Principles of Investments Lecture NotesSet 4: Stock Valuation

IntroductionThe purpose of understanding how the value of stock is determined is be able to identify whether a stock, at its current price, is undervalued or overvalued.

If a stock is undervalued, this means the current price is less than the “long-run” price, and we would expect the price to rise. Hence it would be good to buy the stock now.

If a stock is overvalued, this means the current price is more than the “long-run” price, and we would expect the price to fall. Hence it would be good to sell the stock now.

In the discussion below we will consider two related methods of stock valuation that are based on fundamentals of the company (such as divident payments and earnings), and we will also consider deviations in the market price from that price implied by the fundamentals.

Present ValueDefinition of Present ValueAny discussion of the price of any asset must begin with the concept of present value.

Present Value is the current market value of an amount that is to be received in the future.

The market value for any good is simply the price people are willing to pay for the good. Hence, one can say, the

present value is the amount people are willing to pay today to receive an amount of money in the future.

For example, suppose someone may pay you 1100 in one year. How much would you pay today to receive 1100 in one year? The answer to that question is the present value of 1100 in one year.

How do we determine the present value? That is, what is the amount the you would pay today to receive some amount in the future? The answer to that question is as follows:

The amount you would pay today to receive some amount in the future (i.e. the present value) is simply the amount you would have to invest today at current interest rates to receive that amount in the future.

To understand this, consider the amount of 1100 paid in one year. If the interest rate is 10% you would have to invest 1000 today to receive 1100 in one year. So you would never pay more than 1000 to receive 1100 in one year. Of course you would like to pay

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less than 1000 today to receive 1100, but there is no reason for anyone to agree to that since the maximum you, or anyone else, would pay is 1000. Hence the market price (the amount people are willing to pay) for a promise of receiving 1100 in one year is 1000.

Finding Present ValueHow can we generally find this present value? That is, how can we find the amount that must be invested today, at current interest rates, to yield a particular amount in the future?

Let PV be the amount invested today, FV be the amount in the future, and let r be the interest rate. In this case, if PV is invested today at interest rate r, then the amount received in one year is

FV = PV + PV*r

The above equation simply says that the future amount is the sum of two terms. The first term represent that fact that the principle of the investment (PV) is returned to you. The second term is the interest payment to you. However, notice that if we factor out PV, we have

FV = PV(1+r)

So if PV = 100 and r = 5%, then in one year you have

FV = 100(1.05) = 105.

But recall, the goal is to find out how much must be invested today (PV) to yield a specific future amount (FV). Now, notice the formula FV = PV(1+r) can be solved for the present value, PV, to yield

PV= FV1+r

This is the most fundamental equation of present value, and all of finance, including investments and stock valuation is based on this formula. Hence it must be understood thoroughly. Let us consider some examples:

Example 1Using our above numbers, what is the present value of 105 received in one year, assuming the interest rate is 5%? The answer can be found from the PV formula above as

PV =1051.05

=100

As you can see we find the PV to be 100; the amount that would have to be invested today to yield 105 in one year.

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Example 2What is the present value of 1100 received in one year, assuming the interest rate is 10%? The answer can be found from the PV formula above as

PV =11001. 1

=1000

Example 3What is the present value of 5200 received in one year, assuming the interest rate is 4%? The answer can be found from the PV formula above as

PV =52001.04

=5000

Present Value for Amount Received Many Years in the FutureIn the above analysis we assumed that the future value was coming in one year. But what if it comes in 2 years? Or 3 years? Or n years?

It turns out this does not add a great deal of difficulty. The basic logic is the same. The present value of an amount received n years in the future is still just the amount that would have to be invested today to yield that amount in n years.

Again, suppose we invest PV today at the interest rate of r that applies over the next n years. How much would you have in n years?

After 1 year: FV =PV (1+r )

The amount you receive after one year is reinvested for another year. That is, PV(1+r) is invested in year 2.

After 2 years: FV =( PV (1+r )) (1+r )=PV (1+r )2

Similarly, the amount after two years, PV(1+r)2, is invested for another year.

After 3 years: FV =(PV (1+r )2)(1+r )=PV (1+r )3

In general,

After n years: FV=PV (1+r )n

In this case then, the present value of receiving FV in n years is simply

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PV= FV(1+r )n

We will now use this formula to determine the fundamental value of a stock.

