Equity-Bonds Equilibrium and Inflation Uncertainties François Oustry, May 3rd, 2015 Bonds-Equity correlation is positive again! The correlation between Bonds and Equity returns is moving fast these days: in a few weeks it has moved from -70% in February to +12% this week.

Is this move into positive territory abnormal? What are the factors driving this change of sign? Does it mean that investors should change their Bonds-Equity allocation? Is that a warning signal for the fixed-income products? Back to the pre-2000 regime? A positive correlation between the returns of bonds prices and equity prices means a negative correlation between bonds yields and equity returns; this inverse relationship between equity prices variations and yields had been actually prevailing before 2000. As explained by Ranking & Shah Idil, in their paper “A Century of Stocks-Bonds Correlations”, Royal Bank of Australia, 2014:

The recent [last 14 years] period of positive correlations has been commonly

ascribed to the emergence of a ‘risk-on, risk-off’ paradigm, whereby US Treasuries and equities have served as proxies for ‘safe-haven’ and ‘risk’ assets, respectively, and broad shifts in risk sentiment have had an unusually large role in determining asset price movements. However, it is notable that the stock-bond yield correlation had already been positive for most of the decade before the global financial crisis. Fundamental Drivers of Stock Prices and Bond Yields Again, as described by the two economists of the Australian Central Bank: Yields on longer-term government bonds are determined by the expected path of the risk-free rate (over the life of the bond), plus a term premium that compensates investors for uncertainty about potential future changes in the value of the bond stemming from changes in real interest rates and/or inflation. A firm’s stock price is determined by the present value of expected future dividend payments, with future payments discounted by the expected path of the risk-free rate and an equity risk premium (the additional return over the risk-free rate that investors require in order to hold riskier stocks). At a more fundamental level, these variables reflect expectations for, and uncertainty about, growth and inflation. In particular, changes in investors’ central expectations for growth and inflation determine their forecasts for dividends and interest rates; stronger economic growth and higher inflation increase interest rates (via actual or expected future policy tightening) while also raising dividends via increased corporate profits. The extent to which there is uncertainty about these variables influences stock prices and bond yields via the equity risk and term premia. An increase in uncertainty about growth raises the equity risk premium while plausibly. Uncertainty on rates and inflation are clearly growing these days: The “core” inflation rate, which strips out energy and food prices, unexpectedly ticked back up to 1.8 per cent in March, and many economists expect wage growth to pick up further in the coming year as the US labour market becomes tighter. As a result, some investors are reappraising the outlook for prices and shifting some of their bets on interest rate policy. Ahead of the US central bank meeting on Wednesday, daily deposits into global inflation-protected bond funds monitored by EPFR Global indicate that inflows have hit a weekly record, with US funds dominating investor interest. Financial Times, April 27th 2015 - http://ow.ly/3xOFlX To what extent this uncertainty on rates and inflation are impacting the Equity-Bonds Equilibrium? The Minimum Variance Equity-Bonds Equilibrium Let us consider an investor willing to find the best equity-bond allocation to face uncertainties in this period of rapid changes. What would be that allocation and how stable is this optimal decision in the current changing environment? Let α∈[0,1] be the proportion of the capital to be allocated on Bonds and Risk (α)=Variance (α R B+(1−α) R E ) , where R B and R E are random variables representing respectively the returns on Bonds and Equity for the investor. Introducing Γ the covariance matrix between Bonds and Equity returns, the ex-ante Risk function becomes: Risk (α)=(α ,1−α)Γ (α ,1−α)T . Introducing σ B , σ E and ρ , respectively the volatility of Bonds, Equity and the Equity-Bonds correlation, we can expand the quadratic form Risk (α) into:

Risk (α)=σ E σ B [(κ+κ−1−2 ρ)α2+2(ρ−κ)α+κ] , where κ=σ E /σ B is the Equity/Bonds volatility (or uncertainty) ratio. We note that as long as σ E >σ B , we also have (σ E −σ B )2=σ 2E +σ 2B −2σ E σ B >0 . Together with ρ⩽1 , we obtain the strict positivity of the curvature parameter (κ+κ−1−2ρ)>0 as well as the strict convexity of the quadratic risk function α → Risk (α) . Optimal allocation is therefore determined by the first order optimality condition: ∂ Risk (α) =2 σ E σ B [(κ+κ−1−2 ρ)α+(ρ−κ)]=0 , ∂α which leads to an optimal allocation on bonds and equity: κ−1−ρ κ−ρ and ᾱE =1− ᾱB= . ᾱB= κ+κ−1−2ρ κ+κ−1−2ρ For a fixed correlation ρ=−0.5 , a decreasing κ=σ E /σ B , i.e a situation where uncertainty on bonds is growing faster than the uncertainty on equity, would lead to a fast decrease on the bonds:

The local Risk-implied Equity-Bonds Correlation The uncertainties on bonds are currently growing faster than uncertainties on equities: the ration of the 60-days volatilities κ=σ E /σ B has recently dropped from 2.75 in October 2014 to 1.62 on April 29th, 2015. The iShares 7-10 Year Treasury Bond ETF is used as a proxy for the bonds market; the

Yet during the same time the realized 60-days correlation ρ E / B is compensating this decrease so that the Equity-Bonds equilibrium is varying much slower. Whereas the volatilities ratio has experienced a 20% drop during the last three months, the correlation has move at a rapid pace to stabilize the Equity-Bonds local equilibrium around a 30-70 position:

For

α>0.5 , the Risk-implied or Equilibrium Correlation is given by the formula: (1−α)κ−α κ−1 . ρ E/ B= 1−2 α

Interpretation of the Minimum Variance Equity-Bonds Equilibrium: early signal on fixed-income risk Does our simplified 2 assets allocation model imply that investors are in average holding a 30-70 Equity-Bonds portfolio? Of course not, there exist a strong diversity of portfolios depending on investors various investment horizons and associated expected returns. Yet an interesting fact can be observed: each time uncertainties on inflation are growing fast enough to produce a very steep slope on ρ E/ B , some outflows in fixed-income funds can be observed.

This Equity-Bonds correlation can be used as a complimentary estimation of the Breakeven Inflation rate

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