Financial Mathematics Text

Thursday, December 19, 2013

Stock Valuation and Anchoring: AF2P Contest Results

Thanks to everyone that participated in my little study. I'm going to outline what questions I wanted to answer with this study. Hopefully, you will find this information useful.


As investors, we're trying to find good companies trading at a good price. But how do we assess what to pay? Obviously there is a good deal we don't know; we must make decisions under conditions of uncertainty.

In Priceless: The Myth of Fair Value (and How to Take Advantage of it), William Poundstone reviews the literature on how people ultimately settle on what is a fair price for consumer goods, salary agreements, real estate and more. All sorts of factors can influence one's ultimate assessment of what something is worth. Even seemingly unrelated numbers can actually inform your judgement.

This latter aspect in psychology is referred to as Anchoring. Per the Wikipedia article, anchoring is "a cognitive bias that describes the common human tendency to rely too heavily on the first piece of information offered (the "anchor") when making decisions."

So that leads us to consider what a stock is actually worth. After all, the prices of stocks are updated in fractions of a second. We can sit at our computers and watch as the price bounces around as each transaction takes place. To what extent does the current stock price actually influence our judgement as investors?

If our goal is to assess the fair value of a stock, how can we do so without being influenced by that price?

The Study

A few participants actually already figured out what this was all about. I wanted to answer mainly two main questions:

1) To what extent does the current market price influence an investor's judgement as to what it's worth?

2) To what extent can we overcome anchoring bias in the context of valuing a stock?

To study this, I used what Geoff Gannon calls a blind valuation. In a blind valuation, you're given a limited amount of information from a company's financial statements and asked to value the stock from that limited information.

Participants in the study were then randomly assigned to one of three groups:

Group A: This group only received the financial figures for the stock. This group received only the black text (see below). If your contest entry number (the 9 digit number) began with 1, 2 or 3, you were placed in this group.

Group B: This group received the same financial figures as Group A but were also told a "current stock price" which just happened to be a number I made up. This group received the black and red text. If your contest entry number began with 4, 5 or 6, you were placed in this group.

Group C: This group received the same information as Group A but were also instructed to attempt to overcome anchoring bias. This group received the black, red and blue text. If your contest entry number began with a 7, 8 or 9, you were placed in this group.

Here's what it looked like:
Your job will be to value the following stock (financial statements below). You will estimate the value per share assuming a 12% cost of equity (discount rate or rate of return). This means that the price you estimate at should be sufficient to give you a 12% annualized rate of return.

As a tie breaker, you will be asked to estimate the next three years of earnings per share (EPS).

The stock is currently trading at:

$87.42 per share

The current share price serves as a psychological anchor. It will influence your judgement about the value of the stock.

An anchor is something, often a number, which has an effect on your judgement. Research shows that even entirely random numbers can influence one's estimate of something uncertain.

Part of your goal is to value the stock independently from the current stock price. Hopefully you can overcome the bias to prevent the $87.42 price tag from influencing your final judgement.

Here are the Financial Statements: 

Note: Year 'T' corresponds to the most recent annual statement. 'T-1' and 'T-2' are the previous two annual statements.

Income StatementTT-1T-2
Net Sales$107,714$107,257$112,883
Gross Profit$49,915$47,671$46,319
Operating Income$18,745$17,373$15,974
Nonoperating Income$1,740$3,514$2,925
Interest Expense$1,388$2,068$1,533
One-time Writedown$4,989$0$2,750
Net Income$7,709$12,041$8,617
Shares Outstanding6,9956,9957,922

