An introduction to portfolio optimization

7/30/2019 An introduction to portfolio optimization

1/55

An introduction to portfolio optimisation

Susan Thomas

IGIDR, Bombay

October 21, 2011

Susan Thomas An introduction to portfolio optimisation
http://goforward/http://find/http://goback/


2/55

Goals

What is a portfolio?

Asset classes that define an Indian portfolio, and their

markets.

Inputs to portfolio optimisation: measuring returns and risk

of a portfolioOptimisation framework: optimising a risky portfolio using

the Markowitz framework

Outcome 1: passive portfolio management

Outcome 2: active portfolio managementEvaluating portfolio performance

http://find/


3/55

A portfolio

Investments in a set of assets together make a portfolio.As opposed to an investment in a single asset.

A portfolio is defined as:

The value of the investment (portfolio).Assets held in the portfolio.

Fraction of the value invested in each asset.These are called the weights of each asset in the portfolio,and are typically denoted as wi for asset i.

Choosing a portfolio is equivalent to choosing the weightsof assets in a portfolio.

1 How are these weights chosen?2 Would the same weights serve the best interests of all

investors?

http://find/http://goback/


4/55

Portfolio choice centers on risk vs. return

Every portfolio outcome is finally measured by the amount

of return it gave for a certain amount of risk taken.

Therefore, portfolio choice follows two stages:

1 capital allocation: how much of riskless vs. risky assets tohold?

2 security section:

What underlying assets constitute a riskless portfolio?

What constitutes a risky portfolio?

http://find/


5/55

What assets can we access?

Riskless Govt. bonds

Risky Corporate bonds, firm equity, commodities,

foreign exchange, foreign equity, foreign gov-ernment bonds

Leverage/Risk

management

Derivatives

http://find/


6/55

Measurement of risk and return


?
http://find/


7/55

Why emphasise the measurement?

Portfolio optimisation depends upon the tradeoff between

the future risk and expected return of the portfolio.

The higher the future risk, the higher the expected return

demanded by a risk-averse investor.

Thus, future risk and expected return have to be modelled

and forecasted as inputs into portfolio choice..

Problem: key inputs are observed (past) returns vs. risk

behaviour for the portfolio.

These pose severe challenges.


Ch ll i i d i k
http://find/


8/55

Challenges in measuring return and risk

Past returns are easy to measurewhen assets are traded

and prices are observed.

Challenge: There is no consensus on good forecasting

models for expected returns.

Challenge: how to measure risk?

Unlike the dynamics of prices, which we measure as

returns, there is no separate instrument to observe risk?

Once they are measured (typically by proxy), there appear

to be reasonably good forecasting models for risk.


H d l tilit ?
http://find/


9/55

How do we measure volatility?

Volatility is 2 of returns, or the standard deviation ofreturns instead, .

The dependent variable used in the modelling and

forecasting exercise is .However, this is static measure of volatility.

Real world observation: variance changes.

We need a time series of volatility.


A h t t ti i f l tilit
http://find/


10/55

Approaches to create time series of volatility

Traditional approaches:1 Time series of r2t ,2 Moving window average ,3 Range of prices as a measure of .

Recent approaches using new markets and improvedmarket information:

1 Implied volatility from options markets,2 Realised volatility using high-frequency data.


T diti l h S d t
http://find/http://goback/


11/55

Traditional approach: Squared returns

Given a time series of returns at a certain frequency (daily,

weekly, monthly, etc), rft.

Time series of volatility at the same frequency, ft wascreated as square of the returns.

2ft = r2ft


Traditional approach: rolling windows
http://find/


12/55

Traditional approach: rolling windows

Moving window average: Out of a total of T observations

on returns, use k < T observations to calculate one value

of t.Then, the volatility time series will be:

21 =1

(k

1)

ki=1

(ri r)2

22 =1

(k 1)k+1i=2

(ri r)2

. . . . . . . . .

