If we asked investors how stocks are likely to perform relative to bonds, what would they generally say? Historically, stocks have produced higher excess returns^{1} over the long run, both on an absolute and a volatility-adjusted basis. Going forward, stocks are likely to deliver a higher Sharpe ratio (excess return divided by volatility) relative to bonds, as holding stocks involves assuming risk that is correlated with the business cycle. But how much higher will the Sharpe ratio of stocks be versus bonds?
While we can’t ask everyone this question, it turns out one can infer an investor’s relative views from their asset allocation. This topic has been extensively studied in the academic literature on mean-variance portfolio construction,^{2} with the key insight being that the amount of risk allocated to a given asset is related to the expected return of that asset and other correlated assets.
Take, for example, the popular 60/40 stock/bond portfolio, which is a good approximation of many institutional investors’ asset allocation. Given a set of assumptions consistent with long-term market characteristics,^{3} e.g., 16% volatility for stocks, 4% for bonds and a 0% stock-bond correlation, it can be shown that stocks contribute 97% of the estimated excess return above cash, with bonds contributing the remaining 3%. Investors holding this allocation mix are essentially expressing the view that stocks are likely to deliver a Sharpe ratio six times higher than bonds.^{4 }Intuitively, this result should make sense: Why else would investors allocate 97% of their risk budget to a single factor unless they believed it offered a significantly better risk-reward ratio?
As the current economic expansion ages further, we believe asset owners should conduct a thorough analysis to ensure their portfolios are sufficiently diversified and, importantly, aligned with their own views.
We walk through an analytical framework that can be used to gauge the level of diversification in a portfolio by estimating the total volatility contribution from each position. This framework can then be used to calculate investors’ forward-looking excess return estimates for each position, providing an informative “gut check” on whether those estimates are aligned with their actual views.^{5}
Assessing diversification of risks
In a mean-variance-efficient framework, portfolio construction is about the allocation of volatility (i.e., risk), not capital, in a diversified manner. An effective way to gauge diversification in a portfolio is by estimating net volatility contribution, which quantifies how much each position contributes to the total portfolio volatility after accounting for its covariance with other assets (i.e., diversification benefits).
The relationship between the net volatility contribution and the stand-alone volatility of an asset is a function of the correlation of the asset’s excess return to the overall portfolio excess return. In particular, for a portfolio containing multiple assets, net volatility contribution of an asset “s” is equal to:
Net Volatility Contribution(s)=Weight(s)×Vol(s)×ρ(s),
where,
Weight (s) = Weight of the asset in the portfolio
Vol (s) = Volatility estimate for the asset
= Estimate of the correlation of the asset “s” with portfolio excess return^{6}
The sum of net volatility contribution from all assets in a portfolio mathematically is equal to the overall portfolio volatility. It is important to highlight that if two assets have equal net volatility contributions, it does not necessarily imply the investor has the same level of conviction. In the next section, we will show how investors can determine their implied views on the relative attractiveness of various positions in their portfolio.
Gauging conviction in investment views
Calculating the Sharpe ratio for a single asset or risk factor is straightforward: It is the ratio of expected excess return to its assumed volatility. However, coming up with expected Sharpe ratios across complex institutional portfolios isn’t always feasible given they span multiple asset classes, managers and potentially thousands of underlying securities.
But what if the problem were turned on its head? Given the expected Sharpe ratio for the entire portfolio, what if one could derive the implied Sharpe ratio of each underlying holding? Investors could then use this information to decide if those implied views on risk and reward for each security were congruent with their own thinking. That is exactly what the implied Sharpe ratio (ISR) measure does.
Given a portfolio, the ISR is the implied expected excess return of a given position, s, after adjusting for its volatility:
ISR(s)=Expected Sharpe ratio of the portfolio × ρ(s)
The correlation, ρ(s), as defined in the previous section, is the ratio of the net volatility contribution of an asset to its stand-alone volatility.
ρ(s)=(Net Volatility Contribution)/(Weight(s)× Vol (s))
Under the hypothesis that the portfolio is mean-variance-efficient without any constraints, this ratio is proportional to the ex ante Sharpe ratio of the position. For this reason, the correlation may be interpreted as a measure of an investor’s conviction in active positions.
