Risk Parity, a relatively new portfolio construction method, took Wall Street by storm overcoming the traditional mean-variance and 60/40 methods. Why this method is better and when?
4. Description
Risk parity is a portfolio construction
method where each asset contributes
equally to portfolio risk.
It allocates capital to each asset inversely
proportionally to its risk (measured by the
standard deviation of its returns)
5. History
Markowitz, 1952, “Portfolio selection.” Last page
To use the E-V rule in the selection of securities we must have
procedures for finding reasonable and i,j. These procedures, I
believe, should combine statistical techniques and the judgment of
practical men. My feeling is that the statistical computations
should be used to arrive at a tentative set of and i,j. Judgment
should then be used in increasing or decreasing some of these
and i,j on the basis of factors or nuances not taken into account by
the formal computations. Using this revised set of and i,j the set of
efficient E, V combinations could be computed, the investor could
select the combination he preferred, and the portfolio which gave
rise to this E, V combination could be found.
8. History: Definition
The Risk Parity (RP) portfolio’s weights v, vT
= (v1, … , vn)
are by definition inversely proportional to the asset
volatilities:
σ−1 ≝
1
σ1
, … ,
1
σn
T
, σi = σi,i , i = 1, … , n
Taking into account the normalizing constraint i vi = 1,
we have:
vi =
σi
−1
σi
−1 , i = 1, … ,
13. History: Diversification Example
Returns Asset 1 Asset 2 50/50 25/75
Year 1 100% -20% 40% 10.00%
Year 2 -50% 25% -13% 6.25%
Total 0% 0% 23% 17%
m 13% 1% 7% 4%
vol 75% 23% 26% 2%
Sharpe 0.17 0.06 0.26 2.17
Risk Parity Concepts:
1. Allocates
less weight to
greater risk asset
2. Produces smaller
returns
3. Produces greater
Sharpe ratio
14. Mystery: Performance
“We find that unleveraged risk-parity strategies generally
compounded returns around 7% per year since 1990 with
annualized volatility around 4%.”
Roberto Croce, director of quantitative research
Lee Partridge, chief investment officer of Salient
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15. "This has nothing to do with bets. It has to do with how to make all the assets
the same risk parity.“
"What most [of] the investor[s] need to do is have a balanced, structured
portfolio. A portfolio that does well in different environments.“
Ray Dalio, founder of
Bridgewater with $142 billion
under management
“All Weather” a first risk
parity fund started in 1996, $46
billions, one of the largest funds
in the U.S.
"You're not going to win by,
sort of, trying to get what
the next tip is," he said.
“We spend hundreds of
millions of dollars on
research … and we don't
know that we're going to
win. We have to have
diversified bets.“
Mystery: Performance
16. Mystery: Growth of Risk Parity Funds
Bridgwater (All Weather) (1996) swelled from $25bln to $60bln
PanAgora (2006)
AQR : first risk parity fund AQRIX (2010), $1.1 billion, two more
risk parity funds in (2012) total invested in risk parity >$20 bln
Aquila (2008)
Northwater
Wellington
Putnam (2012)
ATP (2006)
AllianceBernstein (2010)
Clifton (2011)
Invesco (2009) $1.5 billion
2009: 10.38%, 2010: 12.98%, 2011: 10.30%, 2012: 10.56%
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17. Mystery: Positive Response to Risk Parity from
Institutional Investors
Examples:
The Wisconsin State Investment Board
The Pennsylvania Public Schools Employees’ Retirement System
The Alaska Permanent Fund Corp
The San Diego County Employees Retirement Association
Danish pension fund ATP has been running its entire portfolio on
a risk parity basis since 2006
18. Mystery - Reason
• Modern Optimal portfolio theory: Builds the best portfolio when
all parameters are known.
• n assets having random excess returns:
• XT
= X1, … , Xn
• E X = μ and Var X = Σ
• where
• μT = μ1, … , μn and Σ = σi,j = σiρi,jσj , i, j = 1, … , n.