2Tk =1

(k 1)T

i=Tk+1

(ri r)2

Using overlapping windows, we can calculate a time series

with T k volatility observations, from T observations forreturns data.


Traditional approach: Range
http://find/


13/55

Traditional approach: Range

Range is calculated as the difference between the high

and the low prices over a given period it tells us how

variable prices have been during some period.One way of calculating the volatility of returns using the

range

Ranget = 100 (Hight Lowt)/LowtRange is a measure that also uses multiple observations

at a given frequency.

If we have T points of observation of high and low prices,

then we get T observations for the time series of volatility.

Formal definition of a range-based volatility measure is

0.361 (100 (log High log Low))2

(Source: Michael Parkinson, 1980. The extreme value

method for estimating the variance of the rate of return,

Journal of Business, 53 (January):61-65.)


Recent measures of volatility: implied volatility
http://find/


14/55

Recent measures of volatility: implied volatility

Implied volatility: Apply an option pricing model to an

existing market price for an option to reverse calculate the

volatility of the asset.

What we need:1 Option price: from the market, daily2 Pricing model: Black-Scholes, Binomial lattice, etc.

What we get: a daily time series of volatility.What is remarkable: IV is an average of the expected

volatility.

It is already a forecast of the asset volatility.

Issues: If IV measures the volatility of the asset, then all

option prices on the same asset should give the same

volatility.

This is not so: IV changes with strike (volatility smile);

with type of option (put/call).


Recent measures of volatility: Realised volatility
http://find/


15/55

Recent measures of volatility: Realised volatility

Realised volatility (RV): calculated using high-frequencyreturns. In order to be useful, RV requires that:

The returns data is synchronised (say returns data at every5-minutes, or at every half-hour).For example, 66 5-minute returns can be used to calculate5-minute 5min.

This can be used to calculate daily as:

daily =

665 minutes

11 half hour returns can be used to calculate 0.5hours.

This can be used to calcalate daily as:

daily =

1130 minutes

http://find/


16/55

Setup for the portfolio problem: definitions,concepts


The problem of portfolio optimisation
http://find/


17/55

The problem of portfolio optimisation

We are given a (competitive, liquid) market where n assets

are traded.

The market is efficient - there are no easy forecasts for

tomorrows returns.

Given an individuals utility function, how does she allocate

her wealth among these n assets?


Rules of the game
http://find/


18/55

Rules of the game

ra,t can follow any distribution.

Typical model choice 1: Suppose ra,t N().In a gaussian world, returns on one asset is a random

variable x needs:

x, 2x

For returns on two assets, (x, y), additional requirement:

xy or xy


Interest rates vs asset rates of return
http://find/


19/55

Interest rates vs. asset rates of return

Risk-free interest rate is rf.

rf is assumed to be known upfront and fixed.

ra

is the return obtained in investing in an asset.

It is a sum of capital gains and cashflow/dividend

payouts.

ra is a random variable when the investment is done.


Example: E(rp) p calculation
http://find/


20/55

Example: E(rp), p calculation

Suppose there are two assets:

Asset 1: E(r1) = 0.12%, 1 = 0.20%.Asset 2: E(r2) = 0.15%, 2 = 0.18%.

1,2 = 0.01.

Portfolio weights = 0.25, 0.75

What is E(rp), p?


Example of E(rp), p calculation
http://find/


21/55

Example of E(rp), p calculation

Expected returns is a weighted average of individual

returns.

E(rp) = wE(ra)

= (0.25 0.12) + (0.75 0.15)= 0.1425

Variance of the portfolio is 2

w

is a little more complicated:

2p = ww

= (0.252 0.202) + (0.752 0.182) + 2 (0.25 0.75 0.01= 0.024475

p =

2p = 0.15644

Here, we also require the correlation between the two

assets, 1,2 = 0.01.