Present Value of Dividends and Stock ValueNow we want to apply the above concept to a stock. A stock does not just pay one amount in the future, but it provides a “stream” of dividends paid at various dates in the future.

Let D0, D1, D2,…Dn be the dividends today to n periods in the future. Then we can say the present value of the stock is the sum of the present value of all dividends.

Let us express the present value of the dividends in the following table (where note the present value of D0 received today is just the amount D0 ):

Period0 1 2 … n

Dividend D0 D1 D2 Dn

PV of Dividend

D0 D1

1+rD1

(1+r )2

Dn

(1+r )n

Hence the value, or price, of the stock is simply the sum of all the present value of dividends. That is,

P=D0+D1

(1+r )+

D2

(1+r )2+D3

(1+r )3 +. ..+Dn

(1+r )n

Now notice this can be more succinctly written as

P=∑i=0

n Di

(1+r )i

This is the value of a stock based on the fundamentals of its future stream of dividends.

The Case of Constant Growing DividendsIn the real world we do not know the future dividends of a stock. So often times we assume dividends grow at a constant rate g. That is, we assume dividends follow the following pattern:

Dt=Dt−1(1+g )

Where 0<g<1.

This means D1=D0 (1+g ), D2=D1(1+g ), D3=D2 (1+g ), etc.

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However, given D2=D1(1+g ), we can substitute for D1 and write

D2=D0 (1+g )(1+g)=D0(1+g )2

In the same way, given D3=D2 (1+g ) we can substitute for D2 and write

D3=D0 (1+g )2(1+g )=D0 (1+g)3

Or, in general we have

Di=D0(1+g )i

If we then substitute into the above equation for the stock price we have

P=D0+D0(1+g )(1+r )

+D0 (1+g )2

(1+r )2 +D0(1+g )3

(1+r )3 +. . .+D0 (1+g)n

(1+r )n

Expressing this with the summation sign we then have

P=D0∑i=0

n (1+g )i

(1+r )i =D0∑i=0

n

( 1+g1+r )

i

,

Where we have factored out the D0 that is common to all terms.With this price equation, it says the fundamental value of the stock is given by the current dividend, the interest rate, and the growth rate of the stock.

Letting n go to infinity However, though the equation is expressed succinctly with the summation operator, it still would involve adding together many terms. It turns out that if we assume that n→∞ the pricing equation becomes much simpler (as long as g < r).

To understand this let us first introduce a mathematical principle. Consider a number x such that 0<x<1 . Now consider the following sum

J=1+x+x2+x3+ .. .

Where the sum goes on to infinity. It turns out that sum converges to the following

J= 11−x .

Now consider our pricing equation. We have

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P=D0+D0 ( 1+g1+r )+D0( 1+g

1+r )2+D0 (1+g

1+r )3+. ..

Which when we factor out D0 becomes

P=D0[1+( 1+g1+r )+( 1+g

1+r )2+(1+g

1+r )3+. ..]

Now if we assume g < r, then the ratio (1+g)/(1+r) < 1. Hence the mathemacial principle above implies the the price can be written as

P=D0[ 1

1−1+g1+r ]

This can be rearranged to yield our final pricing equation for the fundamental value of stocks

P=D0[ 1+rr−g ]

Thus rather than adding together so many terms, we simply need to know the current dividend, the growth rate and the interest rate. Then with one simple formula we can find the value of the stock.

Note that the textbook assumes the price of a stock depends not on current dividends but only future dividends. That is, it assumes the dividend D0 has already been paid. In that case we need to subtract the dividend D0 from our fundamental value equation. Hence we would have

P0=D 0[ 1+rr−g ]−D0

Putting under a common denominator we have

P0=D 0[ 1+rr−g ]−D0[ r−g

r−g ]

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P0=D0[ 1+r−r+gr−g ]

P0=D 0[ 1+gr−g ]

However, since D1=D0(1+g) we have

P0=D1

r−g

Which is the formula given in the textbook/

Using the Pricing FormulaThe above pricing equation tells what the “long-run” price of the stock should be based on fundamentals. If the actual market price of the stock differs from this long-run price, then we should either buy or sell the stock.

To express this more completely, let Pc be the current price of the stock.

If Pc<D 0[ 1+r

r−g ] then we say the stock is undervalued and would expect the price of the stock to rise over time. Hence you should buy the stock (i.e. take a long position).

