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

An annuity pays €1 500 at the end of each year for 4 years. The interest rate is 8% per year. Find the present value PV of the annuity.

A = €1 500i = 8%n = 4

PV = €

2.Time value of moneyHard

A mortgage of €200 000 is repaid with equal monthly payments over 25 years at 3.6% per year, compounded monthly. How much is still owed just after the payment at the end of year 13? Find the outstanding balance B.

P = €200 0003.6% / 12N = 300k = 156

B = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays a 3% annual coupon in two half-yearly payments and matures in 7 years. The yield to maturity is 2.5% per year. Find the price P of the bond.

F = €1 000coupon = 3%YTM = 2.5%n = 7

P = €

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 7% matures in 4 years and sells for €1 055. Find its yield to maturity YTM in percent.

F = €1 000coupon = 7%n = 4P = €1 055

YTM = %

5.CAPM & WACCMedium

A company has a debt-to-equity ratio (at market values) of 0.4. Its cost of equity is 9%, its pre-tax cost of debt is 4% and the tax rate is 25.8%. Find the WACC.

D/E = 0.4r_e = 9%r_d = 4%T = 25.8%

WACC = %

6.CAPM & WACCHard

A company's business has an unlevered (asset) beta of 1. It plans a debt-to-equity ratio of 0.75. The tax rate is 30%, the risk-free rate is 4% and the market risk premium is 6%. Find the cost of equity r_e at the new capital structure.

β_U = 1D/E = 0.75T = 30%r_f = 4%r_m − r_f = 6%

r_e = %

7.NPV & IRRMedium

An investment of €30 000 gives these cash flows at the end of years 1 to 4: €10 000, €11 000, €6 000, €14 000. The cost of capital is 10%. Find the net present value NPV.

r = 10%CF₀ = −30 000CF₁ = 10 000CF₂ = 11 000CF₃ = 6 000CF₄ = 14 000

NPV = €

8.NPV & IRRHard

A project costs €100 000 today. Its cash flow is €22 000 at the end of year 1 and grows by 4% per year until the last cash flow at the end of year 5. The cost of capital is 12%. Find the net present value NPV.

C₀ = €100 000CF₁ = €22 000g = 4%n = 5r = 12%

NPV = €

3Solutions

1.Time value of money

Answer: PV = 4968 €

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.

Annuity factor

(1 − (1 + i)⁻ⁿ)/i = (1 − 0.735)/0.08 = 3.312

Present value

PV = 1 500·3.312 = €4 968.19

Answer

PV = €4 968.19

2.Time value of money

Answer: B = 118192 €

Monthly payment

i = 0.003, N = 300

A = 200 000·0.003/(1 − (1 + 0.003)⁻³⁰⁰) = €1 012.01

Present value of the remaining payments

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

N − k = 300 − 156 = 144

B = 1012·(1 − (1 + 0.003)⁻¹⁴⁴)/0.003 = €118 192.33

Answer

B = €118 192.33

3.Bond pricing & yield

Answer: P = 1032 €

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.

Per period

C = €15.00, i = 0.0125, N = 14

Present value of the coupons

€15.00·(1 − 0.8404)/0.0125 = €191.56

Present value of the face value

€1 000·0.8404 = €840.37

Price

P = €1 031.93

Answer

P = €1 031.93

The yield is below the coupon rate, so the bond sells at a premium.

4.Bond pricing & yield

Answer: YTM = 5.433 %

Equation for the yield

€1 055 = €70·(1 − (1 + y)⁻⁴)/y + €1 000/(1 + y)⁴

Solve numerically

P(5%) = €1 070.92, P(6%) = €1 034.65

YTM = 5.433%

Answer

YTM = 5.433%

5.CAPM & WACC

Answer: WACC = 7.277 %

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 from D/E

E/V = 1/(1 + 0.4) = 0.7143

D/V = 0.4/(1 + 0.4) = 0.2857

WACC

WACC = 0.7143·9% + 0.2857·4%·(1 − 0.258) = 7.277%

Answer

WACC = 7.277%

6.CAPM & WACC

Answer: r_e = 13.15 %

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.

Levered beta

β_L = 1·(1 + 0.7·0.75) = 1.525

Cost of equity (CAPM)

r_e = 4% + 1.525·6% = 13.15%

Answer

r_e = 13.15%

7.NPV & IRR

Answer: NPV = 2252 €

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 + 0.1)¹ = 10 000/1.1 = 9 090.91

CF₂/(1 + 0.1)² = 11 000/1.21 = 9 090.91

CF₃/(1 + 0.1)³ = 6 000/1.331 = 4 507.89

CF₄/(1 + 0.1)⁴ = 14 000/1.464 = 9 562.19

Add them up

NPV = €2 251.90

Answer

NPV = €2 251.90

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

8.NPV & IRR

Answer: NPV = −14849 €

Present value of the growing annuity

((1.04)/(1.12))⁵ = 0.6904

PV = 22 000/(0.12 − 0.04)·(1 − 0.6904) = 85 150.58

NPV

NPV = 85 150.58 − 100 000 = −€14 849.42

Answer

NPV = −€14 849.42

NPV is negative, so the project earns less than the required return.

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