Example 2: E(rp), p calculation for a 3-asset portfolio
http://find/


22/55

Example 2: E(rp), p calculation for a 3 asset portfolio

Suppose we add one more asset to our set of 2:

Asset 1: E(r1) = 0.12%, 1 = 0.20%.Asset 2: E(r2) = 0.15%, 2 = 0.18%.

Asset 3: E(r2) = 0.10%, 3 = 0.15%.1,2 = 0.01, 1,3 = 0.005, 2,3 = 0.008

Portfolio weights = 0.25, 0.25, 0.5

What is E(rp), p?


Example: E(rp), p calculation
http://find/


23/55

p ( p), p

We calculate it using the same equations as before:

E(rp) = wE(ra)

= (0.25 0.12) + (0.25 0.15) + (0.5 0.10)= 0.1175

2p = ww

= (0.252 0.202) + (0.252 0.182) + (0.52 0.152) +2 (0.25 0.25 0.01) + 2 (0.25 0.5 0.005) +2 (0.25 0.5 0.008)

= 0.015

p =

2p = 0.12


E(rp), p for different w
http://find/


24/55

( p), p

For an alternative portfolio, w = 0.25, 0.5, 0.25 we haveE(rp, p) as:

E(rp) = w

E(ra)= (0.25 0.12) + (0.5 0.15) + (0.25 0.10)= 0.13

2p = ww

= (0.252 0.202) + (0.52 0.182) + (0.252 0.152) +2 (0.25 0.5 0.01) + 2 (0.25 0.25 0.005) +2

(0.25

0.5

0.008)

= 0.017

p =

2p = 0.13

For this portfolio, we have got both higher returns and higher

risk. Susan Thomas An introduction to portfolio optimisation

Observations
http://find/


25/55

Multivariate rather than univariate distributions We need more

than an understanding of returns on a single

asset: we need to understand their relationship

with all the other assets in the portfolio.

Diversification The variance on the portfolio is much lower thanthe variance on either asset.

0.15644 < 0.20

0.15644 < 0.18


Issues in diversification
http://find/


26/55

Diversification is the reduction in variance of the portfolioreturns.

This is achieved by:1 Holding a large number of assets, such that the weights on

each become smaller and smaller.Then, the effect of asset i in the portfolio variance is w2i .The smaller the w2i , the larger the reduction in variance.

2 Holding uncorrelated assets, ie, create a portfolio withassets (i,j) such that i,j is very small.The lower the i,j across the n assets, the higher the

reduction in variance of the portfolio


Matrix notation in portfolio optimisation
http://find/


27/55

p p

There are k assets, where each looks like it comes from a

normal distribution.Jointly, the returns on all these K assets can be written as

a vector:ra MVN(, )

is K 1 is K K and is a positive definite symmetric matrix.If portfolio invests in nassets with a set of weights w, the

portfolio is then a linear combination of the n assets.

Then, the portfolio features are calculated as:

rp N(w, ww)


Summary: portfolio characteristics and portfolio choice
http://find/


28/55

y p p

Expected return on the portfolio: E(rp) =n

i=1 wiE(ri).

Variance of the portfolio:

2p =n

i=1 w2i

2i + 2

ni=1

nj=1 wiwjij


Values of w
http://find/


29/55

w sums to one.

Typically, we consider each wi to fall between 0 and 1.

In a country where short-selling is permitted, wi can be

less than 1, or greater than 1.Choosing the portfolio means choosing a vector of weights

w.

Once w is chosen, we know the E(rp) and p for theportfolio.


The E(rp) p graph
http://find/


30/55

Every choice w induces two numbers - E(rp) and 2p.

A key analytic tool to choose w: E(ra) a graph.A graph with E(rp) on the yaxis and on the xaxis.

For all possible portfolios containing the same assets, butin different proportion, plot the portfolio as a point on the

E(r) graph.This is a simple 2-D graph, regardless of how many assets

you have!


