Financial Products and Markets

Lecture 6

Model with N risky assets

• Assume to invest one unit of wealth in a set of N risky assets, with expected returns given by an arrray and covariance matrix V.

• The best portfolio allocation is chosen in the set of vectors

w that solve the problem (e is the unit vector)

1'

'

'min *

ew

w s.t.

Vwww

prE

The solution

• The solution is

…and we must recover the Lagrange multipliers from the contraints.

eVV

eVw

12

11

2

11*

Fund separation theorem

• Re-write the optimal portfolio

eVe

eVeVw

Ve

VVw

ww

eVVw

1

11

11

11

0

1201

1

2

1

1

''

*

cb

cbc

cb

b

Two-fund separation

• It is immediate to verify that– both w0 and w1 are portfolios (e’w0 = e’w1 =1)– both w0 and w1 are on the efficient frontier.

• More precisely,– w1 is the global minimum variance portfolio– w0 is the portfolio on the efficient frontier

obtained linking the origin and crossing the efficient frontier on the minimum variance portfolio.

Which funds, and why two?

• Which funds can be used to replicate the effficient portfolios? Only w0 and w1?

– No, obviously every pair of combinations of w0 and w1 could be used

• Why two funds?– From the math, because we had two constraints

Portfolio allocation with hedging fund

• Assume to include other constraints in the portfolio allocation strategy (i.e., we require some sensitivity to some risk factors). We then obtain

Bw

w

ew

BwewwVww

'

'

1'

..

'21'2'2' 21

p

p

rE

ts

rEL

The solution

The solution is

Bw0

w

ew

b-eVw=

*'

*'0

1*'0

*02

221

p

S

sss

rE

S+2 fund separation theorem

• Proceeding as before we obtain

2....3,2'

''

'*

1

1

1

11

11

11

0

2

2

11100

Ss

cb

cb

s

ss

S

ssss

bVe

bVw

eVe

eVeVw

Ve

VVw

wbVewww

Model with N risky and one risk-free asset

• Assume to invest one unit of wealth in a set of N risky assets, with expected returns and covariance matrix V, and a risk-free asset with return Rf

• The portfolio allocation problem

fpf RrER ew s.t.

Vwww

'

'min *

The solution

• The solution is obtained with the same technique

e'1w*

whichfrom

0

*0

*0

1 weVw

e

eVw

f

fpf

f

R

RrER

R

Efficient frontier

• Recover the Lagrange multiplier substituting the constraints

21 2' fffffp cRbRaRVRRrE ee

2

2

22

112**2

2

2

''

ff

fp

ff

ffp

cRbRa

RrE

cRbRa

RR

eVVVeVww

Two-fund separation

• We find again a two-fund separation theorem (two we remind that formally we still have two contraints).

• Each portfolio allocation can be represented as

f

ft

ftf

cRb

R

cRbcRb

eV

w

ww

1

0 1w*

Model with N risky assets and a risk-free asset

• In the model with N risky assets and a riskless one, the efficient frontier is given by a linear relationship .

• Even in this case, we have a fund separation theorem. Each portfolio can be replicated by a portfolio invested in:

1. A share of the portfolio in the riskless assets2. The remaining share in a portfolio entirely made up

by risky assets.

A market model

• All agents have the same information and ompute risk and return in the same way. Investors have different preferences.

• The equilibrium requires that in the whole economy the net investment in risk-free asset is zero (bond zero net supply) so that thre risky asset fund coincides with the overall supply of risky assets (market portfolio).

• Because of this, we obtain that:– The market portfolio is in the frontier efficient – Every portfolio can be replicated by a portfolio of the

risk free asset and the market portfolio.

CAPM

– CAPM (Capital Asset Pricing Model).

– The model imp0plies a precise relationship between the expected return of all securities and the market return

– Every portfolio can be replicated by a portfolio of teh two funds: a risk-free asset and the market portfolio.

MP rErrE 1

Beta

• The value of the replicating portfolio represents the regression coefficient with respect to the market return:

i = cov(ri, rM)/var(rM)The expected return of each security is

E(ri) = r + i E(rM – r ) in a model similar to the APT model. The quantity of risk, different for every asset, is given by tge coefficient iwhile the market price of risk is represented by the excess return of the market portfolio, that is E(rM – r ).

Collective investment: products

• Collective investment: mutual funds, Sicav, closed-end funds, hedge funds.

• Individual investment: GPM, GPF, customized products

• Fund management for institutional investors: small banks, insurance companies, SIM,foundations…

• Private banking: complete service of wealth management.

Why funds? The economic rationale

• Goal: – Efficient portfolio: has the lowest possible risk,

measured with standard error

• Constraints– All wealth must be invested in financial assets – Expected return of the portfolio must be E(R).

