A report on a modified Tail Risk Parity asset allocation model written for the Sprott Student Investment Fund.

1. Greg Poapst
Quantitative Analyst
B.Comm Finance, 2017
Greg.poapst@carleton.ca
Tail Risk Parity – A Quantitative Framework
Investment Philosophy
The Sprott Student Investment Fund’s
equity portfolio takes a fundamental
approach to investing with a value
orientation, seeking to maximize the
value of its assets over the very long
term. The primary focus is on building a
portfolio of wealth-creating firms, with
unique competitive advantages, strong
financial positions, and proven
management teams. We strive to make
investments only in firms that trade at
discounts to their intrinsic value.
April 7, 2016
Report Highlights
In this report we provide a background on Tail Risk Parity, and a Quantitative
model built on the backbone of this theory. We have broken up the report into
four sections:
 Primer on Tail Risk Parity
A brief background on the Tail Risk Parity approach. The theory and history
behind Tail Risk Parity and empirical evidence in financial literature.
 Primer on Differential Evolution Optimization
Empirical results of our own model, and modifications made to the Tail Risk
Parity process.
 Empirical Tail Risk Parity Tests
An explanation of how we tested our model for robustness, and some of the
major benefits in using such an approach with our portfolio.
 Optimization Results
We explain the results of our model, and differential evolution process of
optimization.
TailRiskParity–AQuantitativeFramework

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Figure 1 – Fat Tailed Kurtosis
Primer on Tail Risk Parity
Before we begin to describe Tail Risk Parity, it is important to understand both our goals
and how risk based asset allocation came to light. Typically, investment managers would use
a static asset allocation strategy to determine the weights of asset classes in their portfolio,
however, the stock market is not static. One of the most common and yet least
sophisticated asset allocation schemes used in the industry is the 60/40 balanced portfolio.
The 60/40 balanced portfolio (or any arbitrary weighting) says you should hold 60% of your
assets in equities and 40% in bonds. The riskiness of such a “balanced” portfolio is not
balanced at all. Equities are much riskier in terms of volatility than bonds. This creates a
problem where the risk of the portfolio is extremely dependent on the equities portion of
the portfolio.
Risk Parity attempts to provide a solution to this problem of static and risky portfolios by
weighing each asset class to contribute an equal amount of risk to the total portfolio. The
idea here, is that you would weigh your asset classes by risk as opposed to dollars invested,
and reduce your overall risk. While this provides some value in terms of risk, it does come at
a trade-off (which we will touch a bit on later).
The idea of Risk Parity (or equal risk contribution) was thrown around over the years, but
never truly used in practice until 1996 when Ray Dalio developed an “All Weather Portfolio”
based on the Risk Parity ideaology. Today Ray Dalio’s hedge fund – Bridgewater Associates –
is the largest hedge fund in the world. Much of this success can be attributed to the success
of his flagship all weather fund, and the performance specifically through what we call black
swan events.
Black swan events are described as events that happen unexpectedly, have massive
reprecussions, and are rationalized to some extent after the fact. In the investment
universe, black swan events happen all too often. This has led many academics to believe
and subsequently prove that the stock market return distribution exhibits large kurtosis.
That is to say that stock market returns do not follow a normal distribution, and in essence
have more extreme outcomes than it should (as shown in figure 1).

