Tuesday, April 21, 2015

Looking Back at Risk Parity's Golden Age

My initial goal of this post was to share why risk parity was less likely to be a free lunch going forward using historical data back to the 1950's (the last time we saw rates at current levels), but it became more of a risk parity 101 piece. I'll save much of those comments for another day.

What is Risk Parity?

Risk parity was a relatively unknown strategy until Bridgewater's All Weather Fund powered its way through the financial crisis ("risk parity" doesn't even show up on Google Trends until 2009). Since that time, the success has spawned numerous imitators. But first... what is Risk Parity?

Per Investopedia:

A portfolio allocation strategy based on targeting risk levels across the various components of an investment portfolio. The risk parity approach to asset allocation allows investors to target specific levels of risk and to divide that risk equally across the entire investment portfolio in order to achieve optimal portfolio diversification for each individual investor. 
The objective of allocating equally to 2+ asset classes by risk is to receive the benefit of diversification and materially improve the Sharpe ratio of a blended allocation.

Backdrop: The Importance of Sharpe Ratios and Correlation

There are two main factors that drive the relative performance of a risk parity allocation vs stocks; (1) the relative Sharpe Ratio of the new asset class introduced (the higher, the better) and (2) the correlation between the new asset class introduced and stocks (the lower, the better).

As a reminder, per Investopedia:
The Sharpe ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk. 
For example…. if cash returns are 2.5%, then stocks returning 10% with a 15% standard deviation has the same 0.50 Sharpe ratio as bonds returning 5% with a 5% standard deviation: 
  • Stocks: (10% - 2.5%) / 15% = 0.50 Sharpe ratio 
  • Bonds: (5% - 2.5%) / 5% = 0.50 Sharpe ratio 
An identical Sharpe ratio means an investor would be indifferent as to whether they held either stocks or bonds (in isolation), irrespective of their risk appetite. If an investor prefers a 7.5% risk target, a 50% stock / 50% cash allocation cuts the initial 15% risk in half, while returning 6.125% (50% x 10% + 50% x 2.5% = 6.125%). Similarly, a bond allocation gets to the same 6.125% return at a 7.5% risk by leveraging up bonds by 50% (150% x 5% - 50% x 2.5% = 6.125%).

If viewed in isolation (if you can only allocate to one asset class), an investor should allocate to the asset class with the highest expected Sharpe ratio, then lever up / down that asset class to match their desired risk target (an either / or question of stocks vs. bonds). Looking at the rolling three-year Sharpe ratio of stocks and bond since 1976, we see that from 1976 through the mid-1990's stocks had consistent positive performance relative to cash (the Sharpe was consistently above zero) and bonds moved largely in sync with stocks. Since the mid 1990's, stocks have had two material periods of negative excess performance, while bonds have provided consistent positive excess return.

Few investors are restricted to holding only stocks and bonds, but many are restricted from applying leverage. For those with the flexibility to allocate to both stock and bonds, as well as apply leverage, correlation between asset classes plays an even greater factor in determining whether the additional asset class should be added, broadening the decision from an initial 'stocks or bonds' question to 'stocks and/or/no bonds' question.

As a reminder per Investopedia:
Correlation is computed into what is known as the correlation coefficient, which ranges between -1 and +1. Perfect positive correlation (a correlation co-efficient of +1) implies that as one security moves, either up or down, the other security will move in lockstep, in the same direction. Alternatively, perfect negative correlation means that if one security moves in either direction the security that is perfectly negatively correlated will move in the opposite direction.
The huge shift in the relationship between stocks and bonds (hugely positive correlation to hugely negative) starting in the late 1990's can be seen in the chart above (the Sharpe ratio of bonds spiked when stocks crashed) and below (a shift that has been hugely beneficial for investors allocating to levered bonds to balance their stock allocation).

Taken together, the mid 1990's started what has been the Golden Age for risk parity. Instead of requiring a high Sharpe ratio in order for bonds to be added a stock allocation, the negative correlation has meant bonds have largely benefited a risk parity allocation even in periods when bonds have materially underperformed stocks.

As a reminder for those that took the CFA and have blacked out all memories, the formula for determining whether a new allocation improves a Sharpe ratio is as follows:

New asset class Sharpe ratio > Current Sharpe ratio x correlation with new asset class

This means that an allocation to bonds makes sense if the Sharpe ratio of bonds > the Sharpe ratio of Stocks x correlation. When correlations are sharply negative, there are much fewer instances when an allocation to bonds won't improve the Sharpe ratio relative to a stock only allocation (something to keep in mind for commodities even if your expected excess return to cash is 0%).

If the figure in the below chart is > 0, an allocation to bonds (at some level) improves the Sharpe ratio. Note that any notional size allocation to bonds may not improve the Sharpe (i.e. a 10% allocation may improve the Sharpe, a 50% allocation may not - see comment section for more of this discussion).

Given the added benefit of materially positive excess performance of bonds relative to cash since the mid-1990's, risk parity has been a home run (in the example below, risk parity is defined as 25% stocks and 75% bonds). These strong results since 1996 coincided with the year Bridgewater launched their All Weather iteration (hats off to Ray Dalio and his timing). As a comparison, predating the Golden Age was a ten year period from 1974-1984 when risk parity underperformed a simple stock allocation, despite an annualized bond return north of 8% (they key being cash yielded even more). Note the slope of the lines below is equal to the Sharpe ratio over each period.