Varying w from 0 to 1 in a E(rp) p graph


31/55


Optimisation
http://find/


32/55

Now we are ready to understand the problem space of

portfolio optimisation.Start with a two-asset universe:

With two assets, (A, B) where

(A, B

MVN(, )

is 2 1 with A, B.and is 2 2 with 2A, 2B, AB.

The investment in one asset as w.

Then, the investment in the other is 1 w.Porfolio optimisation problem: for the parameters in , ,find the optimal w.


Varying w from 0 to 1 - interpretation
http://find/


33/55

The previous graph shows how E(rp) p combinationchanges as w changes.

There is some value of w for which E(rp) is the maximum.There is some value of w for which p is the minimum.

w for maximum E(rp) and minimum p is not the same.


What happens if changes?


34/55

E(r)

sigm

A

B

P


Changing - interpretation
http://find/


35/55

How does the E(rp), p graph vary with changing values of ?

AB is (E(rp), p) for all nonnegative linear combinationsof A and B, when AB = 1.PA and PB define the boundaries of (E(rp), p)(1 < AB < 1).The curve AB defines (E(rp), p) for all nonnegativelinear combinations of A and B for some intermediate fixed

value of AB.


Choosing the correct portfolio
http://find/


36/55

Given this graph,1 We first choose a desired E(rp).2 We next choose w such that the variance is minimised for a

selected value of E(rp).This is the optimal portfolio with (1 w) in asset A and win asset B.

http://find/


37/55

Special case of capital allocation


Capital allocation between risky and risk-free
http://find/


38/55

We earlier considered any two assets A and B.

What if A is the risk-free asset? Then,E(rp) = wrf + (1 w)E(rB)This is a linear combination of expected returns as usual.

But,

(2

p

) = w22

rf+ (1

w)22

B

+ 2w(1

w)covrf,B, where we

know:

rf = 0covrf,B = 0

Then,

(2p) = (1 w)22B, orp = (1 w)B


Interpreting the E(rp) p graph for risky and risk-free
http://find/


39/55

Remarkable result: A portfolio of the risk-free and one

risky asset has an E(rp) p graph which is linear in bothE(rp) and .

This means that as you increase w, p decreases linearly.

For example, at w = 1, p = 0

At w = 0, p is the maximum.

Solution to the capital allocation problem: risk-return

across risky and risk-free assets is linear in w.



40/55

Leverage


Leverage in w


41/55

In the above example, p is bounded to be equal to Bbecause w falls between 0 and 1.

By implication, the investor cannot access risk (or returns)

that are higher than B.

This can change if we have short-selling.

Short-selling means w can be negative.

If w is negative, it means you borrow at the risk-free rate to

invest in the risky security, B.

This is leverage.

With leverage, you can create portfolios with p > B.


Example of leverage
http://find/


42/55

An investor has Rs.1 million to allocate between the

risk-free asset and risky asset B.

The data shows that rf = 7%, rB = 35%, B = 40%

What combinations of E(rp)

p can she achieve?

At w = 0.5, the portfolio is the riskless asset: E(rp) = 0.5 7 + 0.5 35 = 21%; p = 0.5 40 = 20%;At w = 1, the portfolio is purely risky asset, B.

E(rp) = 35%;

p = 40%


With leverage


43/55

At w = 0.5, the portfolio has:E(rp) = 0.5 7 + 1.5 35 = 49%p = 1.5 40 = 60%

How is this achieved?

At w =

0.5, the investor borrows half a million from thebank to invest in B.

This is a leveraged portfolio: the size of the investment is

more than the size of the initial wealth.

Note: The risk in the last portfolio is the highest, but the

return is also (linearly) high.


The capital allocation problem redux


44/55

The capital allocation problem remains linear:

Higher the investment in the risky portfolio, return and risk

are linearly higher.

The level of return desired drives the quantum ofinvestment in risky vs. riskless asset.

Next question: out of all possible combinations of risky

assets, what is the optimal risky portfolio?



