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

Buy the full version of these notes or essay plans and more in our Principles of Finance Notes.