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
<.*

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 − g

(1 − plowback ratio) × EPS*
rT − ROE × plowback ratio

ROE* =

EPS*
BV) (Equity) per share

g = ROE × Plowback ratio

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

Present Value of

𝑔 = 𝑟= − 𝑑𝑖𝑣. 𝑦𝑖𝑒𝑙𝑑 = 𝑟= −

•

Growth
Opportunities
(PVGO)

•

•

•

P=

'>?
@

+ PVGO

PVGO: =

𝑑𝑖𝑣
𝐸𝑃𝑆

(*2BCDEFGHI @G:?!"#
@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 <.*

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

Modified duration (volatility)

D
1+y

%ΔP =
Spot rate and forward interest rate

ΔP
≈ −DR Δy
P

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

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

(1 + r0 )0
−1
(1 + r* )

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

(1 + 𝑟T )T
−1
(1 + 𝑟T2* )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
σ = 8 P< (R < − R

0 <.*

Sample variance
Expected return of portfolio

Portfolio variance

s0 =

1 v)0 8(R < − R
K−1
U

<.*

S

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

S

<.*

S

Var(R > ) = 8 8 x< xV ρ<,V σ< σV = 8 x<0 σ0< + 8 x< xV ρ<,V σ< σV
<.* V.*

<.*

) = 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

<.*

Portfolio variance

σ0,

= β0, σ0+ + σ0X,

0 = 𝜌,Y

σ0, 0
σ + σX,
σ0+ +

= Cρ,Y σ, D + σ0X, , 0 < ρ < 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

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

sensitivity of the security returns to changes in market returns

• a measure of stand-alone risk

• In a well-diversified portfolio, unique risks tend to

• useful in the context of a well-diversified

cancel each other out and only the market risk remains.

portfolio.

• Market portfolio has a beta of one.

Portfolio theory and CAPM
Feasible set solution equation

𝑚𝑖𝑛 σ0+ 𝑠. 𝑡. ∑𝑥, = 1, 𝐸(𝑅+ ) = 𝐸

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

CML

𝑅+ = 𝑅\ +

𝐸(𝑅3 ) − 𝑅\
× σ+ = 𝑅\ + 𝑆𝑅3 × σ+
𝜎3

𝑆𝑅3 = sharpe ratio of the market
CAPM

ECR V D = R % + βV (E(R R ) − R % )

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

𝛽] =
𝑃) =

𝐶𝑜𝑣C𝑅] , 𝑅^ D
𝑉𝑎𝑟(𝑅^ )

𝐷*
𝐷*
=
𝑟 − 𝑔 (𝑅\ + βŠ𝐸(𝑅^ ) − 𝑅\ D − 𝑔

Excess return

𝐸C𝑅] D − 𝑅\ = 𝛼] + 𝛽] C𝐸(𝑅^ ) − 𝑅\ D

APT

𝐸C𝑅] D = 𝑅\ + 𝛽*,] C𝑅_* − 𝑅\ D + 𝛽0,] (𝑅_0 − 𝑅\ )

Farma-French 3 factor model
Sharpe ratio

𝐸(α) = 0

E(R V ) − R % = βR,V (E(R R ) − R % ) + β?,V (E(R ?R` )) + βP,V (E(R aR` ))
𝑠h𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 =

𝑒𝑥𝑐𝑒𝑠𝑠 𝑟𝑒𝑡𝑢𝑟𝑛
𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦

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.

•

in a CAPM world: the market portfolio is an efficient portfolio •

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

•

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

•

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

•

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

•

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

Forwards and futures
Long payoff

S9 − F

Short payoff

F − S9

Making to market

•

Gain = margin balance - initial margin

•

Loss = total amount in margin calls

•

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

Covered interest rate parity

𝐹\ =
b

𝑆C1 + 𝑟\_c8d,ef D

1 C1 + 𝑟\gc3d6h,i D

1

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