If Pc>D0[ 1+r

r−g ] then we say the stock is overvalued and would expect the price of the stock to fall over time. Hence you should sell the stock (take a short position).

Criticisms of the Pricing MethodWhile the above method of pricing stocks is appealing since it is based on fundamentals (dividends) and ends with a very simple formula, there are a few criticisms of the model. I give the criticisms below. After each I will give a response to the criticism.

Criticism 1: Stock dividends do not grow at a constant rate. Hence the model is very inaccurate.

Response 1: This is true, but if you are looking at holding the stock for the long-run, then the long-run expected growth in dividends is a good approximation for determing the long-run fundamental value of a stock.

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Criticism 2: If compare stocks, what growth rate do we use? Certainly all stocks do not grow at the same rate.

Response 2: Again, this is correct. For this reason, when you compare stocks it is very important to choose a growth rate that makes sense for that stock. It could be a long-run average growth for a sector of the economy, or something based on a particular company, but one does have to be careful in choosing the growth rate, and some errors will exist.

Criticism 3: This pricing method does not take risk into consideration. Certainly more risky stocks will have a lower price than less risky stocks.

Response 3: Again, this is true. In principle, the model could be adjusted for risk factors. If we could measure how much risk exists in a stock (by the standard deviation of past returns for example), and we had a market measure of how much investors dislike risk, then we could calculate a “risk factor” in the price equation, and the price equation could become

P=D0[ 1+rr−g ]− λ

,Where λ is the risk factor. This basically says the price is the present value of expected stream of dividends, minus a risk factor.

Price-Earning Ratio and Stock ValuationAnother factor investors look at to determine over or undervalued stocks is the price-earnings ratio; or the P/E ratio. The P/E ratio is simply the price of a stock divided by the earnings per share per year. For example, suppose the price of a stock is 50 and the earnings per share is 10, then the P/E ratio is

P/ E=5010

=5

The way to interpret the P/E ratio is that it tells one the amount of money it takes to get 1RO of earnings from the stock. In the above example, it takes RO5 to get 1RO in earnings.

Now suppose there is a long-run market average P/E ratio; call it R̄ . And let any stock i P/E ratio be given by R

i. Then,

If Ri< R̄ , this means it costs less to get 1RO of earnings from stock i than it does from

the average stock. Hence this stock is undervalued and one should buy the stock.

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If Ri> R̄ , this means it costs more to get 1RO of earnings from stock i than it does from

the average stock. Hence this stock is overvalued and one should sell the stock.

Is the P/E Ratio a Good Method of Stock Valuation?In one sense it is better than discounted dividends because dividends do not tell the whole story about the performance of a stock. Investors also care about capital gains (i.e the increase in the price of the stock). Sometimes high earnings are used to pay out dividends, but other times high earnings are used to buy outstanding shares of stock, reducing the supply of stock on the market, driving up the price of a stock, and creating capital gains for shareholders. Hence earnings would capture either method of benefiting stockholders, whereas dividends do not.

For other reasons, there are problems with this measure. First, current earnings are not necessarily a good measure of future earnings. For companies in decline, or companies that are new and growing, the P/E ratio is not reliable. However for companies that are mature and stable, it is a reliable method of stock valuation. As with our discounted dividends model of stock valuation, one must be careful when applying it.

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More on Stock Valuation

We earlier saw the dividend growth model yielded a stock value given by

P=D 01+rr−g ,

Where D0 is the dividend is in the initial period, r is the discount rate, and g is the growth rate, which is assumed constant from now to infinity.This set of notes addresses two issues related to this method of stock valuation. First we consider how to operationalize this equation by discussing the estimate of both r and g. This is important as the stock value given will be shown to be sensitive to the values chosen. Second we address two common criticisms of the model; namely that dividends do not grow at a constant rate and that many stocks do not pay dividends.Operationalizing Stock Valuation Equation by Choice of r and gSensitivity of Stock Valuation to r and g Let us first consider that the above equation is sensitive to the choice of r and g. Suppose the initial stock value is 10. The table below calculates the stock value for different values of r and g.

Stock Value (P) r gP = 350 .05 .02P = 214 .07 .02P = 1050 .05 .04P = 357 .07 .04

As one can see, small changes in either r or g can cause large changes in the stock value. For instance, in comparing the first two rows, an increase in the discount rate from 5% to 7% (while g = 2%) causes the stock value to fall from 350 to 214, which is a fall of about 39%. Similarly, in comparing rows 1 and 3, an increase in the g from 2% to 4% causes the stock value to triple from 350 to 1050. Hence clearly, small change in either r or g can cause substantial differences in our estimation of the fundamental value of the stock, and thus correctly estimating r and g is critical to estimating the fundamental value.

