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FM213 Formula Sheet

Statistics

πΆππ£(π, π) = π!,# = π!,# π! π#

General calculation notes

β’

In order to construct a zero-risk, zero-cost, positive income in perpetuity portfolio the investor needs to make sure that his total cash flow today is zero and the future cash flows are risk free.

β’

β’

Risk-free debt: Ξ²$ = 0, r$ = r%

Unlevered (all equity) firm: Ξ²& = Ξ²' , r& = r'

r = r% + Ξ²(r( β r% )

CAPM

r& = r% + Ξ²& (r( β r%)

r' = r% + Ξ²' (r( β r% )

Gordon's growth model

P) =

DIV*

rβg

Present value calculations

PV formula

Perpetuity

W/o growth

ππ+ = 8,.*

Annuity

πΆ

πΆ

=

,

(1 + π)

π

ππ/ = ππ* β ππ0 =

=

w/ growthππ+ = 8

,.*

πΆ

πΆ(1 + π),2*

=

(1 + π),

πβπ

πΆ

1 <1 β

=

(1 + π)1

π

ππ/ =

Requires r > g, otherwise CF grows more quickly than discount factor will

=

πΆ

πΆ

1 β

1 π (1 + π) π

1+π 1 πΆ

πΆ

β?

@

1+π πβπ

πβπ

1+π 1

πΆ

@ B

A1 β ?

1+π

πβπ

cause the sum to be infinite

Converting rates with different payment frequencies

Continuous discounting

Real interest rates

Approximate real interest rates

(1 + π3 )*0 = C1 + π4 D + (1 + π6 )0 = 1 + π7 5

ππ = ππ 281

(1 + π) =

π βπβπ

1+π

1+Ο Stocks and bonds

Bond price and YTM

P=

F

cF

cF

cF

cF

+

+

+. . +

=8

(1 + y)9

(1 + y): (1 + y)9 1 + y (1 + y)0 9

:.*

cF

1 F

= ?1 β

@+

(1 + y)9

(1 + y)9 y

Semi-annual coupon bond

Expected return on a share

P=

r=

cF

cF

cF

+F

2 2 2

+

+.

.

+

0 y y 09

O1 + P O1 + yP

O1 + 2P

2 2

E: [D:;* + P:;* β P: ] E: (D:;* ) E: (P:;* β P: )

=

+

P:

P:

P:

Expected return = CF at t+1/current price = expected div. yield + expected capital gain

(expected %+ in the share price)

Share price,

assuming perpetualP: = 8

<.*

E: (D:;< )

(1 + rT)<

stream of dividends

Share price,

assuming perpetualP: = 8

<.*

D

D

=

(1 + rT)< rT

stream of constant dividends

Gordon growth formulaP: = 8

<.*

P) =

Return on Equity

Earning's growth

D(1 + g)<2* D:;*

=

(1 + rT)<

rT β g

(1 β plowback ratio) Γ EPS*

rT β ROE Γ plowback ratio

ROE* =

EPS*

BV) (Equity) per share

g = ROE Γ Plowback ratio

If ROE and plowback ratio are constant, then earning's growth = dividend's growth

Dividend's growth

Present Value of

π = π= β πππ£. π¦ππππ = π= β

β’

Growth

Opportunities

(PVGO)

β’

β’

β’

P=

'>?

@

+ PVGO

PVGO: =

πππ£

πΈππ

(*2BCDEFGHI @G:?!"#

@M

Without growth, the price will equal the earnings for next period capitalised at r

'>?

>

= r(1 β

>PQO

>

)

PVGO - while is the stock price higher when a company plows back its earnings.

The PVGO comes from the fact that the firm is retaining earnings that are generating a return of 20% (the ROE)

while the discount rate is only 10%. Thus the value of the firm rises. Valuing government bonds

Macaulay duration (negative elasticity)

Shortcoming of duration: use of a linear

1 C<

D = 8i

(1 + y)<

P

I <.*

approximation for a non-linear convex relationship between bond prices and yields

DR =

Modified duration (volatility)

D

1+y

%ΞP =

Spot rate and forward interest rate

ΞP

β βDR Ξy

P

(1 + r0 )0 = (1 + r* )(1 + f0 )

(1 + rS )S = (1 + rS2* )S2* (1 + fS ) = (1 + r*)(1 + f0 ) β¦ (1 + fS )

f0 =

(1 + r0 )0

β1

(1 + r* )

πT =

Inferring one- and two-year forward rates with CF for 2 bonds with different CF

(1 + πT )T

β1

(1 + πT2* )T2*

P = C* d* + C0 d0 , d* =

1 1

, d0 =

(1 + r* )(1 + f* )

1 + r*

streams

Interpretation of forward rates:

1. no-arbitrage one-year rate agreed today for a loan to start in one year's time

