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Fm213 Formula Sheet Notes

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FM213 Formula Sheet
Statistics

πΆπ‘œπ‘£(𝑋, π‘Œ) = 𝜎!,# = 𝜌!,# 𝜎! 𝜎#

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( βˆ’ r% )

CAPM

r& = r% + Ξ²& (r( βˆ’ r%)
r' = r% + Ξ²' (r( βˆ’ r% )

Gordon's growth model

P) =

DIV*
rβˆ’g

Present value calculations
PV formula

Perpetuity

W/o growth

𝑃𝑉+ = 8,.*

Annuity

𝐢
𝐢
=
,
(1 + π‘Ÿ)
π‘Ÿ

𝑃𝑉/ = 𝑃𝑉* βˆ’ 𝑃𝑉0 =
=

w/ growth𝑃𝑉+ = 8
,.*

𝐢
𝐢(1 + 𝑔),2*
=
(1 + π‘Ÿ),
π‘Ÿβˆ’π‘”

𝐢

1 <1 βˆ’
=
(1 + π‘Ÿ)1
π‘Ÿ

𝑃𝑉/ =

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

=

𝐢
𝐢

1 βˆ’

1 π‘Ÿ (1 + π‘Ÿ) π‘Ÿ

1+𝑔 1 𝐢
𝐢
βˆ’?
@
1+π‘Ÿ π‘Ÿβˆ’π‘”
π‘Ÿβˆ’π‘”

1+𝑔 1
𝐢
@ B
A1 βˆ’ ?
1+π‘Ÿ
π‘Ÿβˆ’π‘”

cause the sum to be infinite

Converting rates with different payment frequencies
Continuous discounting
Real interest rates

Approximate real interest rates

(1 + π‘Ÿ3 )*0 = C1 + π‘Ÿ4 D + (1 + π‘Ÿ6 )0 = 1 + π‘Ÿ7 5

𝑃𝑉 = 𝑋𝑒 281
(1 + 𝑖) =
𝑖 β‰ˆπ‘Ÿβˆ’πœ‹

1+π‘Ÿ
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 βˆ’
@+
(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:;* βˆ’ P: ] E: (D:;* ) E: (P:;* βˆ’ 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 βˆ’

>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 ρ ) = x*0 Οƒ*0 + x00 Οƒ00 + 2x* x0 ρ*,0 Οƒ* Οƒ0 = x*0 Οƒ*0 + x00 Οƒ00 + 2x* x0 cov(x* , x0 )
Covariance

π‘π‘œπ‘£(𝑋, π‘Œ) =

Beta

Ξ²< =

βˆ‘(𝑋, βˆ’ 𝑋~)(π‘Œ, βˆ’ π‘Œ~)
π‘βˆ’1

Cov(R < , R > ) 𝜌,,+ Οƒ, Οƒ+
Οƒ<
=
= ρ
Οƒ>
Var(R > )
Οƒ0+ 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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