• σi are standard deviations, ρi,j are correlations
19. Mystery – Reason
• E 𝑋 = 𝜇 and Var 𝑋 = Σ
Merton, Robert C., 1980. “On estimating
the expected return on the market: An
exploratory investigation.”
• estimating expected returns requires a
longer time period
• estimating variance requires finer
observations of returns
• Even finer observations is not enough:
– High frequencies
– Momentum/mean reversion
– Spurious correlations
20. Mystery – Definition
• Equal Risk Contribution
• Sometimes is called Risk Parity
• And Risk Parity sometimes is called Naïve Risk Parity
• Define:
21. Mystery – Definition
Define Equal Risk Contribution:
𝜎 𝑤 is the volatility of a portfolio with weights 𝑤 =
(𝑤1, … , 𝑤𝑛)
Define the risk contribution of asset 𝑖 as:
𝜎𝑖 𝑤 ≝ 𝑤𝑖
𝜕𝜎 𝑤
𝜕𝑤𝑖
The risk (volatility) of the portfolio is the sum of its asset risk
contributions:
𝜎 𝑤 =
𝑖=1
𝑛
𝜎𝑖 𝑤
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22. Mystery – Definition
Define Equal Risk Contribution (or Beta Parity):
• The Equal Risk Contribution portfolio is defined by requiring that all
assets’ risks are equal:
• 𝜎𝑖 𝑢 =
𝜎 𝑢
𝑛
, 𝑖 = 1, … , 𝑛
• And the normalizing constraint 𝑢𝑖 = 1
• and the no-short-selling constraint 0 ≤ 𝑢𝑖 ≤ 1, 𝑖 = 1, … , 𝑛
• Could be shown that
• 𝑢𝑖 =
𝛽𝑖
−1
𝑛
=
𝛽𝑖
−1
𝛽𝑖
−1 , 𝛽𝑖 is beta of asset 𝑖 with the portfolio
23. Mystery – Who’s naïve?
• For n assets, Risk parity needs to know n standard deviations.
Equal Risk Contribution needs to know about n^2 correlations.
• When we observe T monthly returns of n assets, say, T = 24
months, total number of observations is n*T
• Risk parity needs n estimates of standard deviations
• Equal Risk Contribution needs n*(n-1)/2 estimates of correlation + n
standard deviations.
• Who’s naïve?
24. Risk Parity Criticism
Leverage
it is our view that investors are typically better off increasing their
holdings of a well-diversified portfolio — even if that means employing
leverage — than concentrating them in assets with higher volatility
and expected returns. In other words, we believe the modest and
prudent use of leverage is often times preferable to pinning our ability
to meet future obligations on hopes that equity markets will rally
indefinitely. We think this holds true regardless of whether investors
are targeting annual returns of 8% or 18%. In practice, we find that a
risk-parity strategy targeting 10% volatility generally requires notional
exposures that fluctuate between 200% and 400%, with median
exposure around 300%.
Croce, Partridge from Salient.
25. Risk Parity Criticism
Rising interest rates
• Aquila study examines how a Risk Parity portfolio would have performed in
three testing market environments: the Great Depression, the inflationary
period of the 1970s and the bond market drawdowns between 2005 and
2008.
• The conclusion is that Risk Parity strategies can still deliver attractive risk-
adjusted returns across different types of market uncertainty, including
periods of rising interest rates.
• Aquila Capital, founded in 2001, is one of Europe's leading independent
Alternative Investment managers with over $5.3 billion in assets under
management. The company, headquartered in Germany,
• AQR says, for example, that during a nearly three-year period of high
inflation in the early 1970s, a risk-parity portfolio would have outperformed
a 60/40 portfolio by more than 45%, because the risk-parity fund would
likely have included commodities. (WSJ)
26. Mystery – explained?
“AQR, a global leader in risk parity investing”
www.aqrfunds.com
Leverage Aversion and Risk Parity, 2012
Clifford Asness, Andrea Frazzini, and Lasse Heje Pedersen
Abstract.
We show that leverage aversion changes the predictions of modern portfolio theory:
It implies that safer assets must offer higher risk-adjusted returns than riskier assets.