45/55

Constructing the optimal risky portfolio


Step2: Optimising the risky portfolio


46/55

Next question: among all the risky assets, what is the

optimal risky portfolio?

We know that the return-risk trade-off among only risky

assets is not linear: some combinations of assets give alower return for a higher level of risk.

What does this combination of risk-return look like?

We find out using a simulation with real data.

http://find/


47/55

Simulating the E(rp) p graph for a set of7-stocks


Example: A dataset of weekly stock returns
http://find/


48/55

load(file="10.rda")

colMeans(r)print(cov(r), digits=3)


Example: Mean-variance of weekly returns


49/55

> colMeans(r)

RIL Infosys TataChem TELCO TISCO TTEA Grasim0.31392 0.15702 0.40761 0.29796 0.44311 0.12302 0.32051

> print(cov(r), digits=3)

RIL Infosys TataChem TELCO TISCO TTEA Grasim

RIL 29.08 11.01 9.2 12.85 15.93 11.27 8.13

Infosys 11.01 61.23 10.1 4.06 8.72 7.48 7.60

TataChem 9.19 10.11 33.9 15.75 14.51 13.75 12.25

TELCO 12.85 4.06 15.7 38.17 20.16 16.60 7.64

TISCO 15.93 8.72 14.5 20.16 32.94 13.95 11.79

TTEA 11.27 7.48 13.8 16.60 13.95 29.64 8.02

Grasim 8.13 7.60 12.2 7.64 11.79 8.02 34.17


Example: Correlations of weekly returns
http://find/


50/55

> print(cor(r), digits=3)

RIL Infosys TataChem TELCO TISCO TTEA Grasim

RIL 1.000 0.261 0.293 0.386 0.515 0.384 0.258

Infosys 0.261 1.000 0.222 0.084 0.194 0.176 0.166

TataChem 0.293 0.222 1.000 0.438 0.434 0.434 0.360TELCO 0.386 0.084 0.438 1.000 0.569 0.493 0.212

TISCO 0.515 0.194 0.434 0.569 1.000 0.446 0.351

TTEA 0.384 0.176 0.434 0.493 0.446 1.000 0.252

Grasim 0.258 0.166 0.360 0.212 0.351 0.252 1.000


Example: Creating portfolios randomly
http://find/


51/55

load(file="10.rda")

mu = colMeans(r)

bigsig = cov(r)

m = nrow(bigsig)-1N = 2 0

w = diff(c(0,sort(runif(m)), 1));

rb = sum(w*mu);

sb = sum(w*bigsig*w);

for (j in 2:N) {

w = diff(c(0,sort(runif(m)), 1));

r = sum(w*mu); rb = rbind(rb,r);

s = sum(w*bigsig*w); sb = rbind(sb,s);

}

d = data.frame(rb, sb);

d$sb = sqrt(d$sb);

pdf("10_2.pdf", width=5.6, height=2.8, bg="cadetblue1", points

plot(d$sb, d$rb, ylab="E(r)", xlab="Sigma", col="blue")


Example: E(r)- graph, N=20 portfolios


52/55

qq qq

q

q

q

q

qq

qq

q

qqq

q

q

0 2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

Sigma

E(r)

http://find/


53/55

Observations from the simulations
http://find/


54/55

The characteristics of random portfolios (where theweights on the securities are randomly selected) show a

convex curve for different w.

There is a minimum value of p: no matter whatcombination of w, there is no way of reaching a lower pwith this set of assets.

There are a set of portfolios which no-one would want to

hold: where E(rp) decreasesas p increases.

We need to focus on those portfolios where E(rp)

increases as p increases.


The Markowitz optimisation
http://find/


55/55

We observe that the return-risk trade-off among only risky

assets is not linear: some combinations of assets give a

lower return for a higher level of risk.

Is there a closed form solution to what is the optimal riskyportfolio?

Solution: the Markowitz framework.


Documents

An introduction to portfolio optimization