Extension

• Goal: – Effficient portfolio: has the lowest possible

risk, measured by standard error

• Vincoli– tAll wealth must be invested in financial assets – Expected return of the portfolio must be E(R).– target of sentitivity to other risk factors

K+2 Funds

• The investment problem can bee solved by investing in k+2 baskets of assets, investiment funds.

• The result is that all preferences for risk and return, conditional on sensitivities to risk factors, can be realized using a limited number of funds.

Perché i fondi? altre spiegazioni...

• Economie di scala nei costi di transazione. Poiché ogni operatore può realizzare il suo obiettivo combinando l’acquisto di due panieri dei titoli, è più efficiente incaricare un intermediario che li acquisti in blocco

• Gestione professionale del risparmio. Gli investitori non hanno tempo o skills per seguire in maniera professionale i mercati.

Active and passive management

• Economies of scale justify the market of passive fund management, o indexed, that have the goal of “replicating” a market or an index as close as possible.

• Professional fund management justifies a market of active management, with the goal of “beating” the market.

Management styles

• Active and passive management

• Active management can use– market timing– stock picking

• Management “styles”– value/growth– momentum/contrarian

Fund management: players

• SIM: can manage individual funds

• Sicav: can supply collective management services

• Società di gestione del risparmio (SGR): can supply both collective and individual fund management services.

Mutual funds

• Open end funds: – Money can be withdrawn in every moment, and

are allowed to invest in securities, mostly listed, and bank deposits

• Closed end funds– Money can be withdrawn after a longer period

of time. Can invest in real estate, unlisted securities, credit risky bonds, other assets with at least semi-annual valuation.

Open end funds

• Main specialization– Money market funds (liquid assets)

– Bond market funds

– Equity funds

– Special investment (Euro area, US, emerging market)

– Flessible or multi-asset (no investment constraint)

• Passive management: funds replicating markets and indexes, if are listed they are the so called ETF (Spyders in the US market).

Closed end funds

• Private equity funds– Venture capital– Vulture funds

• Real estate funds

• Funds investing in unlisted securities

• Vehicles

Hedge funds

• Speculative funds (hedge fund) do not have limits in the choice of investments and use of leverage. They can freely invest in derivastives.

• Taxonomy– Macro

– Long/Short

– Relative value

– Event

– Funds of funds

Value of fund quotes

• The value of every quote of investment in a managed fund is computed by dividing the NAV (Net Asset Value) by the number of quotes.

• Quotes can be withdrawn with two day notice (for open end funds)

• They are sold by banks, but the EU is requiring that they be listed in the stock exchange.

Performance

• How to measure performance?– Management return

– Investment return

• Performance attribution– How much is due to stock picking

– How much is due to market timing

• Style analysis?– In which assets is a fund manager investing?

Returns time and money weighted

• Time weighted– Compute the return without taking into account

investments and withdrawals– Are a measure of management performance

• Money weighted– Take into account investments and withdrawals– Are a measure of overall performance.

Time and money weighted methods

• Time weighted methods– Daily return methods: requires daily cash flows– Quote methods: change in the percenatage

value of the quote value

• Money weighted methods– Fisher method: compute internal rate of return– Dietz methods: return on average invested

capital

An example

• Investment of 100 mio in a fund. 6 months later the portfolio is worth 220 mio, plus other 440 mln of new investment. Final value after 6 more months: 330 mio

• Time weighted: (220/100)(330/660)-1=10%• Money weighted (Dietz): - 65.6%

– Return: 330 - 100 - 440 = - 220– Average capital: 100 + 440/2 =320

• Thanks Riccardo Cesari and Fabio Panetta

Performance measurement

• Allows to evaluate if the manager is doing better than the market.

• The measure can be either in absolute terms or for unit of risk.

’s Jensen

– Jensen’s

–

– It is a measure of excess return with respect to an efficient portfolio with the same systematic risk features, In this sense, it is a measure of stock picking activity

fmfp RRRR

Sharpe ratio

– Sharpe ratio

–

– It is a measure of the market price of risk. It must be compared with that of the market

p

fp RR

Treynor ratio

– Treynor ratio

–

– Same as Sharpe ratio, but measured with respect to systemic risk.

p

fp RR

Performance attribution

– Let us estimate the regression

–

and measure the stock picking anf market timing activities respectively.

– The idea is that the manager switches between the market and the risk free asset and the market depending on his forecast.

ptftmtftmtftpt RRRRRR ,0max

Style analysis

• Sharpe quadratic optimization method– We collect time series of returns of funds and

asset classes. – We estimate the portfolio that minimizes the

tracking error under the constraint that all capital is invested and there are no short positions.

An example of style analysis Azionario Italia ( 7/98-7/99)

Quote stimate Indice Comit Media FondiIndustriali 26.3821% 23.9146%Assicurativi 15.6680% 15.7658%Bancari 25.7639% 22.2674%Comunicazioni 26.5817% 24.1912%Altri settori 5.6583% 4.7949%Liquidità 0% 9.0661%