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The underlying theory here is that because we know that we cannot predict these black
swan events by definition, we should employ some sort of hedge against them. The
problem has been that hedging againt mass market corrections generally comes at a cost.
Alankar, DePalma and Scholes (2012) prove that using a modified tail risk parity approach,
they are able to hedge against these large corrections with about 50% less cost than
hedging using options.
While a risk parity approach provides volatility protection, it comes at a high cost.
Essentially we are talking about building a portfolio that uses Expected Tail Loss as a risk
measure instead of a traditional standard deviation of returns. Using a Tail Risk Parity
approach we are able to protect ourselves from large drawdown events, but without giving
up as much of the upside (the part of volatility that we want!).
Alankar, DePalma and Scholes (2012) found that a Tail Risk Parity portfolio performed at
almost the same level of return as the 60/40 portfolio, but with only a third of the tail loss
risk (shown in figure 3 and 4 on the left). This paper provides the academic rationale to build
a Tail Risk Parity portfolio for the Sprott Student Investment Fund, and if interested it is
highly encouraged that you read it.
We conclude that due to the risk of large drawdowns during times of economic turmoil, the
inherent kurtosis of the markets and the skewed risk/reward characteristics of the Tail Risk
Parity portfolio, it provides the right quantitative backbone for our portfolio weighting to
build on.
Figure 2 - Historical Monthly Returns from 1950 to Present
Source: Alankar, DePalma and Scholes
(2012)
Figure 4 – Tail loss from 2003 to 2012
Figure 3 – Returns from 2003 to 2012
Source: Alankar, DePalma and Scholes
(2012)

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Primer on Differential Evolution Optimization
Differential Evolution Optimization (DE) was first developed by Rainer Storn and Kenneth
Price (1997). DE is a method of computational optimization used to find the most optimal
solution to a given set of objectives and constraints. Basically, the DE algorithm attempts to
find an optimal solution without making any assumptions about the problem at hand. This is
why is is used for many complex problems in physics, science, math and now finance. It is
specifically useful when you either don’t know all of the functions in the optimization
problem, or when you know them but are unable to differentiate (take the derivative) as
you would in classical optimization problems such as the lagrange method in
microeconomics.
Given the massive amount of flexibility and ease of use the DE method provides, it has
many applications. The actual algorithm itself is run iteratively (meaning it finds optimal
levels sequentially and eventually converges to the true optimal level of weights). While this
essentially will only ever provide an extremely close approximation of the true optimal level,
it does consistently provide a solution to portfolio optimization problems. There is always a
chance however, that it will not find a solution, although that is usually due to a lack of
constraints or objectives to follow.
To simplify the explanation above, the optimization continues to run new scenarios, and
drops the ones that don’t make sense. It continues this process until the changes in
portfolio weights from one scenario to the next are so small that it assumes optimality.

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Empirical Tail Risk Parity Results
In order to get a better idea of how Tail Risk Parity weighting would affect our portfolio,
we ran a backtest. Using our current holdings, we ran the Tail Risk Parity model on our
portfolio for the years 2013 and 2014. Using the data for these two years, we developed our
Tail Risk Parity weights going forward. After we had our weights we calculated return, risk,
and volatility for the hypothetical portfolio over 2015, and a very simple equal weighted
portfolio using the same holdings. While there are many factors that could have change our
results (such as rebalancing and changing holdings), this backtest does give us some
indication of the positive aspects of the Tail Risk Parity portfolio approach.
What we observed was quite interesting and motivating. The Tail Risk Parity portfolio
showed both higher returns and lower standard deviation than the equal weight portfolio
(see figure 6 on the left). The figure below (figure 7) shows the relative starting weights of
the Tail Risk Parity portfolio.
To get a better idea of the true level of volatility of the two portfolio methods, we use a
GARCH(1,1) model. GARCH stands for generalized autoregression conditional
heteroscedasticity. Because GARCH is somewhat complicated, I will leave Kent Osband to
explain using the theory of wiggles.
“To translate [meaning of GARCH], skedasticity refers to the volatility or wiggle of a time
series. Heteroskedastic means that the wiggle itself tends to wiggle. Conditional means the
wiggle of the wiggle depends on something else. Autoregressive means that the wiggle of
TRP EQ
Returns -3.70% -5.30%
Std Deviation 0.143 0.149
Sharpe Ratio -0.259 -0.357
Results
Figure 6 – 2015 Test Results
Figure 7 – Tail Risk Parity weights as of January 1, 2015

