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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

?!"#

@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 ) = x* Β΅* + x0 Β΅0 + . . . + xS Β΅S = 8 x< Β΅<

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

S

S

) = 8 8 x< xV Ο ) = 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

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