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 €20 000 at 2% per year with continuous compounding. Find the value FV after 15 years.

PV = €20 000r = 2%t = 15

FV = €

2.Time value of moneyHard

A mortgage of €250 000 is repaid with equal monthly payments over 25 years at 2.4% per year, compounded monthly. How much is still owed just after the payment at the end of year 19? Find the outstanding balance B.

P = €250 0002.4% / 12N = 300k = 228

B = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays an annual coupon of 6% and matures in 5 years. Its yield to maturity changes from 8% to 6.5%. By how much ΔP does the bond price change?

F = €1 000coupon = 6%n = 5y₁ = 8%y₂ = 6.5%

ΔP = €

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 4% matures in 4 years and sells for €1 035. Find its yield to maturity YTM in percent.

F = €1 000coupon = 4%n = 4P = €1 035

YTM = %

5.CAPM & WACCMedium

A company has equity worth €8 000 000 and debt worth €5 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 7% and the tax rate is 25.8%. Find the WACC.

E = €8 000 000D = €5 000 000r_e = 10%r_d = 7%T = 25.8%

WACC = %

6.CAPM & WACCHard

A company's shares have a beta of 1.2 at its current debt-to-equity ratio of 0.4. It plans to raise its debt-to-equity ratio to 1. The tax rate is 30%, the risk-free rate 3.5% and the market risk premium 5%. Find the cost of equity r_e after the change.

β_L = 1.2D/E₁ = 0.4D/E₂ = 1T = 30%r_f = 3.5%r_m − r_f = 5%

r_e = %

7.NPV & IRRMedium

An investment of €30 000 gives these cash flows at the end of years 1 to 4: €10 000, €12 000, €11 000, €12 000. The cost of capital is 15%. Find the net present value NPV.

r = 15%CF₀ = −30 000CF₁ = 10 000CF₂ = 12 000CF₃ = 11 000CF₄ = 12 000

NPV = €

8.NPV & IRRHard

Project A costs €20 000 and returns €11 500, €8 000, €4 000 at the end of years 1 to 3. Project B costs €30 000 and returns €8 000, €12 500, €15 500. Find the crossover rate r* at which both projects have the same NPV, in percent.

A: −20 000, 11 500, 8 000, 4 000B: −30 000, 8 000, 12 500, 15 500

r* = %

3Solutions

1.Time value of money

Answer: FV = 26997 €

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.

Exponent

r·t = 0.02·15 = 0.3

e^0.3 = 1.35

Future value

FV = 20 000·1.35 = €26 997.18

Answer

FV = €26 997.18

2.Time value of money

Answer: B = 74296 €

Monthly payment

i = 0.002, N = 300

A = 250 000·0.002/(1 − (1 + 0.002)⁻³⁰⁰) = €1 108.99

Present value of the remaining payments

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

N − k = 300 − 228 = 72

B = 1109·(1 − (1 + 0.002)⁻⁷²)/0.002 = €74 295.73

Answer

B = €74 295.73

3.Bond pricing & yield

Answer: ΔP = 59.08 €

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 = 0.08, N = 5

€60.00·(1 − 0.6806)/0.08 + €1 000·0.6806 = €239.56 + €680.58 = €920.15

Price at the new yield

C = €60.00, i = 0.065, N = 5

€60.00·(1 − 0.7299)/0.065 + €1 000·0.7299 = €249.34 + €729.88 = €979.22

Change

ΔP = 979.22 − 920.15 = €59.08

Answer

ΔP = €59.08

4.Bond pricing & yield

Answer: YTM = 3.057 %

Equation for the yield

€1 035 = €40·(1 − (1 + y)⁻⁴)/y + €1 000/(1 + y)⁴

Solve numerically

P(3%) = €1 037.17, P(4%) = €1 000.00

YTM = 3.057%

Answer

YTM = 3.057%

5.CAPM & WACC

Answer: WACC = 8.152 %

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 = 0.6154·10% + 0.3846·7%·(1 − 0.258) = 8.152%

Answer

WACC = 8.152%

6.CAPM & WACC

Answer: r_e = 11.47 %

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 = 1.2/(1 + 0.7·0.4) = 0.9375

Relever at the new D/E

β_L = 0.9375·(1 + 0.7·1) = 1.594

Cost of equity (CAPM)

r_e = 3.5% + 1.594·5% = 11.47%

Answer

r_e = 11.47%

7.NPV & IRR

Answer: NPV = 1863 €

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 + 0.15)¹ = 10 000/1.15 = 8 695.65

CF₂/(1 + 0.15)² = 12 000/1.322 = 9 073.72

CF₃/(1 + 0.15)³ = 11 000/1.521 = 7 232.68

CF₄/(1 + 0.15)⁴ = 12 000/1.749 = 6 861.04

Add them up

NPV = €1 863.09

Answer

NPV = €1 863.09

NPV is positive, so the project earns more than the required return.

8.NPV & IRR

Answer: r* = 7.154 %

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 = −30 000 − (−20 000) = −10 000

ΔCF₁ = 8 000 − 11 500 = −3 500

ΔCF₂ = 12 500 − 8 000 = 4 500

ΔCF₃ = 15 500 − 4 000 = 11 500

IRR of the difference

NPV_Δ(7%) = 46.87

NPV_Δ(8%) = −253.65

r* = 7.154%

Answer

r* = 7.154%

Check: at 7.154% both NPVs are €950.93.

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