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## Fm213 Formula Sheet Notes This is an extract of our Fm213 Formula Sheet document, which we sell as part of our Principles of Finance Notes collection written by the top tier of LSE students.

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FM213 Formula Sheet
Statistics

𝐶𝑜𝑣(𝑋, 𝑌) = 𝜎!,# = 𝜌!,# 𝜎! 𝜎#

General calculation notes

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.

Risk-free debt: β\$ = 0, r\$ = r%

Unlevered (all equity) firm: β& = β' , r& = r'
r = r% + β(r( − r% )

CAPM

r& = r% + β& (r( − r%)
r' = r% + β' (r( − r% )

Gordon's growth model

P) =

DIV*
r−g

Present value calculations
PV formula

Perpetuity

W/o growth

𝑃𝑉+ = 8,.*

Annuity

𝐶
𝐶
=
,
(1 + 𝑟)
𝑟

𝑃𝑉/ = 𝑃𝑉* − 𝑃𝑉0 =
=

w/ growth𝑃𝑉+ = 8
,.*

𝐶
𝐶(1 + 𝑔),2*
=
(1 + 𝑟),
𝑟−𝑔

𝐶

1 <1 −
=
(1 + 𝑟)1
𝑟

𝑃𝑉/ =

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

=

𝐶
𝐶

1 −

1 𝑟 (1 + 𝑟) 𝑟

1+𝑔 1 𝐶
𝐶
−?
@
1+𝑟 𝑟−𝑔
𝑟−𝑔

1+𝑔 1
𝐶
@ B
A1 − ?
1+𝑟
𝑟−𝑔

cause the sum to be infinite

Converting rates with different payment frequencies
Continuous discounting
Real interest rates

Approximate real interest rates

(1 + 𝑟3 )*0 = C1 + 𝑟4 D + (1 + 𝑟6 )0 = 1 + 𝑟7 5

𝑃𝑉 = 𝑋𝑒 281
(1 + 𝑖) =
𝑖 ≈𝑟−𝜋

1+𝑟
1+π 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 −
@+
(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:;* − P: ] E: (D:;* ) E: (P:;* − 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 −

>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* µ* + x0 µ0 + . . . + xS µS = 8 x< µ<
Two asset p/f: E(R > ) = x* µ* + x0 µ0
S

S

) = 8 8 x< xV ρ ) = x*0 σ*0 + x00 σ00 + 2x* x0 ρ*,0 σ* σ0 = x*0 σ*0 + x00 σ00 + 2x* x0 cov(x* , x0 )
Covariance

𝑐𝑜𝑣(𝑋, 𝑌) =

Beta

β< =

∑(𝑋, − 𝑋~)(𝑌, − 𝑌~)
𝑁−1

Cov(R < , R > ) 𝜌,,+ σ, σ+
σ<
=
= ρ
σ>
Var(R > )
σ0+ Beta of a

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

portfolio

β> = 8 x< β<

S

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