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 €12 000 at a nominal rate of 4% per year, compounded monthly. Find the value FV of the investment after 10 years.

PV = €12 0004% / 12n = 10

FV = €

2.Time value of moneyHard

A loan of €200 000 is repaid with equal monthly payments over 30 years. The interest rate is 4.8% per year, compounded monthly. Find the monthly payment A.

P = €200 0004.8% / 12N = 360

A = €

3.Bond pricing & yieldMedium

You buy a bond with 5 years to maturity (face value €1 000, annual coupon 2%) when its yield to maturity is 5.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 6.5%. Find your holding period return HPR in percent.

F = €1 000coupon = 2%n = 5y₀ = 5.5%y₁ = 6.5%

HPR = %

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 3% matures in 6 years and sells for €875. Find its yield to maturity YTM in percent.

F = €1 000coupon = 3%n = 6P = €875

YTM = %

5.CAPM & WACCMedium

A share with a beta of 1.5 has a required return of 10.75%. The risk-free rate is 4%. What expected market return r_m does CAPM imply?

β = 1.5r_e = 10.75%r_f = 4%

r_m = %

6.CAPM & WACCHard

A company has 1 000 000 shares trading at €21.50; the book value of its equity is €12 900 000. It just paid a dividend of €1.50 per share, and dividends grow by 3% per year. Its debt has a market value of €12 900 000 and a pre-tax cost of 3%. The tax rate is 28%. Find the WACC, using the dividend growth model for the cost of equity.

N = 1 000 000P₀ = €21.50book equity = €12 900 000D₀ = €1.50g = 3%D = €12 900 000r_d = 3%T = 28%

WACC = %

7.NPV & IRRMedium

A project costs €50 000 today and returns €18 000 at the end of each year for 5 years. The cost of capital is 5%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.

r = 5%CF₀ = −50 000CF₁ = 18 000CF₂ = 18 000CF₃ = 18 000CF₄ = 18 000CF₅ = 18 000

DPB = years

8.NPV & IRRHard

Project A costs €40 000 and returns €22 000, €16 000, €9 500 at the end of years 1 to 3. Project B costs €45 000 and returns €13 000, €18 000, €23 500. Find the crossover rate r* at which both projects have the same NPV, in percent.

A: −40 000, 22 000, 16 000, 9 500B: −45 000, 13 000, 18 000, 23 500

r* = %

3Solutions

1.Time value of money

Answer: FV = 17890 €

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 = 0.04/12 = 0.003333

N = m·n = 12·10 = 120

Future value

FV = 12 000·(1 + 0.003333)¹²⁰ = 12 000·1.491 = €17 889.99

Answer

FV = €17 889.99

2.Time value of money

Answer: A = 1049 €

Monthly rate and number of payments

i = 0.048/12 = 0.004

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 = 0.2376

A = 200 000·0.004/(1 − 0.2376) = €1 049.33

Answer

A = €1 049.33

3.Bond pricing & yield

Answer: HPR = 1.799 %

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.

Purchase price

C = €20.00, i = 0.055, N = 5

€20.00·(1 − 0.7651)/0.055 + €1 000·0.7651 = €85.41 + €765.13 = €850.54

Selling price

C = €20.00, i = 0.065, N = 4

€20.00·(1 − 0.7773)/0.065 + €1 000·0.7773 = €68.52 + €777.32 = €845.84

Holding period return

HPR = (20 + 845.84 − 850.54)/850.54 = 1.799%

Answer

HPR = 1.799%

4.Bond pricing & yield

Answer: YTM = 5.502 %

Equation for the yield

€875 = €30·(1 − (1 + y)⁻⁶)/y + €1 000/(1 + y)⁶

Solve numerically

P(5%) = €898.49, P(6%) = €852.48

YTM = 5.502%

Answer

YTM = 5.502%

5.CAPM & WACC

Answer: r_m = 8.5 %

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 = (10.75% − 4%)/1.5 = 4.5%

Market return

r_m = 4% + 4.5% = 8.5%

Answer

r_m = 8.5%

6.CAPM & WACC

Answer: WACC = 7.176 %

Cost of equity

D₁ = 1.5·1.03 = €1.54

r_e = 1.545/21.5 + 0.03 = 10.19%

Market values

E = 1 000 000·€21.50 = €21 500 000

V = €34 400 000, E/V = 0.625, D/V = 0.375

WACC

WACC = 0.625·10.19% + 0.375·3%·(1 − 0.28) = 7.176%

Answer

WACC = 7.176%

7.NPV & IRR

Answer: DPB = 3.066 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: 18 000/1.05 = 17 142.86 → Σ = 17 142.86

year 2: 18 000/1.103 = 16 326.53 → Σ = 33 469.39

year 3: 18 000/1.158 = 15 549.08 → Σ = 49 018.46

year 4: 18 000/1.216 = 14 808.64 → Σ = 63 827.11

Fraction of the last year

DPB = 3 + 981.54/14 808.64 = 3.066

Answer

DPB = 3.066 years

The discounted payback is always longer than the simple payback (2.778 years).

8.NPV & IRR

Answer: r* = 6.155 %

Crossover rate: the discount rate at which two projects have the same NPV. It is the IRR of the difference in their cash flows (B − A). Below it the project with the later, larger cash flows has the higher NPV; above it the other one wins.

Difference in cash flows

ΔCF₀ = −45 000 − (−40 000) = −5 000

ΔCF₁ = 13 000 − 22 000 = −9 000

ΔCF₂ = 18 000 − 16 000 = 2 000

ΔCF₃ = 23 500 − 9 500 = 14 000

IRR of the difference

NPV_Δ(6%) = 44.10

NPV_Δ(7%) = −236.17

r* = 6.155%

Answer

r* = 6.155%

Check: at 6.155% both NPVs are €2 864.57.

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