Investments and Valuation: cram pack

Formula sheets, the key method steps for every topic and 8 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.

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

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.

2Practice problems

1.Time value of moneyMedium

You invest €2 000 at a nominal rate of 3% per year, compounded monthly. Find the value FV of the investment after 6 years.

PV = €2 0003% / 12n = 6

FV = €

2.Time value of moneyHard

A loan of €250 000 is repaid with equal monthly payments over 30 years. The interest rate is 6% per year, compounded monthly. Find the monthly payment A.

P = €250 0006% / 12N = 360

A = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 matures in 7 years and pays annual coupons. It sells for €849 when the yield to maturity is 9%. Find the coupon rate c in percent.

F = €1 000n = 7P = €849YTM = 9%

c = %

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 6% matures in 9 years and sells for €875. Find its yield to maturity YTM in percent.

F = €1 000coupon = 6%n = 9P = €875

YTM = %

5.CAPM & WACCMedium

A company has equity worth €8 000 000 and debt worth €5 000 000. Its cost of equity is 13%, its pre-tax cost of debt is 7% and the tax rate is 28%. Find the WACC.

E = €8 000 000D = €5 000 000r_e = 13%r_d = 7%T = 28%

WACC = %

6.CAPM & WACCHard

A company has 5 000 000 shares trading at €12.50; the book value of its equity is €25 000 000. It just paid a dividend of €1.50 per share, and dividends grow by 2% per year. Its debt has a market value of €37 500 000 and a pre-tax cost of 6%. The tax rate is 30%. Find the WACC, using the dividend growth model for the cost of equity.

N = 5 000 000P₀ = €12.50book equity = €25 000 000D₀ = €1.50g = 2%D = €37 500 000r_d = 6%T = 30%

WACC = %

7.NPV & IRRMedium

A machine costs €60 000 today, lasts 6 years and costs €4 000 per year to run (paid at the end of each year). The cost of capital is 8%. Find its equivalent annual cost EAC.

C₀ = €60 000M = €4 000n = 6r = 8%

EAC = €

8.NPV & IRRHard

A project costs €40 000 and returns €17 500, €26 000 at the end of years 1 to 2. Find its internal rate of return IRR in percent.

CF₀ = −40 000CF₁ = 17 500CF₂ = 26 000

IRR = %

3Solutions

1.Time value of money

Answer: FV = 2394 €

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 = 0.03/12 = 0.0025

N = m·n = 12·6 = 72

Future value

FV = 2 000·(1 + 0.0025)⁷² = 2 000·1.197 = €2 393.90

Answer

FV = €2 393.90

2.Time value of money

Answer: A = 1499 €

Monthly rate and number of payments

i = 0.06/12 = 0.005

N = 12·30 = 360

Payment

The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.

(1 + i)^−N = 0.166

A = 250 000·0.005/(1 − 0.166) = €1 498.88

Answer

A = €1 498.88

3.Bond pricing & yield

Answer: c = 6 %

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.

Present value of the face value

1000·(1 + 0.09)⁻⁷ = €547.03

Coupon

C = (849 − 547.03)/5.033 = €60.00

Coupon rate

c = 60/1000 = 6%

Answer

c = 6%

4.Bond pricing & yield

Answer: YTM = 8.001 %

Equation for the yield

€875 = €60·(1 − (1 + y)⁻⁹)/y + €1 000/(1 + y)⁹

Solve numerically

P(8%) = €875.06, P(9%) = €820.14

YTM = 8.001%

Answer

YTM = 8.001%

5.CAPM & WACC

Answer: WACC = 9.938 %

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).

Weights

V = €13 000 000, E/V = 0.6154, D/V = 0.3846

WACC

WACC = 0.6154·13% + 0.3846·7%·(1 − 0.28) = 9.938%

Answer

WACC = 9.938%

6.CAPM & WACC

Answer: WACC = 10.47 %

Cost of equity

D₁ = 1.5·1.02 = €1.53

r_e = 1.53/12.5 + 0.02 = 14.24%

Market values

E = 5 000 000·€12.50 = €62 500 000

V = €100 000 000, E/V = 0.625, D/V = 0.375

WACC

WACC = 0.625·14.24% + 0.375·6%·(1 − 0.3) = 10.47%

Answer

WACC = 10.47%

7.NPV & IRR

Answer: EAC = 16979 €

Equivalent annual cost: spread the present value of all costs over the life as an equal yearly amount (an annuity): EAC = PV·r/(1 − (1 + r)⁻ⁿ). It lets you compare machines with different lives.

Annuity factor

(1 − 1.08⁻⁶)/0.08 = 4.623

Annualised purchase price

60 000/4.623 = 12 978.92

EAC

EAC = 12 978.92 + 4 000 = €16 978.92

Answer

EAC = €16 978.92

8.NPV & IRR

Answer: IRR = 5.413 %

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.

Bracket the IRR

NPV falls as the rate rises, so the IRR lies between a rate with positive NPV and one with negative NPV.

NPV(5%) = 249.43

NPV(10%) = −2 603.31

Narrow down

Linear interpolation gives a first estimate; repeating it (or a calculator's IRR function) gives the exact rate.

IRR ≈ 5 + 5·249.43/(249.43 − (−2 603.31)) = 5.437%

exact: NPV(5.413%) = 0 → IRR = 5.413%

Answer

IRR = 5.413%

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