1- Client Constraints and Investments-
2- Assumptions-
-
Financial data is
stochastic –assuming randomness, i.e. price moves do not have memory (“random
walk”/GBM with drift), fully reflect all available information.
-
Returns (“random
steps”), or logs of returns are normally distributed (check for higher moments,
normality tests).
-
i.i.d –returns
are presumed to be independent and identically distributed, which leads to
normality of their aggregation (Central Limit Theorem).
-
Return
distributions are stationary –past samples tell us a lot about future possible
values; note that prices are not stationary. Stationarity assumes that mean and
variance are stable through time.
This specifically
means that we can take the average return from history, and the volatility
associated with past sample and extend it into the future.
Time Varying Volatility by GARCH (Volatility
is mean reverting), EWMA etc. models.
Market Inefficiencies-
-
Small Cap
outperformance
-
Low P/E
outperformance
-
Returns via
technical analysis.
3- Risk Budget-
Refers
to how much pain the Client is willing to accept either at the end of or during
the investment period to stay invested.
4- Measuring Return-
a.
Time-Weighted Rate of Return – compound rate of
growth in a portfolio. Because this method eliminates the distorting effects
created by inflows of new money.
b.
IRR –Internal Rate of Return – rate at which the
net present value of all the cash flows (both positive and negative) from a
project or investment equal zero.
c.
Geometric- Already includes risk.
d.
Log-Returns-
-
Log-returns
approximate raw returns
-
Log-normality is
a nice assumption for prices, making log-returns normal
-
Log-returns
additive in time leading to normality of compounded returns
-
Log-returns help
with calculus
Note:
log-returns do not aggregate across securities.
5- Risk-Reward Ratios-
4.1 Sharpe Ratio-
Measures risk reward portfolio
efficiency.
Excess return over per unit of
standard deviation.
4.2 Sortino Ratio-
Focuses on downside volatility. MAR is
Minimum Acceptable Return.
4.3 Information Ratio-
Ratio of portfolio
returns above the returns of a benchmark
-- usually an index -- to the volatility
of those returns.
6- Maximum Loss given probability-
6.1 VaR –
Measures the potential loss in value over a defined period for a
given confidence interval.
6.2 cVaR / ETL –
Estimated Tail Loss, is frequently defined as
the mean (expected value) of the distribution of potential losses beyond VaR.
6.3 Stress loss
7- Equity Modeling-
7.1.1 Utility function (Risky And risk-free asset)-
Utility Function-
Utility function U(W) is a
relationship between a level of wealth (W) and your perceived satisfaction from
it
ÊŽ = risk-aversion factor of investor.
Risk-free asset has zero volatility
and zero correlation to the risk asset, we have the following situation:
Maximizing Utility Score
7.1.2 CAPM-
Expected return of a security is driven entirely by risk-free rate, return
of the market portfolio, and security’s sensitivity to the return of the market.
7.1.3 Modern Portfolio Theory-
In the presence of risk free asset rational
investors will figure out that the best strategy is to combine one optimal
risky portfolio (which becomes the “market portfolio”) with a long or short
risk-free position.
Assumptions-
Asset Assumptions
|
Market Assumptions
|
Investor Assumptions
|
-
Asset can
be described by returns, deviations and pairwise correlations.
-
Returns
are normally distributed and are stationary.
|
-
Risk free
borrowing and lending
-
No taxes
and transaction costs.
-
Markets
are efficient in information and trading
|
-
Investors are rational and risk
averse.
-
Investor balance their portfolio
frequently.
|
7.1.4 Factor Models-
-
Helps in
identifying systematic risk.
-
Thousands of
securities can be expressed in meaningful factors.
-
Factors are
generally considered close to Normal Distribution.
-
Linear
aggregation of factor betas provides a tool for bottom-up and top-down analysis
of portfolios.
Optimizing Factor model via Regression-
E.g.- We have 5 assets and number of macro-economic
factors and indexes-
a.
