In Part 1, I described how to calculate monthly portfolio returns net of on-going contributions. In this part, I will compare my returns to benchmark indices. Some readers may find this post overly technical but I sincerely hope that still some finds it useful. There are a couple of things before we proceed:

- I’m using my own returns as an example to demonstrate the mechanics of calculating risk-adjusted returns. The numbers are decent, but not great. If you think they’re high, I didn’t mean to gloat. If you think they’re low, well, that’s why I’m still learning.
- I understand that many readers will not have the time or desire to do a similarly detailed analysis; however, for someone interested in getting a professional money manager, seeing how the performance metrics are derived may be of value.
- My portfolio is divided into an asset allocation (AA) portion and an actively managed (AM) portion. Each is calculated separately as well as the combined total. The two portions were roughly equal for most of 2006.
- My own allocation in the AA portion, as described in the series on asset allocation, was 30% domestic stocks + 30% foreign (24% developed + 6% emerging) + 10% commodities including PM and 30% fixed income.
- Tracking started on Jan. 11, 2006, the inception date of this blog. That’s why the numbers for the benchmark indices do not agree with what you read elsewhere. The stock market did very well in the first 10 days of 2006: the S&P gained around 3.5% and my portfolios gained 4-4.5%. Adding 4% to my cumulative numbers gets you 20.6% for the AM accounts, 16.59% for the AA accounts and 19.11% combined for the whole of 2006.

Most data is contained in the big table below. The first section lists monthly gains for the AM, AA and total accounts. To the right are monthly numbers for index ETFs that I use as proxies to the benchmark indices, such as VTI (Wilshire 5000), SPY (S&P 500), IWM (Russell 2000), etc. EFA (Foreign Developed), EEM (Emerging Markets) and AGG (Lehman total bond index) were also tracked. All these numbers were adjusted for dividend payouts. Finally, there are three columns representing three very basic asset allocations: 70% VTI + 30% AGG (AA1), 35% SPY + 35% IWM + 30% AGG (AA2), and 20% SPY + 20% IWM + 24% EFA + 6% EEM + 30% AGG (AA3).

The returns are plotted below for a more graphical presentation. I’m using SPY as a proxy for S&P 500. From Jan 11 to year end, SPY returned 11.4% which I beat handily, but the ups and downs (volatility) in my accounts also stands out.

**Alpha and beta**

Regular viewers of CNBC can attest that alpha and beta have become a fixed part of the vocabulary for the financial “in” crowd. They came from Markowitz’s Capital Asset Pricing Model (CAMP). Essentially, they describe the relative performance of a given portfolio with respect to an index (usually the S&P) with a linear approximation. To find my alpha and beta, I plot my monthly returns vs. that of SPY using a scatter plot in Excel (see chart below). Then it’s a just click of a button to do the linear regression.

Alpha is simply the y-intercept of the line. According to CAMP, it represents of the manager’s skill. A positive alpha means the manager is providing extra return beyond the risk he takes.

Beta is the slope of the line. According to CAMP, beta represents the risk in the portfolio. The market by definition has a beta of 1. A positive beta less than 1 means less risk than the market. A positive beta greater than 1 means greater risk; hence “high beta” is synonymous with “high risk” (in a period of market advance high beta stocks should outperform on average). A negative beta refers to a negative correlation with the market.

Now let’s take a look at the red line which corresponds to my AA accounts. It has a slope of 0.8 which means my AA accounts bore only 80% of the risk of S&P. The alpha is 0.0027 or 0.27% per month. That is how much extra return my asset allocation has provided. I’m happy with the way it worked out since my AA accounts include some high fee mutual funds in my 401(k) plan. The parameter R^{2} describes how well the line fits the scatter points. A value of 0.5 is actually fairly low. It means my return is poorly correlated with the market (mostly due to the commodity allocation). CAMP will also say that 50% of my return is due to luck.

My AM accounts (black line) tell a completely different story. Firstly, the beta is negative. This is not surprising as I was highly concentrated in PM and energy stocks. I also timed the market and had significant short exposure such as in September. I still managed an alpha of 1.58% but the R^{2} is so low as to make the whole thing meaningless.

The combined accounts (green line) naturally lie somewhere in between. By adjusting the relative weighting of AA and AM I can make the green line flat with a positive alpha. If I could make increase the R^{2}, it would mean that I can reliably generate a positive return irrespective of market conditions. Oh, if only that were true!

**Sharpe Ratio**

As you can see, alpha and beta are not perfect descriptors of portfolio performance especially when the correlation with the market is poor. There are other ways to describe volatility such as max draw-down (MaxDD) which is simply the worst performance in a given period. My worst month in AM was -7.96% in September when I was long PMs and short the market and suffered a double whammy. In AA, it was -4.06% in May. 20% of my wife’s 401(k) was in her company stock that took a swoon then. In comparison, MaxDD in SPY was only -3% in May.

The most recognized metric for risk adjusted return is, however, the Sharpe ratio (SR), named after William Sharpe, co-recipient of the Nobel prize in Economics with Markowitz. It is defined as

SR = (R – RWhere R is the average return; R_{f})/StdDev(R)

_{f}is the risk-free rate; and StdDev(R) is the standard deviation of R. The numerator is the actual return above the risk-free rate, also called the excess return. The denominator is the standard deviation of the return, the technical measure of volatility. The SR, therefore, gives the return per unit of risk. There is a nice collection of web-based calculators including detailed instructions for this calculation at Sharpe's website.

I have seen people using 0 for the risk-free rate (cash not earning interest). More common is to use the short term treasury rate. My Sharpe ratios are shown in the bottom part of the table above. The risk-free rate was assumed to be either 0 or a constant rate corresponding to 4.5% per annum.

Although my absolute returns were respectable, on a risk-adjust basis they fell short. Looking at the annualized Sharpe ratio with a finite risk-free rate (the last line), EFA at 1.42 was by far the best. My AA accounts (1.12) were better than SPY or VTI (tied at 1.05), but very slightly worse than the simple allocation AA3 (1.13). For all my meddling in my AM accounts (0.70), I achieved a high return but at the expense of even higher volatility. The Sharpe ratio also shows EEM to be fairly poor on a risk-adjusted basis despite having the best cumulative return.

**Looking Ahead**

It’s clear from this exercise that my asset allocation plan worked well. I rebalance my allocation at the end of last year. The overall assignment remains the same but I overweighed domestic large caps and added some REITs despite my misgivings about the residential real estate market. I also re-jigged my AM/AA balance to favor AA by 10% or so. This exercise doesn’t completely invalidate my work in AM accounts as the absolute return was still greater. Rather it points to a need for better money management so as not to repeat the disaster in September. I’m hoping 2007 to be an even better year.