I'm using Idzorek's version of the Black-Litterman model for estimating asset returns. Idzorek's version bypasses the need to estimate directly the covariance matrix $\Omega$ of errors in the various views by allowing an investor to specify confidence levels for each view. The entries in $\Omega$ can then be calculated from those confidence levels.

The confidence $C_k$ in view $k$ is expressed as a percentage between $0\%$ and $100\%$. Thus, for example, an investor can say that she is $50\%$ confident that international bonds will outperform US bonds by 25 basis points. This way of indirectly obtaining $\Omega$ has the advantage of being intuitive for the individual investor, but it's also fairly subjective, as the value of $C_k$ is really just the investor's opinion.

What are some more quantitative ways to obtain these confidences $C_k$?


1 Answer 1


The primary alternative to Bayesian subjective probabilities is the frequentist approach. This would involve measuring the % of times where international bonds outperformed US bonds by 25 bps over the relevant period in market history and using that as your confidence level.

A quantitative view in-between the Bayesian and frequentist approaches would be a regression model that forecasts the spread on international bonds vs. US treasuries using some macro factors (e.g. interest rate differentials, yield curve level and twist differences, growth forecasts, balance of payments data, monetary policy stance dummy variables, currency swap rates, etc.). Once you have the forecasted spread you can take a z-score of the predicted value and measure the probability of the forecast being above 25 bps given the standard error of the forecast.

Alternatively you could create a binary logistic model or a probit model that cranks out a likelihood of the spread being greater than 25 bps. In terms of speed and simplicity I would opt for the logistic model as a starting point as you won't have to be as concerned about serial correlation and heteroskedasticity.

Sometimes fixed-income spreads have long-term mean reverting levels or exhibit patterns that lend themselves to time-series analysis. For example, check out this chart of EMEA bond-spreads over the US treasury spot curve. There appears to be a long-run avg spread in the 3-4% range (an auto-regressive effect) and a shock (moving average) effect. An ARIMA model might help you predict - perhaps directionally - the spread in the next period(s). This is the easiest method to test but the performance will be lousy the more periods in the future you project and might not do better than a random walk.

A more (perhaps overly) complex approach would be to develop a Markov regime switching model. The regimes might correspond to your outcomes of interest, or to states of the world that drive your outcomes of interest (i.e. global recession/flight to safety vs. growth/concern for inflation). Regime switching models return the probability of various states being true, as well as the transition probabilities amongst these states. The sum of the probabilities of the states that are favorable to your 25bps spread can be treated as a confidence.

Another technique would be to define some assumptions around the volatility of interest rates and changes in spreads and perform a monte-carlo simulation to measure the % of times the international bonds out-performed by 25 bps. You can calibrate the Monte Carlo assumptions based on your own assumptions or use historical distributions. This approach is difficult unless you have access to international and domestic interest rate volatility models.

  • $\begingroup$ No prob - just threw in a time-series model idea as well. Good luck! $\endgroup$ Mar 8, 2012 at 5:14

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