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

A credit card charges a nominal interest rate of 24% per year, compounded semi-annually. Find the effective annual rate EAR in percent.

r = 24%m = 2

EAR = %

2.Time value of moneyHard

You save €100 at the end of every month for 18 years. The account pays 7.2% per year, compounded monthly. How much have you saved at the end? Find FV.

A = €1007.2% / 12N = 216

FV = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 matures in 3 years and pays annual coupons. It sells for €972 when the yield to maturity is 3%. Find the coupon rate c in percent.

F = €1 000n = 3P = €972YTM = 3%

c = %

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 6% matures in 6 years and sells for €1 105. Find its yield to maturity YTM in percent.

F = €1 000coupon = 6%n = 6P = €1 105

YTM = %

5.CAPM & WACCMedium

A company is financed with ordinary shares worth €8 000 000, preferred shares worth €1 500 000 and debt worth €6 000 000. The cost of equity is 14%, the cost of preferred stock 6.5% and the pre-tax cost of debt 5%. The tax rate is 21%. Find the WACC.

E = €8 000 000P_s = €1 500 000D = €6 000 000r_e = 14%r_p = 6.5%r_d = 5%T = 21%

WACC = %

6.CAPM & WACCHard

A company's shares have a beta of 1.4 at its current debt-to-equity ratio of 0.5. It plans to raise its debt-to-equity ratio to 1.5. The tax rate is 25.8%, the risk-free rate 1% and the market risk premium 4.5%. Find the cost of equity r_e after the change.

β_L = 1.4D/E₁ = 0.5D/E₂ = 1.5T = 25.8%r_f = 1%r_m − r_f = 4.5%

r_e = %

7.NPV & IRRMedium

A machine costs €20 000 today, lasts 6 years and costs €1 000 per year to run (paid at the end of each year). The cost of capital is 8%. Find its equivalent annual cost EAC.

C₀ = €20 000M = €1 000n = 6r = 8%

EAC = €

8.NPV & IRRHard

A project costs €40 000 and returns €23 500, €20 000, €22 500 at the end of years 1 to 3. Cash flows can be reinvested at the cost of capital of 6%. Find the modified internal rate of return MIRR in percent.

r = 6%CF₀ = −40 000CF₁ = 23 500CF₂ = 20 000CF₃ = 22 500

MIRR = %

3Solutions

1.Time value of money

Answer: EAR = 25.44 %

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 per period

r/m = 0.24/2 = 0.12

Compound over one year

EAR = (1 + 0.12)² − 1 = 1.254 − 1 = 25.44%

Answer

EAR = 25.44%

The EAR is always above the nominal rate when interest compounds more than once a year.

2.Time value of money

Answer: FV = 44009 €

Monthly rate and number of deposits

i = 0.07200000000000001/12 = 0.006

N = 12·18 = 216

Future value of the deposits

FV = A·((1 + i)^N − 1)/i = 100·(3.641 − 1)/0.006 = €44 008.72

Answer

FV = €44 008.72

You paid in €21 600 yourself; the rest is interest.

3.Bond pricing & yield

Answer: c = 2.01 %

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 + 0.03)⁻³ = €915.14

Coupon

C = (972 − 915.14)/2.829 = €20.10

Coupon rate

c = 20.1/1000 = 2.01%

Answer

c = 2.01%

4.Bond pricing & yield

Answer: YTM = 3.997 %

Equation for the yield

€1 105 = €60·(1 − (1 + y)⁻⁶)/y + €1 000/(1 + y)⁶

Solve numerically

P(3%) = €1 162.52, P(4%) = €1 104.84

YTM = 3.997%

Answer

YTM = 3.997%

5.CAPM & WACC

Answer: WACC = 9.384 %

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 = 8 000 000 + 1 500 000 + 6 000 000 = €15 500 000

E/V = 0.5161, P_s/V = 0.09677, D/V = 0.3871

WACC

WACC = 0.5161·14% + 0.09677·6.5% + 0.3871·5%·(1 − 0.21) = 9.384%

Answer

WACC = 9.384%

6.CAPM & WACC

Answer: r_e = 10.71 %

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 = 1.4/(1 + 0.742·0.5) = 1.021

Relever at the new D/E

β_L = 1.021·(1 + 0.742·1.5) = 2.158

Cost of equity (CAPM)

r_e = 1% + 2.158·4.5% = 10.71%

Answer

r_e = 10.71%

7.NPV & IRR

Answer: EAC = 5326 €

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

20 000/4.623 = 4 326.31

EAC

EAC = 4 326.31 + 1 000 = €5 326.31

Answer

EAC = €5 326.31

8.NPV & IRR

Answer: MIRR = 20.57 %

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

23 500·1.06² = 26 404.60

20 000·1.06¹ = 21 200.00

22 500·1.06⁰ = 22 500.00

TV = 70 104.60

MIRR

MIRR = (70 104.60/40 000)^1/3 − 1 = 20.57%

Answer

MIRR = 20.57%

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