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Quantitative Portfolio
Management Strategies
A Systematic Approach
“Successful investing is anticipating the anticipations
of others.”
John Maynard Keynes (Economist)

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Why Portfolio Management
What are we missing here?
Uncertainties: `Best` stocks picked by anyone right now are just our best guesses given
our current understanding (model) of the world. It is not perfect.
Changing market: Best stocks, good strategies and winning signals are not static, they
change over time, even if our current understanding was perfect!
Context: Best stocks and good strategies are not unique. They are different for different
investors even if our world model was perfect and we have a full 20/20 future vision!
The technical terms are 1) risks 2) time dependence and 3) utility

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Which is the Best Strategy?
Strategy-I: 18.4%, Strategy-II: 15.0%, Strategy-III: 19.1%, Strategy-II: 3.8%

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Returns from the Best Strategy
Risk adjusted returns: 21.6% (scaled)

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The Value from a Naive Portfolio
Risk adjusted returns: 36.2% (scaled)

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What is Portfolio Management
Investment/ Strategy: A single expected source of returns streams
Portfolio Management: A meta strategy to allocate capital based on risks and context,
over a period of time.
Strategy value: v= 𝟏 + 𝝓𝒕−𝟏 𝒔 𝒕 × 𝟏 + 𝝓𝒕 𝒔 𝒕+𝟏 … = (𝟏 + 𝝓𝒕−𝟏 𝒔𝒕)𝑻
𝟏
Portfolio value: 𝐩 = 𝟏 + 𝒘 𝒕−𝟏
𝒌
𝒓 𝒕
𝑵
𝟏
𝑻
𝟏
Building Portfolio: max
𝒍≤𝒘≤𝒖
𝔼𝑼 , 𝑼 = 𝒇(𝒑)
Different types: How we define 𝑈 = 𝑓(𝑝) , (mean-variance optimization, Kelly
optimization, risk-parity etc.) How we optimize it – static (per period and rebalance) vs
dynamic (continuous rebalance), how we define constraints (long only, long-short,
leveraged), and how we chose the set of returns streams (alpha strategies, factor
portfolio, value portfolio etc.)

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How to Build a Portfolio
Building Portfolio: max
𝒍≤𝒘≤𝒖
𝔼𝑼 , 𝑼 = 𝒇(𝒑)
Equally portfolio makes sense if
#1: All the stocks (or assets or strategies) are exactly equivalent (a trivial case!)
#2. If your 𝑈 function is product of the weights (for some reasons!)
#3. You have a very high uncertainties around those returns stream. Remember
regularization from data science? This is equivalent to a L2 regularization with a very
high penalty. So high that maximization problem becomes effectively minimization of the
penalty term.
#4. Empirically, equal weighted portfolio does fairly good!
But, if we have a good model (better than random), then we can do better!

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How to Build a Portfolio
We can do better if we can have good model of the world and define what we care about
(utility function) more precisely in terms of our model variables
Let’s be reasonable and assume
#1. at least we can predict the expected returns and risks of our assets reliably and
#2. We prefer higher returns for a given amount of risks, or a lower risks given a return
This sets us up for the famous mean-variance optimization (MVO). It can also be shown
this is (somewhat) equivalent to assume a quadratic utility.

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How to Build a Portfolio
A Simple Game of Luck:
You pay $X dollars to play a betting game against the “house”. The game starts with $1
in the pot. At each turn the dealer toss a fair coin. If it is tails, you get whatever in the
pot, else we move to the next turn with the house doubling the pot amount (i.e. $4, then
$8 then $16 and so on). AKA St. Petersburg paradox.
What is X. How much you will pay to play this game?

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How to Build a Portfolio
Risks: A measure of uncertainty. Ambiguity: Uncertainty in risks itself. In finance we
typically use a “deviation risk measure” like standard deviation.
Utility: a mathematical formulation of how we feel about our wealth – how much an
extra dollar is worth to you, given your current dollar wealth.
Risks vs Utility: They are related through “risk premium”, how much lower certain
returns one is ready to accept to avoid uncertainty (certainty equivalence). Risk premium
exists because we assume most of us are risk averse (given same returns, prefer a
lower risk.
If most investors in an economy are indifferent between a 3% p.a. certain returns (like a
bank deposit or treasury bill), compared to an expected 8% p.a. returns from the broad
equity market, the risk premia is 5%.

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How to Build a Portfolio
This leads us to 𝔼𝑼 as a of mean of W (𝝁) and variance of W(𝝈 𝟐
)! The central
optimization problem becomes a mean-variance optimization (MVO) problem! AKA
Modern Portfolio Theory or MPT.
Risk Aversion: 𝔼𝑼 𝑾 + 𝝐 < 𝑼 𝑾
Risk Premium: 𝝅 | 𝔼𝑼 𝑾 + 𝝐 = 𝑼 𝑾 − 𝝅 , depends on 𝝐, shape of 𝑼 and current
wealth! But for small risks, a function of variance of 𝝐
A special case: 𝐔 𝑾 = 𝑾 −
𝝀
𝟐
𝑾 𝟐
, 𝝀 > 𝟎
Alternatively, if you do not like quadratic utility, we can just assume all asset returns are
normal, then wealth distribution is also normal. So finding max of 𝔼𝑼 again becomes a
mean-variance game, i.e. MPT!