Consuming the high risk-adjusted returns offered by safer assets requires leverage,
creating an opportunity for investors with the ability and willingness to apply leverage.
A Risk Parity (RP) portfolio exploits this in a simple way, namely by equalizing the
risk allocation across asset classes, thus overweighting safer assets relative to
their weight in the market portfolio. Consistent with our theory of leverage
aversion, we find empirically that RP has outperformed the market over the last
century by a statistically and economically significant amount.
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27. Mystery – explained!
Risk Parity Optimality
Gregg S. Fisher, Philip Z. Maymin, Zakhar G. Maymin
Abstract
We show that the probability of risk parity beating any other portfolio is
more than 50 percent. We also prove that if portfolio performance is
measured by Sharpe ratio, risk parity is the only maximin portfolio
when (1) all assets’ future Sharpe ratios are greater than an unknown
constant and all correlations are less than another constant, or (2)
when the sum of all assets’ future Sharpe ratios is greater than some
constant. If portfolio performance is measured by expected return, risk
parity is the only minimax portfolio when the sum of assets' Sharpe
ratios is greater than a constant.
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28. $$
$
Risk Parity
Mystery – explained
The probability of risk parity beating any other portfolio is
more than 50 percent.
29. Mystery - explained
“Life is a game - play it. ”
Sathya Sai Baba (1926-2011)
2-player zero sum game.
𝑏1 𝑏2
𝑎1 1 2
𝑎2 3 4
𝑉1 = max
𝑎𝜖𝐴
min
𝑏𝜖𝐵
𝜙(𝑎, 𝑏)
𝑉2 = min
𝑏𝜖𝐵
max
𝑎𝜖𝐴
𝜙(𝑎, 𝑏)
30. Mystery - explained
If 𝑉1 = 𝑉2, then 𝑉 = 𝑉1 is called the value of the game
𝑉 ≝ max
𝑎𝜖𝐴
min
𝑏𝜖𝐵
𝜙(𝑎, 𝑏) = min
𝑏𝜖𝐵
max
𝑎𝜖𝐴
𝜙 𝑎, 𝑏
It turns out that a matrix game always has a solution among pure
strategies if the matrix has a saddle point, i.e. the matrix of payoffs has
at least one element that is the minimum in its row and the maximum in
its column.
In our example, the matrix has a saddle point in row 2 and column 1,
namely the value 3.
31. Mystery - explained
𝑏1 𝑏2
𝑎1 1 3
𝑎2 4 2
V1=2
V2=3
no saddle point, no solution among pure strategies.
32. Mystery - explained
The solution, the mixed strategies of player 1
and player 2, is the Nash equilibrium following
Nash (1951) who generalized von Neumann
result for non-zero-sum games.
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However, for a matrix game, a solution
always exist among mixed strategies.
This is the famous result of von
Neumann (1928).
33. Mystery - explained
If the sum of assets non−negative expected Sharpe ratios is
greater than a certain (unknown) constant, then the risk parity
portfolio is the only minimax portfolio among all no−short−sales
portfolios.
In other words, if the portfolio manager knows that the sum of
all non−negative assets Sharpe ratios is greater than a certain
(unknown) constant, then, regardless of the constant, the
minimax portfolio is the risk parity portfolio.
This is the portfolio that will have the greatest expected value
under the worst possible scenario.
We also proved that this is the portfolio that has the greatest
Sharpe ratio under the worst possible scenarios. The standard
game theory can’t be applied in this case.
34. Mystery explained. How trading works
According to Modern Portfolio Theory:
1. Market chooses expected values of assets.
2. Portfolio manager chooses the weights for his portfolio.
3. The best portfolio is the tangency portfolio.
According to our results:
1. Portfolio manager chooses the weights for his portfolio.
2. Market chooses expected values of assets.
2. The best portfolio is the risk parity portfolio.
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40. Risk Parity Revolution
Other ideas
Life lessons
Serial actions
Time axis has the direction
Important not to fail on each step
Think of chess
Effect of income tax Probability of success treatment of losses
Good trade 51% success With 50% taxes game changes looking
for more risks hard to succed
Going up the wall and every step you go 50% back