2. Pure Expectation Hypothesis (the (constant) premium is zero, T = 0, for all time-periods and all bonds,

regardless of time-to-maturity): forward rate is the market forecast of the 1-year rate in one year's time

Risk, return and cost of capital variance

S

v )0

Ο = 8 P< (R < β R

0 <.*

Sample variance

Expected return of portfolio

Portfolio variance

s0 =

1 v)0 8(R < β R

Kβ1

U

<.*

S

E(R > ) = x* Β΅* + x0 Β΅0 + . . . + xS Β΅S = 8 x< Β΅<

Two asset p/f: E(R > ) = x* Β΅* + x0 Β΅0

S

S

<.*

S

Var(R > ) = 8 8 x< xV Ο<,V Ο< ΟV = 8 x<0 Ο0< + 8 x< xV Ο<,V Ο< ΟV

<.* V.*

<.*

) = x*0 Ο*0 + x00 Ο00 + 2x* x0 Ο*,0 Ο* Ο0 = x*0 Ο*0 + x00 Ο00 + 2x* x0 cov(x* , x0 )

Covariance

πππ£(π, π) =

Beta

Ξ²< =

β(π, β π~)(π, β π~)

πβ1

Cov(R < , R > ) π,,+ Ο, Ο+

Ο<

=

= Ο<,>

Ο>

Var(R > )

Ο0+ Beta of a

Beta of the portfolio is the weighted average of betas of individual stocks

portfolio

Ξ²> = 8 x< Ξ²<

S

<.*

Portfolio variance

Ο0,

= Ξ²0, Ο0+ + Ο0X,

0 = π,Y

Ο0, 0

Ο + ΟX,

Ο0+ +

= CΟ,Y Ο, D + Ο0X, , 0 < Ο < 1 0

variance = Systematic/aggregate/market-wide risk + idiosyncratic risk/firm-specific risk

Difference between variance and beta as a measure of risk

Variance

Beta

β’ measures the total risk (both unique risk and market β’ a measure of market risk - measures the risk) of a security

sensitivity of the security returns to changes in market returns

β’ a measure of stand-alone risk

β’ In a well-diversified portfolio, unique risks tend to

β’ useful in the context of a well-diversified

cancel each other out and only the market risk remains.

portfolio.

β’ Market portfolio has a beta of one.

Portfolio theory and CAPM

Feasible set solution equation

πππ Ο0+ π . π‘. βπ₯, = 1, πΈ(π + ) = πΈ

Z# ,Z$ ,...,Z%

CML

π + = π \ +

πΈ(π
3 ) β π
\

Γ Ο+ = π
\ + ππ
3 Γ Ο+

π3

ππ
3 = sharpe ratio of the market

CAPM

ECR V D = R % + Ξ²V (E(R R ) β R % )

If a firm has zero systematic risk, π ] = π \

π½] =

π) =

πΆππ£Cπ
] , π
^ D

πππ(π
^ )

π·*

π·*

=

π β π (π
\ + Ξ²Ε πΈ(π
^ ) β π
\ D β π

Excess return

πΈCπ ] D β π \ = πΌ] + π½] CπΈ(π ^ ) β π \ D

APT

πΈCπ ] D = π \ + π½*,] Cπ _* β π \ D + π½0,] (π _0 β π \ )

Farma-French 3 factor model

Sharpe ratio

πΈ(Ξ±) = 0

E(R V ) β R % = Ξ²R,V (E(R R ) β R % ) + Ξ²?,V (E(R ?R` )) + Ξ²P,V (E(R aR` ))

π hππππ πππ‘ππ =

ππ₯πππ π πππ‘π’ππ

π£ππππ‘ππππ‘π¦

Sharpe ratio = risk/return trade-off

Under the CAPM, we would expect the market portfolio to dominate as it delivers the best possible Sharpe Ratio.

β’

in a CAPM world: the market portfolio is an efficient portfolio β’

A combination of the market portfolio and the risk-free asset (sometimes called a 'leveraged position' in the market) has

o

highest expected return of any portfolio for a given volatility

o

lowest volatility for a given expected return.

Impacts of risk preferences on portfolio selection

β’

The portfolio with a higher Sharpe ratio is preferred by someone who is willing to take a little risk in order to earn more return

β’

The variance minimising portfolio is appropriate for someone who cares only about experiencing the smallest possible risk.

β’

The MVP investor invests most of her cash in the safe asset, while the EWP(equally weighted portfolio)

investor invests more of the cash in the riskier asset as it has greater mean return

β’

As soon as the investor starts to worry about risk, then diversification plays a role.

Forwards and futures

Long payoff

S9 β F

Short payoff

F β S9

Making to market

β’

Gain = margin balance - initial margin

β’

Loss = total amount in margin calls

β’

In total, the investor pays quantity * market price of the asset + loss in margin calls - gain

Covered interest rate parity

πΉ\ =

b

πC1 + π\_c8d,ef D

1 C1 + π\gc3d6h,i D

1

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