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 need €5 000 in 6 years. An account pays 4.8% per year, compounded quarterly. How much must you deposit today? Find the present value PV.

FV = €5 0004.8% / 4n = 6

PV = €

2.Time value of moneyHard

A loan of €30 000 is repaid with equal monthly payments over 5 years at 3.6% per year, compounded monthly. How much interest I is paid in total over the life of the loan?

P = €30 0003.6% / 12N = 60

I = €

3.NPV & IRRMedium

A project costs €60 000 today and returns €24 000, €15 500, €22 500, €20 500, €17 500 at the end of years 1 to 5. The cost of capital is 6%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.

r = 6%CF₀ = −60 000CF₁ = 24 000CF₂ = 15 500CF₃ = 22 500CF₄ = 20 500CF₅ = 17 500

DPB = years

4.NPV & IRRHard

A project costs €10 000 and returns €4 500, €4 000, €3 000, €5 500 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 = −10 000CF₁ = 4 500CF₂ = 4 000CF₃ = 3 000CF₄ = 5 500

MIRR = %

5.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays an annual coupon of 6% and matures in 11 years. Its yield to maturity changes from 4% to 4.5%. By how much ΔP does the bond price change?

F = €1 000coupon = 6%n = 11y₁ = 4%y₂ = 4.5%

ΔP = €

6.Bond pricing & yieldHard

A bond with a face value of €1 000 pays a 2% annual coupon in two half-yearly payments and matures in 12 years. It sells for €543. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.

F = €1 000coupon = 2% (2×)n = 12P = €543

YTM = %

7.CAPM & WACCMedium

A share with a beta of 0.8 has a required return of 4.7%. The risk-free rate is 1.5%. What expected market return r_m does CAPM imply?

β = 0.8r_e = 4.7%r_f = 1.5%

r_m = %

8.CAPM & WACCHard

A company has 2 500 000 shares at €41.00 each and 5 000 bonds trading at €1 050 each. The pre-tax cost of debt is 5% and the tax rate is 25.8%. The share’s beta is 1.2, the risk-free rate 1.5% and the market return 7.5%. Find the WACC.

β = 1.2r_f = 1.5%r_m = 7.5%r_d = 5%T = 25.8%

WACC = %

9.Break-even analysisMedium

A product sells for €28 per unit with a variable cost of €20 per unit. Fixed costs are €4 000 per month. How many units Q must be sold per month to make a profit of €2 000?

F = €4 000p = €28v = €20profit = €2 000

Q = units

10.Break-even analysisHard

A company sells two products. Product A sells for €11 with a variable cost of €8; product B sells for €35 with a variable cost of €20. They sell in a fixed mix of 2 units of A for every 3 of B. Fixed costs are €15 300 per month. Find the total number of units Q (A and B together) needed to break even.

p_A = €11, v_A = €8p_B = €35, v_B = €20A : B = 2 : 3F = €15 300

Q = units

3Solutions

1.Time value of money

Answer: PV = 3755 €

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 = 0.048/4 = 0.012

N = 4·6 = 24

Present value

PV = FV/(1 + i)^N = 5 000/1.331 = €3 755.24

Answer

PV = €3 755.24

2.Time value of money

Answer: I = 2826 €

Monthly payment

i = 0.003, N = 60

A = 30 000·0.003/(1 − (1 + 0.003)⁻⁶⁰) = €547.10

Total interest

I = N·A − P = 60·547.1 − 30 000 = €2 825.81

Answer

I = €2 825.81

3.NPV & IRR

Answer: DPB = 3.288 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.

Discounted and cumulative cash flows

year 1: 24 000/1.06 = 22 641.51 → Σ = 22 641.51

year 2: 15 500/1.124 = 13 794.94 → Σ = 36 436.45

year 3: 22 500/1.191 = 18 891.43 → Σ = 55 327.89

year 4: 20 500/1.262 = 16 237.92 → Σ = 71 565.81

Fraction of the last year

DPB = 3 + 4 672.11/16 237.92 = 3.288

Answer

DPB = 3.288 years

The discounted payback is always longer than the simple payback (2.911 years).

4.NPV & IRR

Answer: MIRR = 18.37 %

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

4 500·1.1³ = 5 989.50

4 000·1.1² = 4 840.00

3 000·1.1¹ = 3 300.00

5 500·1.1⁰ = 5 500.00

TV = 19 629.50

MIRR

MIRR = (19 629.50/10 000)^1/4 − 1 = 18.37%

Answer

MIRR = 18.37%

5.Bond pricing & yield

Answer: ΔP = −47.28 €

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.

Price at the old yield

C = €60.00, i = 0.04, N = 11

€60.00·(1 − 0.6496)/0.04 + €1 000·0.6496 = €525.63 + €649.58 = €1 175.21

Price at the new yield

C = €60.00, i = 0.045, N = 11

€60.00·(1 − 0.6162)/0.045 + €1 000·0.6162 = €511.74 + €616.20 = €1 127.93

Change

ΔP = 1 127.93 − 1 175.21 = −€47.28

Answer

ΔP = −€47.28

6.Bond pricing & yield

Answer: YTM = 7.992 %

Per period

C = €10.00, N = 24

Solve numerically

i = 3.996% per half-year

YTM

YTM = 2·3.996% = 7.992%

Answer

YTM = 7.992%

7.CAPM & WACC

Answer: r_m = 5.5 %

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 = (4.7% − 1.5%)/0.8 = 4%

Market return

r_m = 1.5% + 4% = 5.5%

Answer

r_m = 5.5%

8.CAPM & WACC

Answer: WACC = 8.457 %

Market values

E = 2 500 000·€41.00 = €102 500 000

D = 5 000·€1 050 = €5 250 000

Cost of equity (CAPM)

r_e = 1.5% + 1.2·6% = 8.7%

Weights

V = €107 750 000, E/V = 0.9513, D/V = 0.04872

WACC

WACC = 0.9513·8.7% + 0.04872·5%·(1 − 0.258) = 8.457%

Answer

WACC = 8.457%

9.Break-even analysis

Answer: Q = 750 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 = 28 − 20 = €8

Units for the target profit

Q = (F + profit)/(p − v) = (4 000 + 2 000)/8 = 750

Answer

Q = 750 units

10.Break-even analysis

Answer: Q = 1500 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 = €3, cm_B = €15

Weighted contribution margin

(2·3 + 3·15)/5 = €10.20

Break-even units

Q = 15 300/10.2 = 1 500

A: 600, B: 900

Answer

Q = 1 500 units

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