_e + D/V·r_d·(1 − T)
Firm value: V = E + D

Break-even analysis

Contribution margin: CM = p − v
Break-even quantity: Q_BE = FC/(p − v)
Target profit: Q = (FC + profit)/(p − v)
Profit: π = (p − v)·Q − FC
Margin of safety: (Q − Q_BE)/Q

2Practice problems

1.Time value of moneyMedium

You invest €2 000 at a nominal rate of 6% per year, compounded quarterly. Find the value FV of the investment after 9 years.

PV = €2 0006% / 4n = 9

FV = €

2.Time value of moneyHard

A mortgage of €300 000 is repaid with equal monthly payments over 20 years at 6% per year, compounded monthly. How much is still owed just after the payment at the end of year 2? Find the outstanding balance B.

P = €300 0006% / 12N = 240k = 24

B = €

3.NPV & IRRMedium

An investment of €30 000 gives these cash flows at the end of years 1 to 3: €7 000, €10 000, €8 000. The cost of capital is 5%. Find the net present value NPV.

r = 5%CF₀ = −30 000CF₁ = 7 000CF₂ = 10 000CF₃ = 8 000

NPV = €

4.NPV & IRRHard

A project costs €20 000 and returns €7 000, €9 500, €11 000, €9 000 at the end of years 1 to 4. Cash flows can be reinvested at the cost of capital of 10%. Find the modified internal rate of return MIRR in percent.

r = 10%CF₀ = −20 000CF₁ = 7 000CF₂ = 9 500CF₃ = 11 000CF₄ = 9 000

MIRR = %

5.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays an 8% annual coupon in two half-yearly payments and matures in 12 years. The yield to maturity is 5% per year. Find the price P of the bond.

F = €1 000coupon = 8%YTM = 5%n = 12

P = €

6.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 8% matures in 6 years and sells for €955. Find its yield to maturity YTM in percent.

F = €1 000coupon = 8%n = 6P = €955

YTM = %

7.CAPM & WACCMedium

A share with a beta of 0.9 has a required return of 6.85%. The risk-free rate is 1%. What expected market return r_m does CAPM imply?

β = 0.9r_e = 6.85%r_f = 1%

r_m = %

8.CAPM & WACCHard

A company's shares have a beta of 1.1 at its current debt-to-equity ratio of 0.4. It plans to raise its debt-to-equity ratio to 1.25. The tax rate is 28%, the risk-free rate 2% and the market risk premium 6.5%. Find the cost of equity r_e after the change.

β_L = 1.1D/E₁ = 0.4D/E₂ = 1.25T = 28%r_f = 2%r_m − r_f = 6.5%

r_e = %

9.Break-even analysisMedium

A product sells for €14 per unit with a variable cost of €4 per unit. Fixed costs are €8 000 per month. How many units Q must be sold per month to make a profit of €6 000?

F = €8 000p = €14v = €4profit = €6 000

Q = units

10.Break-even analysisHard

A company sells two products. Product A sells for €18 with a variable cost of €12; product B sells for €30 with a variable cost of €10. They sell in a fixed mix of 3 units of A for every 1 of B. Fixed costs are €30 400 per month. Find the total number of units Q (A and B together) needed to break even.

p_A = €18, v_A = €12p_B = €30, v_B = €10A : B = 3 : 1F = €30 400

Q = units

3Solutions

1.Time value of money

Answer: FV = 3418 €

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.

Rate and number of periods

i = r/m = 0.06/4 = 0.015

N = m·n = 4·9 = 36

Future value

FV = 2 000·(1 + 0.015)³⁶ = 2 000·1.709 = €3 418.28

Answer

FV = €3 418.28

2.Time value of money

Answer: B = 283487 €

Monthly payment

i = 0.005, N = 240

A = 300 000·0.005/(1 − (1 + 0.005)⁻²⁴⁰) = €2 149.29

Present value of the remaining payments

Whatever is still owed must be exactly paid off by the remaining payments.

N − k = 240 − 24 = 216

B = 2149·(1 − (1 + 0.005)⁻²¹⁶)/0.005 = €283 487.21

Answer

B = €283 487.21

3.NPV & IRR

Answer: NPV = −7352 €

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.

Present value of each cash flow

CF₀ = −30 000

CF₁/(1 + 0.05)¹ = 7 000/1.05 = 6 666.67

CF₂/(1 + 0.05)² = 10 000/1.103 = 9 070.29

CF₃/(1 + 0.05)³ = 8 000/1.158 = 6 910.70

Add them up

NPV = €−7 352.34

Answer

NPV = €−7 352.34

NPV is negative, so the project earns less than the required return.

4.NPV & IRR

Answer: MIRR = 20.32 %

Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.

Terminal value of the inflows

7 000·1.1³ = 9 317.00

9 500·1.1² = 11 495.00

11 000·1.1¹ = 12 100.00

9 000·1.1⁰ = 9 000.00

TV = 41 912.00

MIRR

MIRR = (41 912.00/20 000)^1/4 − 1 = 20.32%

Answer

MIRR = 20.32%

5.Bond pricing & yield

Answer: P = 1268 €

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.

Per period

C = €40.00, i = 0.025, N = 24

Present value of the coupons

€40.00·(1 − 0.5529)/0.025 = €715.40

Present value of the face value

€1 000·0.5529 = €552.88

Price

P = €1 268.27

Answer

P = €1 268.27

The yield is below the coupon rate, so the bond sells at a premium.

6.Bond pricing & yield

Answer: YTM = 9.003 %

Equation for the yield

€955 = €80·(1 − (1 + y)⁻⁶)/y + €1 000/(1 + y)⁶

Solve numerically

P(9%) = €955.14, P(10%) = €912.89

YTM = 9.003%

Answer

YTM = 9.003%

7.CAPM & WACC

Answer: r_m = 7.5 %

Cost of capital: CAPM gives the return shareholders require: r_e = r_f + β·(r_m − r_f). The weighted average cost of capital mixes equity and debt by their market values: WACC = E/V·r_e + D/V·r_d·(1 − T), with V = E + D. Interest is tax-deductible, hence the (1 − T).

Market risk premium

r_m − r_f = (6.85% − 1%)/0.9 = 6.5%

Market return

r_m = 1% + 6.5% = 7.5%

Answer

r_m = 7.5%

8.CAPM & WACC

Answer: r_e = 12.55 %

Levered beta (Hamada): debt makes equity riskier. β_L = β_U·(1 + (1 − T)·D/E), where β_U is the beta of the business without debt. To move a beta to another capital structure, unlever it first and then relever it at the new D/E.

Unlever the beta

β_U = 1.1/(1 + 0.72·0.4) = 0.854

Relever at the new D/E

β_L = 0.854·(1 + 0.72·1.25) = 1.623

Cost of equity (CAPM)

r_e = 2% + 1.623·6.5% = 12.55%

Answer

r_e = 12.55%

9.Break-even analysis

Answer: Q = 1400 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).

Contribution margin per unit

p − v = 14 − 4 = €10

Units for the target profit

Q = (F + profit)/(p − v) = (8 000 + 6 000)/10 = 1 400

Answer

Q = 1 400 units

10.Break-even analysis

Answer: Q = 3200 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.

Contribution margins

cm_A = €6, cm_B = €20

Weighted contribution margin

(3·6 + 1·20)/4 = €9.50

Break-even units

Q = 30 400/9.5 = 3 200

A: 2 400, B: 800

Answer

Q = 3 200 units

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