Balance SheetTT-1T-2
Cash and equivalents$8,819$7,969$4,094
Accounts Receivable$13,317$13,253$12,412
Other Current Assets$2,631$1,492$1,568
Total Current Assets$48,935$43,119$37,974
Plants Property & Equipment$28,357$25,233$23,125
Accumulated Depreciation$13,404$11,810$10,814
Net PPE$14,952$13,423$12,311
Other Noncurrent Assets$21,195$27,064$27,405
Total Assets$85,082$83,606$77,690
Accounts Payable$12,324$11,549$11,077
Current portion of LT Debt$69$2,153$66
Other Current Liabilities$4,193$3,322$2,883
Total Current Liabilities$16,585$17,023$14,026
Long-term Debt$9,540$9,607$15,914
Other noncurrent Liabilities$9,348$10,670$9,885
Total Liabilities$35,474$37,299$39,825
Total Equity$49,608$46,307$37,865
Common Equity$48,627$45,325$36,883
Book Equity Per Share$6.95$6.48$4.66

My Hypotheses

I had three working hypotheses going into this.

Hypothesis #1: There would be an anchoring bias in stock estimation. In other words, Group B's estimates would be closer to the anchor than Group A's.

Hypothesis #2: Being aware of anchoring bias can allow one to partially compensate for anchoring bias. In other words, Group C's estimates will be further away from Group B's estimates. I was considering it an open question as to whether or not one could completely overcome the bias (e.g. will Group A's estimates be different from Group C's estimates?).

Hypothesis #3: Investing experience reduces the effects of anchoring bias. This would amount to those with less investing experience giving estimates closer to the anchor than those with more investing experience.

The Winning Answers

So what was the "right" answer?

The financials are actually adapted from Brown-Forman (BF.B). The Annual Report was the 1990 report which you can download here. The actual figures (above) were adapted from the years 1988, 1989 and 1990. I assumed that you purchased in August 1990 (about the time you would have read the annual report) and then held for 20 years (I like even numbers) until August 2010.

Discounting all of the (adjusted) dividends and the final price at a rate of 12% came up with a value of \$26.11.

Since I wanted this to be a fair game, I gave out three prizes. You were only in competition with those within your group. Here were the winning estimates for each group:

Brown-Forman was actually selling for about \$23.25 (adjusted) at the time. So you could have actually bought it for about 11% discount to "fair value".

The Results

So now onto the results of the actual study.1

There were 183 entries to the contest. Unfortunately, only 52 actually completed the whole process. 

We'll start by taking a look at a box plot for the three groups.

Anchoring Bias in Stock Valuation

One striking feature is how much lower Group A is compared with Group B. The third quartile (top of the purple box) of Group A is actually lower than the median values (top of the red box) of both Group B and Group C.

Here's a quick summary of some figures:

Anchoring Bias in Stock Valuation

Methodological Problems

So there are two main problems with the data. For starters, the sample sizes were not as large as I had hoped. I was hoping for at least 30 in each group.

The second problem is that the data is not normally distributed.2 This eliminates the possibility of using all of the techniques taught in basic statistics courses. Fortunately, there are some nonparametric options.

Testing the Hypotheses

So let's take a look at my three hypotheses and see how they did.

Hypothesis #1: Was there anchoring bias?

The answer is clearly yes. Group A and Group B were significantly different3 indicating an anchoring bias.

Hypothesis #2: Can knowledge of anchoring help overcome an anchoring bias?

I'm very sad to say that I cannot answer that with a "yes". I was actually quite hopeful that this would be the case.

Group B and Group C were not significantly different.4 In fact, Group C had a mean and median closer to the anchor than Group B. Perhaps the issue is a matter of small sample size.

The fact that the third quartile was much lower in Group C than in Group B may leave us with some hope.  Further experiments would have to be done.

Hypothesis #3: Can experience help one overcome an anchoring bias?

No significant difference was found.5 But that may just be due to the small sampling size. I enlarged the sample by combining Groups B & C to see if those with more experience were less influenced by anchoring bias. Here are the median estimates for each group:

So overall, only one of my hypotheses are supported by the evidence. The other two were simply not supported. Perhaps further research may give some evidential support for these hypotheses. But as it stands now, we'd have to conclude that there is no evidence that we can compensate anchoring bias.

