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 €8 000 at the end of each year for 6 years. The interest rate is 3% per year. Find the present value PV of the annuity.

A = €8 000i = 3%n = 6

PV = €

2.Time value of moneyHard

You deposit €3 000 at the start of every year for 23 years, the first deposit today. The account pays 5% per year. How much is in the account at the end of year 23? Find FV.

A = €3 000i = 5%n = 23

FV = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays a 2% annual coupon in two half-yearly payments and matures in 9 years. The yield to maturity is 8% per year. Find the price P of the bond.

F = €1 000coupon = 2%YTM = 8%n = 9

P = €

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 8% matures in 10 years and sells for €1 145. Find its yield to maturity YTM in percent.

F = €1 000coupon = 8%n = 10P = €1 145

YTM = %

5.CAPM & WACCMedium

A company has a debt-to-equity ratio (at market values) of 0.25. Its cost of equity is 13%, its pre-tax cost of debt is 6% and the tax rate is 20%. Find the WACC.

D/E = 0.25r_e = 13%r_d = 6%T = 20%

WACC = %

6.CAPM & WACCHard

A company's business has an unlevered (asset) beta of 1.1. It plans a debt-to-equity ratio of 0.25. The tax rate is 21%, the risk-free rate is 3% and the market risk premium is 4%. Find the cost of equity r_e at the new capital structure.

β_U = 1.1D/E = 0.25T = 21%r_f = 3%r_m − r_f = 4%

r_e = %

7.NPV & IRRMedium

A machine costs €80 000 today. It generates €17 500 at the end of each year for 5 years and can then be sold for €16 000. The cost of capital is 12%. Find the net present value NPV.

C₀ = €80 000CF = €17 500S = €16 000n = 5r = 12%

NPV = €

8.NPV & IRRHard

Project A costs €20 000 and returns €11 000, €8 000, €5 000 at the end of years 1 to 3. Project B costs €22 000 and returns €6 000, €8 500, €12 500. Find the crossover rate r* at which both projects have the same NPV, in percent.

A: −20 000, 11 000, 8 000, 5 000B: −22 000, 6 000, 8 500, 12 500

r* = %

3Solutions

1.Time value of money

Answer: PV = 43338 €

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 − 0.8375)/0.03 = 5.417

Present value

PV = 8 000·5.417 = €43 337.53

Answer

PV = €43 337.53

2.Time value of money

Answer: FV = 130506 €

Ordinary annuity factor

((1 + 0.05)²³ − 1)/0.05 = 41.43

Annuity due

Each deposit is invested one year longer.

FV = 3 000·41.43·1.05 = €130 506.00

Answer

FV = €130 506.00

3.Bond pricing & yield

Answer: P = 620.2 €

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 = €10.00, i = 0.04, N = 18

Present value of the coupons

€10.00·(1 − 0.4936)/0.04 = €126.59

Present value of the face value

€1 000·0.4936 = €493.63

Price

P = €620.22

Answer

P = €620.22

The yield is above the coupon rate, so the bond sells at a discount.

4.Bond pricing & yield

Answer: YTM = 6.027 %

Equation for the yield

€1 145 = €80·(1 − (1 + y)⁻¹⁰)/y + €1 000/(1 + y)¹⁰

Solve numerically

P(6%) = €1 147.20, P(7%) = €1 070.24

YTM = 6.027%

Answer

YTM = 6.027%

5.CAPM & WACC

Answer: WACC = 11.36 %

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 + 0.25) = 0.8

D/V = 0.25/(1 + 0.25) = 0.2

WACC

WACC = 0.8·13% + 0.2·6%·(1 − 0.2) = 11.36%

Answer

WACC = 11.36%

6.CAPM & WACC

Answer: r_e = 8.269 %

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·(1 + 0.79·0.25) = 1.317

Cost of equity (CAPM)

r_e = 3% + 1.317·4% = 8.269%

Answer

r_e = 8.269%

7.NPV & IRR

Answer: NPV = −7838 €

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 the yearly cash flows

17 500·3.605 = 63 083.58

Present value of the salvage value

16 000/1.762 = 9 078.83

NPV

NPV = −80 000 + 63 083.58 + 9 078.83 = −€7 837.59

Answer

NPV = −€7 837.59

8.NPV & IRR

Answer: r* = 6.178 %

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 = −22 000 − (−20 000) = −2 000

ΔCF₁ = 6 000 − 11 000 = −5 000

ΔCF₂ = 8 500 − 8 000 = 500

ΔCF₃ = 12 500 − 5 000 = 7 500

IRR of the difference

NPV_Δ(6%) = 25.16

NPV_Δ(7%) = −113.94

r* = 6.178%

Answer

r* = 6.178%

Check: at 6.178% both NPVs are €1 633.19.

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