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MPT, CAPM and Factors
Modern Portfolio Theory: We prefer lower risks for a given amount of returns. We also
can compute expected returns and risks with good confidence.
Let’s randomly allocate a range of weights for each of our assets, and chose the best
portfolio. Let’s see how it works using Excel!
Optimization: min
𝒍≤𝒘≤𝒖
𝝈 𝟐
| 𝝁 = 𝝁∗
, 𝝁 = 𝒘 𝑻
𝑹 & 𝝈 = 𝒘 𝑻
𝚺𝒘, 𝒐𝒓 m𝑎𝑥
𝒍≤𝒘≤𝒖
𝔼𝝁 −
𝟏
𝟐
𝝀𝝈 𝟐
MPT: Tells us given market prices, what an intelligent investors will do (allocate between
a market portfolio and risk free asset)
What if we Flip it? Given intelligent investors and market equilibrium, what that means
for market prices – this leads to the door to an asset pricing model, namely Capital Asset
Pricing Model or CAPM.

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MPT, CAPM and Factors
𝔼𝑅 = 𝑅𝑓 + 𝛽𝑖 𝔼𝑅 𝑀 − 𝑅𝑓
CAPM: gives us the basic mathematical foundation for value investing!
It models the required rate of return or fair value. If we have a good model that tells us
the expected return is higher (lower) we can buy (sell) the asset.
According to CAPM. No excess returns for diversifiable risks. But ONLY idiosyncratic
risks are rewarded! The inspiration for stock picking.
Typical value investors will assess a few investments, bet on their idiosyncratic upside,
and hedge the market (systematic) risk.
Notice how MPT fundamentally changes our concept of risks. Not standard deviation,
but covariance to market portfolio is what matters! 𝛽 is the risk, not 𝜎!
𝑅 = 𝛼 + 𝛽𝑖 𝑅 𝑀 − 𝑅𝑓 + 𝜀

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MPT, CAPM and Factors
𝑅𝑖,𝑡 = 𝑎𝑖 + 𝛽𝑖,𝑗∆𝐹𝑗,𝑡 + 𝜖𝑖,𝑡
𝑗
APT: A framework of (unspecified) factors that explains asset returns. Unlike CAPM it
provides no economic explanation of what these factors can be.
Are the idiosyncratic risks really so? Welcome to factor investing, pioneered in the 70s,
popular after 2008 crisis!
Fama French: Precisely 3 factors. They are called size (large cap vs. small cap), value
(book-to-market rich vs. cheap) in addition to the market factor
𝔼(𝑅𝑖,) = 𝑅𝑓 + 𝛽𝑖,𝑗 𝑅𝑃𝑗
𝑗
What is beta, alpha? Beta – coefficient in the above equation with only market factor.
Alphas are tricky – what if 𝛽 and ∆𝐹 time-invariant and positive? – pure alpha! What if
𝑅𝑃 is stable but 𝛽 varies? – factor allocation! What if 𝛽 stable but 𝑅𝑃 varies? – factor
timing! What if everything is unpredictable? – focus on α, stock picking!

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The Issues with Value
In CAPM world, the job of investors is to find assets with high expected residual returns
(don’t get paid for systematic risks).
The issue with value investing – focus on concentration and idiosyncratic risks.
Factor Models say most of this residuals can be explained by other factors – known as
risk factors. In such a world, the job of the investors is to find factors with high risk
adjusted expected returns and design portfolios to capture this.
We systematically probe the drivers of the markets – than doing in-depth research on a
few companies.
Although this sounds just a continuation, this approach is fundamentally different! We
move from discretionary value research to quantitative factor research (usually called
alpha streams)

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Discretionary vs. Quant
Asset manager A is a discretionary manager and follows 10 stocks diligently. Asset
manager B is a systematic manager and have enough computational power to track 500
stocks. Investors seek outperformance and confidence.
Question #1: What is the hit ratio for A to achieve outperformance in at least 50% (total
5 stocks) of his portfolio with at least 95% probability. What is for B?
-15%
5%
25%
45%
65%
85%
105%
10% 20% 30% 40% 50% 60% 70% 80% 90%
Probabilityofsuccess
Investing Skills
A
B
Pr 𝑛 𝑁, 𝑝) = 𝐶𝑁
𝑛. 𝑝 𝑛
. (1 − 𝑝) 𝑁−𝑛
𝑃 = 1 − Pr 𝑖 𝑁, 𝑝)
𝑛
𝑖=1