The ISR measure is elegant as, without any knowledge of investors’ views on macroeconomic conditions, asset valuations or market technicals, it makes it possible to quantify their forward-looking views. In particular, these views can be inferred at any level of granularity, from the level of asset classes, managers, or individual securities, provided we have a sufficiently granular risk model of correlations and volatilities.
Comparing ISRs of various positions can reveal what a given asset allocation is implying about each position’s relative attractiveness. ISRs can be used to size new positions as well as to detect if a certain position is becoming highly correlated with the rest of the portfolio. Most importantly, ISRs can serve as a diagnostic tool to determine if a portfolio’s positioning is aligned with an investor’s own views.^{7}
From theory to practice
We now apply the proposed framework above to a hypothetical institutional portfolio to demonstrate how investors can use the ISR measure as a tool for portfolio construction and risk management.
Figure 1 shows a typical policy portfolio of an institutional investor. For the purposes of this exercise and for simplicity, we will assume that this portfolio is expected to deliver a Sharpe ratio of 0.5, though it could be a different number that is more consistent with an investor’s forward-looking view.^{8}
Figure 2 shows the assumed correlation matrix to study this hypothetical example consistent with long-term historical correlations.
When analyzing portfolio volatility, many investors look at the stand-alone volatility of each position, which is simply the product of allocation weight and volatility of the asset. For the sample portfolio, stand-alone volatilities for each asset class are shown in Figure 3.
A quick look at the stand-alone volatilities shows global equities dominate the overall risk profile, followed by private equity, hedge funds and real estate. The assets with the smallest stand-alone volatility are core fixed income and high yield corporate bonds.
Comparing stand-alone volatilities of various assets in a portfolio is a good initial step, but in order to develop a better understanding of the true risk posture, correlations also need to be accounted for. The net volatility takes into account both the size of the stand-alone volatility and the correlation with the overall portfolio. However, while the net volatility shows how much a given position contributes to risk, it does not on its own give us a way to determine if a given allocation should be bigger or smaller. For that type of conclusion, we need the implied Sharpe ratio. The ISR, which is the ratio of net to stand-alone volatility times the portfolio-level Sharpe ratio, shows the Sharpe ratio that investors must be expecting in order to allocate the amount of risk they have to a given position.
Applying the ISR concept to portfolio construction
We will now illustrate with a few examples how the ISR measure can be used as a metric for assessing portfolio risk and ensuring alignment with investment views.
First, let’s look at the stand-alone volatility of real estate (102 bps) relative to hedge funds (103 bps), as shown in Figure 3. These assets have almost identical stand-alone volatility, but real estate (approximated through REITs) has a higher net volatility contribution (90 bps) than hedge funds (68 bps) because REITs are more correlated with the rest of the portfolio, equities in particular. Since the investor allocated the same level of stand-alone risk to an asset class (REITs) that was more correlated with the rest of the portfolio, the investor must expect that asset (REITs) to deliver a higher Sharpe ratio.
Second, the investor has sized the positions such that hedge funds have a higher net volatility contribution (68 bps) versus high yield (33 bps). This may lead some to prematurely conclude that this investor has a higher conviction in hedge funds because they have more risk allocated to them. However, as shown in Figure 3, the ISR of 0.40 is higher for high yield bonds even though they have lower stand-alone and net volatility contributions than hedge funds. The reason for that is high yield bonds are highly correlated to other asset classes in the portfolio, in particular, public and private equity. Therefore, by allocating to them, the investor is loading up on a common risk factor.
From a portfolio construction perspective, an allocation to a highly correlated asset is justified only if the level of conviction is extremely high. Hedge funds are a relatively more diversifying asset and thus have a lower ISR of 0.33. In other words, the bar for allocating to them, and the expected Sharpe ratio they will need to deliver, is much lower. This is why we believe investors need to look beyond stand-alone volatility and net volatility contributions; it is the ratio of these two numbers – the ISR – that matters the most in our view.
Let’s put the ISR measure to further use. Notice that ISR of core fixed income in Figure 3 is slightly negative (−0.03). Bonds tend to be negatively correlated with public equities in most market environments, so they have a small negative net volatility contribution to the overall portfolio. The slightly negative ISR means that investors would have allocated this much risk to core bonds even if they expected core bonds to have slightly negative excess returns.