One reason to think this hypothesis may turn out valid is research in real estate. In Priceless: The Myth of Fair Value (and How to Take Advantage of It), William Poundstone notes that in real estate, "the pros were less influenced by the fake listing prices." So perhaps there is still hope we may find evidence that those with investing experience may be able to compensate for anchoring bias.


Earnings Per Share (EPS) Estimates

While I mainly used this to pick a winner in the event of a tie, there may be something interesting with this.

Here are the median estimates for each group:

No significant difference was found.6

This may actually make sense in the context of what the anchor was in this case. In terms of valuing the stock, the "current" stock price was the relevant anchor. In the case of the EPS estimates, the relevant anchor was the last 3 years of EPS data. In this sense, all three groups were given the same anchor.

If you're curious, actual (adjusted) earnings for the years 1991, 1992 and 1993 were  \$1.74, \$1.76 and \$1.88, respectively.

Different Strokes for Different Folks

Some people left some details on how they approached the valuation of the stock. It was interesting to see some of the differences and similarities between valuation techniques.

Many noticed that this was a high return on capital business and so there was some question as to how much more one should be willing to pay for that (depending on whether or not they could continue compounding at a high rate of return.)

Others questioned to what extent the "one-time" charges were really one-time charges. Many normalized earnings to reflect that they may be one-time chargers.

Many found the lack of information to be troublesome. Some of that information would have been important for your valuation techniques. Fortunately, in a real world example, you'd be able to accumulate more detailed information about the company, its products, its industry7 and the like.

Concluding Remarks

We found clear evidence of anchoring bias in stock valuation. Unfortunately, we cannot say there is any evidence that one can compensate for this.

My hope is that the methodologies tried out here will be adapted for further research into this area as I think it's important information for investors to know.

1 All calculations were performed in Microsoft Excel along with Real Statistics add-in.
2 The Shapiro-Wilk Test for Groups A, B and C had p-values of 0.21%, 8.58% and 0.03% respectively.
3 The Wilcoxon rank-sum 1-tailed test was performed with p-value of 4.21%.
4 The Wilcoxon rank-sum 1-tailed test was performed with p-value of 35.11%.
5 All three groups (0-2 years, 2-5 years, 5+ years experience) were tested against each other via Wilcoxon rank-sum test. All had p-values in excess of 40%.
6 All three groups (A, B and C) were tested against each other via Wilcoxon rank-sum test. All had p-values in excess of 16.5%.
7 I've often wondered if one can figure out an industry based on the financial figures. Perhaps some sort of Bayesian analysis could be performed in which, given the financial figures one could estimate how likely that company is in a particular industry. Someone please do this research.


  1. Even thought the company have high return on capital, there is no growth in sales despite increasing capex every year. I am assuming the capex was invested to improve margin (GP margin) but improving GP margin can only go so far. Note the GP margin is already ~50% at year T which is relatively high for any industry. Any thoughts anyone?

    1. @Hin

      I think that's a fair concern. I agree that growing via improving margins isn't going to work for very long.

  2. I forget my specific value, but I think it was $12-13. My method was rather simple and did not include any growth - it was just an average of the actual results even with the one-time charges. Without the qualitative information, I would find it hard to do anything else.

    The one item I'm curious about what others did was how they handled growth. If I had factored in some growth, the valuation could be changed significantly. The challenge was that I didn't know if we were looking at BlockBuster, Coca Cola, or whatever. (Well... with Returns on capital and assets I could figure this out somewhat, but the point is that I had no idea if the business would be doing alright or not a few years later)

  3. The difficult part with these stock valuations always seems to come down to growth.
    From what I've seen so far, a lot of people actually have similar fundamental values, but it's the growth part that really makes the final values different.

    Thanks for this "Study" though.

  4. @Ankit Gupta and Jae,

    Yeah, I don't know how precise you can get with growth estimates. And since this was just one stock, there's a lot of luck involved. It seems to me you could guess a wide range of estimates and they'd all be decent estimates on some level.


Some common OpenID URLs (no change to URL required):