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the wiggle depends on its own past wiggle. Generalized means that the wiggle of the wiggle
can depend on its own past wiggle in all kinds of wiggledy ways.” – Kent Osband
Due to the properties and assumptions of a GARCH model, it has been proven to be one
of the best methods for estimating volatility (and subsequently predicting it). The three
main reasons for forcasting volatility are risk management, asset allocation, and straight up
speculation on the markets or volatility based financial instruments. While there are many
variations of the original ARCH model, we employ a simple GARCH(1,1) model to estimate
the volatility of our portfolio over the course of the year (2015).
As you can see in figure 8 below, volatility over the year was very unpredictable.
Specifically, the spike in volatility around August / September due to worries out of China
shows the dangers of volatility to static asset allocation techniques.
More importantly, we wanted to calculate volatility for both portfolios, and compare the
two. By plotting the difference between the equal weight portfolio and the tail risk parity
portfolio, we are able to show that almost all throughout the year, the tail risk parity
portfolio exhibited lower volatility than the equal weight portfolio (seen in Figure 9 below).
Figure 8 – GARCH Volatility over the year 2015

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Aside from the overwhelmingly positive difference (Equal Weight volatility – TRP
volatility), there is another observation to make regarding figure 9. The largest spike in
volatility difference (when the TRP portfolio worked the best to hedge out volatility) was
actually right before the Chinese market downturn. It was actually predictive of the pending
volatility spike!
To conclude our testing portion of our model, we can say with confidence that our
portfolio can benefit from the quantitative framework developed by many professionals and
academics before us. While most critics of Risk Parity approaches claim that by protecting
yourself against the downside you are limiting your upside, we have seen that Tail Risk
Parity provides much of the protection (especially when most needed) without giving up as
much of the upside. Despite this result, we still want to employ some sort of qualitative
aspect to our model, because we are still stock pickers by nature and we do have a very
concentrated portfolio.

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Optimization Results
We optimized our portfolio holdings using the methods described above. Specifically
using a Tail Risk Parity approach (Expected Tail Loss), Box Constraints setting the weights
have to be between 2% and 9%, contraints stated that we cannot use leverage, and we
cannot short sell. Using a differential evolution approach to optimization on our current
holdings over the past 2 years of daily price data, we are able to determine the optimal
weights of each of our holdings (seen in figure 10 below).
We also wanted to show the risk contribution of each holding under our new proposed
model. Figure 11 below shows the risk contribution of each holding to expected tail loss risk
given our new optimal weights. You may notice that not all holdings are exactly equal risk.
This is due to our box constraint, because certain holdings (North West Company in
particular) are considered very low risk in general. North West Company does not exhibit
signs of high expected tail loss, and is also extremely uncorrelated with the other holdings,
providing even more diversification benefits than the others.
Figure 11 – Risk Contribution by Holding
Figure 10 – Optimal Weights of Holdings

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We plot all 20,000 different portfolio iterations in a risk return space to ensure that our
portfolio is optimized as much as possible (shown in figure 12 below). You may notice that
there are still some portfolios with better expected return for lower risk. This is also due to
our box constraints which by default limit the possible portfolios in our iterative search
algorithm. The light blue lines are the original portfolios found optimal by the algorithm, and
as the algorithm narrowed in on the true optimal portfolio that line moves to a dark blue.
You can truly see the process in which the algorithm converges to the optimal portfolio.
These proposed portfolio weights will provide a solid backbone for our portfolio and a
legitimate solution to asset allocation. By providing a higher return per unit of risk, our
proposed portfolio should outperform others in the long run.
Figure 12 - Possible and optimal portfolios plotted in a risk / return space

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References
 Bollerslev, T. 1986. Generalized Autoregressive conditional heteroscedasticity. Journal of Econometrics 31:307-
327.
 Osband, Kent 2010. Finformatics: SAMURAI GARCH killers. Wilmott Magazine 44-46.
 Alankar, DePalma and Scholes. 2012. An Introduction to Tail Risk Parity. Alliance Berstein Whitepaper.