And the rolling results (just look at the widening gap since 1996). Note that the Sharpe ratio for the risk parity portfolio would not change if you lever it (i.e. a 25% stock / 75% treasury allocation has the same Sharpe ratio as a 50% / 150% portfolio).

What now?

On one hand, it isn't all that difficult to outperform interest rates when they're zero. On the other hand, when interest rates do (eventually) move higher, excess returns of bonds will be impacted (treasury rates are currently VERY low per unit of duration). When interest rates do eventually back up, the relative attractiveness of stocks vs. bonds becomes smaller (that case is outlined here), resulting in a higher correlation between stocks and bonds going forward.

This would equate to a one-two punch against the 20-year Golden Age run that risk-parity has enjoyed.

Source: S&P, Barclays


  1. Looks like the formula for determining whether a new allocation improves a Sharpe ratio is more a rule of thumb rather than mathematically correct according to a couple of simple numerical simulations I run...

  2. Current port ret: 1%
    Current port vola: 1%
    Current port sharpe: 1
    Current port weight: 90%

    New AC ret: -0.5%
    New AC vola: 5%
    New AC sharpe: -0.1
    New AC weight: 10%

    Corr CurrentPort-NewAC: -0.2

    Test: NewACSharpe > CurrentPortSharpe*Corr

    Test is TRUE bat the Sharpe ratio for the new portfolio is actually 0.91 (lower than the CurrentPortSharpe of 1).

    It's much easier with the Excel file I put together...free to request it via email if not clear the above.


  3. In your example, I calculate an optimal allocation of 98% current portfolio, 2% new portfolio - improving the Sharpe ratio from 1.0 to 1.005 (0.97% return and 0.965% risk).

  4. In my understanding the formula is referring to the general case of adding a new asset class to an already existing portfolio without mentioning any particular portfolio optimization scheme.

    Are you then saying that the formula holds true for risk parity only?

  5. The formula just says that at some level, adding the new asset class improves the sharpe. The point was simply that a negative correlation makes it much more likely that an allocation to bonds (via risk parity) is much more likely to add value if:

    A) Sharpe is higher
    B) Correlation is lower

  6. Jake, I do get your main point...I was trying to further understand what the formula is actually proving.

    As such your point that a negative correlation makes it much more likely that an allocation to bonds is much more likely to add value if A and B holds true in general not just for risk parity.

  7. Absolutely... but it has been a material driver of the strong performance of risk parity.

  8. Therefore we cannot conclude that an increase in correlation among stocks and bonds will negatively affect risk parity allocation more than any other asset allocation schemes.

  9. I disagree with that statement. A large shift in correlation will impact investment strategies with higher allocations to the correlating asset than strategies with smaller allocations.

    Example... what would be impacted more by a shift in correlation (all else equal):

    A) a 60% stock / 40% bond allocation
    B) a 60% stock / 180% bond allocation

  10. Risk parit is doing just the opposite by underweighting asset classes with higher correlations. Regarding your example it doesn't make much sense to me as you are comparing a non leveraged portfolio with a leveraged one...

  11. Incorrect on both parts:

    1) you aren't necessarily forced to underweight anything. You can simply add the asset class (i.e. a 100% stock allocation could become a 100% stock / 100% bond allocation if using long bonds instead which are about the same risk as stocks)

    2) A Sharpe ratio doesn't change with leverage (the slope between the risk free rate and the risk/return figure for the unlevered version is the Sharpe ratio), thus a 25% stock / 75% bond allocation has the same Sharpe ratio as a 30/90, 40/120, or 50/150 portfolio

  12. Your comments are not relevant to my main point which is that you can't say that a risk parity asset allocation is negatively affected by an increase in cross-asset correlation MORE than any other portfolio optimization algorithm.

    To put it simply, if both stocks and bonds are going down as usual when you see an increase in correlation, you are losing money in all asset allocation portfolios and you can't predict that the risk parity is going to lose more ex-ante as the result is driven by actual weights.

    As an extreme case if correlation goes up to 1 and equity lose more than bonds as pretty common, you are much better off with a risk parity allocation (say 10% equity, 90% bonds) than a classic 60%equity/40%bond.

  13. With all due respect, the following statement is incorrect "As an extreme case if correlation goes up to 1 and equity lose more than bonds as pretty common, you are much better off with a risk parity allocation (say 10% equity, 90% bonds) than a classic 60%equity/40%bond."

    You are mistaking returns with Sharpe ratio (i.e. excess return per unit of risk). For an apples to apples comparison you need to either:

    A) lever up the 10% stock / 90% bond portfolio
    B) water down the 60% stock / 40% bond portfolio

    So that they are equal risk to one another. Assuming the 10/90 portfolio is 3x less risk than the 60/40 portfolio, the comparison is either:

    A) 30 bond/270 stock vs. the 60 bond/40 stock
    B) 10 bond/90 stock vs. a 20 bond/13 stock/67 cash

    If correlations are literally 1, then you are better off with either stocks or bonds on a stand-alone basis... allocating to whichever has the higher Sharpe.

  14. We are comparing asset allocation schemes with the classic fully invested constraint ie. 100% invested (0-100% in stocks and the remaining in bonds) not portfolios with the same level of risks where you leverage.

    Btw sharpe ratio would still be higher for the risk parity portfolio vs the classic allocation in the above example.

    Anyway thanks for the discussion.