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Value vs. Quant
Discretionary Systematic/ Quant
Small N (usually) Large N (usually)
Edge is deep, proprietary insights Edge is superior info processing
Ideally offers superior upside Ideally offers superior consistency
It is NOT one OR the other, it is one AND the other
The fundamental law of investment management: 𝐼𝑅 ≈ 𝐼𝐶 . 𝑁. 𝑇𝐶

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Optimizing Factor Portfolio
It is very similar than before. We first decompose our returns to factors, then express the
variance in terms of factors covariance + asset covariance, and optimize using a chosen
utility function – like say mean-variance. Note, here we have two parameters (unlike just
one, 𝜆 before! One each for the covariances).
m𝑎𝑥
𝒍≤𝒘≤𝒖
𝑹 𝑻
𝒘 − 𝒘 𝑻
𝝀 𝑭 𝑿 𝑻
𝑭𝑿 + 𝝀 𝑨 𝑫 𝒘
𝒓 = 𝑿𝒇 + 𝒖, 𝒄𝒐𝒗 𝒓 = 𝑿 𝑻
𝑭𝑿 + 𝑫
Theoretically asset covariance 𝑫 is diagonal, under the assumptions that 𝒇 and 𝒖 are
independent, and so are the individual components of 𝒖. These 𝝀 are also known as
price of risk.

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Other Optimization Techniques
The other popular methods are Kelly and Risk Parity
For Kelly, our 𝑼 is different, it is 𝒍𝒏 𝟏 + 𝒓 𝒇 + 𝒖 𝒌 𝒓 𝒌 − 𝒓 𝒇
𝒏
𝒌=𝟏 This is just optimizing
wealth accumulation, than mean vs. variance. In fact both MVO and Kelly follows from a
general class of 𝑼 known as constant relative risk aversion (CRRA). All methods
following this will be independent of starting wealth!!
Mathematically great, but what about practicality?
Interesting points about Kelly: Theoretically dominate all others in minimum time to
reach a target wealth level! In the long run, it almost surely dominates
But this comes at a cost of riskier short terms! Optimization with most aggressive risk
aversion parameter. Also very sensitive to estimation errors in returns. And high volatility
(especially when going is good, may be not too bad!). Most usually follow a factional-
Kelly approach to avoid risk of ruins!

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Other Optimization Techniques
This is mostly empirical, (while it is possibly to construe an utility function here, hard to
make any economic sense!)
Individual risk contribution: 𝜎𝑖 = 𝑤𝑖
𝜕𝜎
𝜕𝑤 𝑖
is equal!, Note 𝜎 = 𝜎𝑖 where 𝜎 = 𝑤 𝑇
𝜎𝑤
We do not talk about expected returns, only risks, which are usually more stable and
easier to estimate with better confidence. Less sensitivity to model errors!

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When to Use What
If we are quite sure about our models, mean-variance or Kelly offers a great choice.
Kelly is best in the long run, but risky in the short term (volatility pumping). So perhaps a
good choice if you are young or can afford risk of ruins!
If we don’t trust any models, as we have seen, equal-weighted portfolio is a choice
On the models – expectation based on CAPM only can be back-ward looking, so we try
to improve with a factor model. Or even a better model if you can imagine and test one.
All these optimization problems are very similar, the right constraints can be on sign of
weights (long-only portfolio), on industry/ sector/ asset class/ factors exposure etc.
If our returns models are NOT good, but risks models are okay, we can go for risk parity
(usually risks are more stable than returns). This is defensive with decent empirical
performance

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Portfolio Under Uncertainty
We can use any methods but we may run in to difficulties. For example, alpha strategies
(say a few intraday trading strategies) are by definition uncorrelated. So an MVO on
them will lead us to whole weight given to the best strategy in the retrospect. What if we
are wrong going forward?
Other, more recent, methods of managing portfolio of assets are strategies may help.
Examples are
Ensemble strategies (we already seen the average portfolio), with (potentially dynamic)
voting weights
Regime-switching portfolio – where we dynamically detect change in model parameters
and switch models (say using a change point analysis)
No-regret Strategy – Adaptive weighing of strategies based on recent performance –
usually based on returns (with possibilities of risk adjusted measures, e.g. Sharpe)

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Portfolio Under Uncertainty
If slot machines were your strategies, how would you play them?
Strike a balance between exploration vs. exploitation (through a learning rate) and
change weights dynamically – a case of online reinforcement learning
𝑤𝑖
𝑡+1
=
𝑤𝑖
𝑡
exp(
𝜂𝑥𝑖
𝑡
𝑾 𝑡. 𝒙 𝑡)
𝑤𝑗
𝑡
exp(
𝜂𝑥𝑗
𝑡
𝑾 𝑡. 𝑿 𝑡)𝑁
𝑗=1
Kind of follow the leaders – will not work in a sideways markets. Multiplicative updates
can lead to concentration (can be controlled by learning rate)

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Open House
Q&A

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Webinar Video Link
https://youtu.be/u5UUjHZQCFk

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THANK YOU!
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