If the investor believes that core bonds are richly valued and that their Sharpe ratios are going to be very negative, then he or she should allocate an even smaller amount to them. On the contrary, if core bonds are estimated to have a positive Sharpe ratio, then they should get a higher allocation, even if that requires using some leverage. The reason is that despite having allocated 15% of market value of the portfolio to core bonds, this allocation is still offsetting the estimated risk in the much larger allocation to equities. The bond allocation is offsetting enough equity risk that it would appear reasonable to allocate to it even if one expected zero returns from core fixed income.^{9}
Let’s look at another example. The ISR of 0.48 for global equities suggests the investor expects them to have the highest Sharpe ratio among assets in the portfolio, and the portfolio is expected to have a high correlation to global equities. Interestingly, asset classes that tend to be highly correlated with equities, like private equity, high yield and real estate, usually also have high ISRs even though their stand-alone volatility is much lower. This should also make intuitive sense as highly correlated assets typically should have similar Sharpe ratios (if not, there should be a great relative value opportunity!).
Coming back full circle, the main question to focus on is whether the investor really believes the Sharpe ratio for core fixed income (with its ISR close to zero) is so much lower than equities (ISR close to 0.5). If that is indeed the investor’s view, then there is nothing further to do in order to align the portfolio’s positioning with that view. If not, the asset allocation needs to be altered so positioning is aligned with the investor’s views.
ISR measure at PIMCO
We use the ISR measure extensively when managing portfolios and sizing positions. We look at ISR both at the individual trade level and at the broad factor or asset class level. When we see a negative ISR we habitually ask, “Should we increase the size of this trade given its diversifying properties?” When ISRs are higher than 0.6–0.7 for multiple positions, we see this as a red flag for potentially insufficient portfolio risk diversification.
In conclusion, the ISR measure is able to encapsulate in a single number an investor’s view of a particular asset, risk factor or trade, and it can be estimated without having any knowledge of the investor’s views on valuations. It has greatly improved the rigor and efficiency of our internal debates on position sizing and portfolio construction, and we believe it is the lens through which investors should approach portfolio construction.
The authors offer special thanks to Masoud Sharif, PIMCO’s head of portfolio analytics, who has led the effort to refine and implement the ideas expressed in this paper to PIMCO’s portfolios.
^{1} In this paper, by excess return, we refer to the total return in
excess of cash equivalent (“risk free”) rates.
^{2} See for instance William F. Sharpe, “Imputing Expected
Security Returns from Portfolio Composition,” Journal of Financial and
Quantitative Analysis, Vol. 9, No. 3 (June 1974), pp. 463–472, and
Fischer Black and Robert Litterman, “Asset Allocation: Combining
Investor Views with Market Equilibrium,” Journal of Fixed Income, Vol.
1, No. 2 (September 1991), pp. 7–18.
^{3} For details on long-term market assumptions, please see the PIMCO
Quantitative Research and Analytics paper, “The Stock‑Bond
Correlation,” by Nic Johnson, Vasant Naik, Niels Pedersen and Steve
Sapra (October 2013).
^{4} For instance, this would be consistent with bonds and equities
having ex ante Sharpe ratios of 0.05 and 0.3, respectively.
^{5} This framework requires several assumptions, including a viable
risk model and a mean-variance-efficient allocation without constraints.
^{6} Estimating the ex ante correlation relies on a risk model that
takes into account the horizon of investment, serial correlation in return of
illiquid assets, and (joint) dynamics of asset returns.
^{7} We should
point out that practical constraints (e.g., liquidity, drawdown) could also
lead to differences of alignment between ISR and investors’ views.
^{8} If an investor has a different ex ante portfolio Sharpe ratio,
then all of the ISRs can be re-scaled accordingly. It is the relative levels
of the ISRs that are relevant.
^{9} There are other interpretations as well. If an investor believes
the correlation of fixed income and equities will be positive going forward,
then the correlation matrix can be adjusted, and an ISR under this assumption
can be calculated and compared to an investor’s estimated Sharpe ratio
going forward.