Find the
macroeconomic factors that are highly correlated with asset returns.
Note- Factors of one asset returns should not have
high correlation with factors of other asset returns.
b.
Regress the
shortlisted factors (two in example) with the corresponding asset returns.
c.
Find the
covariance matrix of all shortlisted factors (C).
d.
Finding the
covariance Matrix of coefficients (B)-
Create a matrix of 5* 10.
-
First row- Column
1 and Column 2 = Slope Regression
coefficients of asset 1 (Intercept is ignored) and all remaining columns with 0
value.
-
Second Row-
Column 3 and Column 4 = Slope Regression
coefficients of asset 2 (Intercept is ignored) and all remaining columns with 0
value.
-
Same process for all remaining rows and
columns.
e.
Perform the
matrix multiplication D = (B’CB).
f.
Covariance
Matrix of Factor Model (Cov)= Add square of regression SE of each asset (b) to
the diagonal of D.
g.
RP = Weight * Expected Returns.
h.
Standard
Deviation = Sqrt( Cov * Weight)
i.
Minimize
Standard deviation.
7.1.4.1 Fama And French Three Factor Model-
Expansion
of CAPM with addition of two factors i.e.
size and Value.
r
= portfolio's expected rate of return, Rf = risk-free return rate,
Km
= return of the market portfolio.
SMB
(size) = "Small [market capitalization] - Big" and
HML
= “High [book-to-market ratio (Value Stock)] -
Minus Low
7.1.5 Arbitrage Pricing Theory / Factor Model-
-
Asset pricing can
be modeled as a function of macro factors and indexes
-
If you can
generate superior predictions of economic development, superior asset pricing
assessment might follow, leading to opportunity of higher returns per unit of
risk
-
Powerful approach
in combination with other allocation approaches
Disadvantage-
It works good only for well diversified portfolio.
7.1.6 Black-Litterman-
-
Start with the
Market Portfolio –markets are efficient and in equilibrium
-
Identify your
views and conviction levels
-
Mix return and
risk projections from the Market with your views, then optimize in MPT fashion
-
There still is an
optimal portfolio for a level of risk, but it will be different based on your
views
7.1.7 Risk Parity-[1]
Weight of asset is inversely proportional to
standard deviation of asset.
-
Adjusted for
risk, various assets produce similar excess returns
-
Allocating risk
“equally” to various risk assets will generate more stable outcomes
-
Solution for
optimal allocation is independent of the risk budget –utilization of budget is
achieved through leverage
Disadvantages- This model is very sensitive to N and
ex-poste and ex-ante returns.
Highly
correlated assets will occupy larger portions of risk.
Covariance
structure is not fully utilized.
7.1.8 “New” Risk Parity –MDP (Maximum Diversification Portfolio)-
Diversification
Ratio:
8- Optimal Allocation: Approaches
8.1 Mean Variance Optimization(MVO)-
Intuition- To maximize sharpe ratio.
Minimize volatility for a given level
of returns-
Return = Weight * retun
Volatility = sqrt (Variance of
Portfolio)
Conditions-
a.
Sum of
weights should be 1 (No short position or leveraging).
b.
Setting
return as per the constraint.
c.
Weights of
assets > Lower bound
d.
Weights of
Assets < Upper Bound.
Minimize Volatility
Disadvantages of MVO)-
-
Highly sensitive
to the unstable part of model i.e. risk and return
-
If assets are
highly correlated, slight changes leads to higher turnover.
-
Maximizing Sharpe
Ratio may lead to large long or short positions.
-
Returns, variance
and correlations are considered stationary.
9- Technical Analysis-
-
Relative Strength
Index
-
Moving Average
Convergence/Divergence (MACD)
-
Bollinger Bands
(BOLL)
-
Stochastics (STO)
-
Directional
Movement Index (DMI)
-
Ichimoku (GOC)
-
Volume at Time
(VAT)