Choice of the Discount RateThe discount rate, r, represents the required rate of return by the investor. It depends on the return required due to the investor not having access to his money; known as the time value of money. It also reflects the fact that the investor will have to be compensated for any risk he takes on due to the investment. The time value of money component is capture by a risk free rate of return in the market and the risk component is captured by a risk premium. Hence the discount rate (or required rate of return) is given by

r = risk free return + risk premiumThe risk free return is usually captured by the return on short-term government bonds. The risk premium is more difficult to understand. So we begin by writing the formula for the discount rate as

r = return on government bonds + Stock Beta*Stock Market Premium

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where we have substituted for the risk premium the term Stock Beta*Stock Market Premium. What is this? Let us begin with the second part; the stock market premium. If one invests in the stock market, they expose themselves to the risk in the overall market. Hence such investment requires a premium to be paid to investors. This premium can be estimated as the difference between the average returns on the stock market and the risk free return. That is,

Stock market premium = Average return in the stock market – risk free returnA stock beta represents the degree to which an individual’s stock return will change with the stock market. The higher the beta, the more sensitive to the overall stock market is that stock, and thus the higher the risk, and the lower the beta, the less sensitive is that stock to the overall stock market, and the lower the risk. Hence estimations of the appropriate r when determining stock valuation, must account for the stock’s beta to correctly incorporate the risk premium in the discount rate.

Choice of gThere are two ways to determine the choice of the growth rate of stocks; g. First, one may rely on historical growth rates of the stock, and second one can compute what is known as the sustainable growth rate. We will discuss each in turn.

Historical Growth Rates: Just as the name implies, the estimation of the growth rate is taken from looking at the history of dividends from the company. There are two methods of calculating historical growth rates; the geometric mean and the arithmetic mean.To understand these consider the following set of historical dividends:

Year Dividends2010 122011 142012 132013 172014 162015 20

The geometric average is based on the dividend growth equation Dt +n=D t (1+g )n. In this case, we know the dividend amounts and must solve for the implied growth rates. That is, we have

(1+g )n=D t+n/Dt

Which can be rearranged to get

1+g=( Dt+n

D t)

1n

Or

g=( Dt+n

Dt)

1n−1

With the above data, we have D2015=D2010 (1+g )5. Substituting in the dividend amounts we have 20=12 (1+g )5. Solving for g we have

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g= (20/12 )15−1

Org=0.1076=10.76 %

Notice with this method we only require the beginning dividend and the last period’s dividend, and we “force” a smooth growth rate between the two. So it is simple to compute as it requires less data.

The arithmetic average is based on averaging the annual growth rate of dividends. Hence on computes the annual growth rates for all years, and then find the average. Below we have the dividend table above, where we have added a column for annual growth rate.

Year Dividends Growth2010 122011 14 16.67%2012 13 -7.14%2013 17 30.77%2014 16 -5.88%2015 20 25.00%

AVERAGE 11.88%

Notice the arithmetic average is more than 1% higher than the geometric average. Hence the two methods can give significant differences in the growth rate.

Sustainable Growth Rate: Apart from the fact that the two methods of historical growth rates can be significantly different, their use also assumes the future growth rate is equal to the historic, but this may not be true. Hence we also have a method known as the sustainable growth rate, which measures the maximum sustainable dividend a company could pay out in the long-run, for a given payout ratio.

Recall the payout ratio is the percentage of earnings paid out as dividends. If not paid out in dividends then we say the company has retained earnings. Hence one can define the retention ratio as the percentage of earnings retained by the company. Thus

Retention Ratio = 1 – Payout Ratio

Hence if the payout ratio is 80%, then the retention ratio is 20%.

Given this, the sustainable growth rate is given bySustainable growth rate = ROE*Retention Ratio.

Where ROE is the return on equity. This measures the percentage return earned on the company’s assets. Return on equity is defined as

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ROE=Net IncomeEquity

Hence the sustainable growth rate says if a company’s ROE is 15%, and the retention ratio is 20% (i.e. 20% of earnings are retained), the sustainable growth rate of the company is 3%.

Addressing Criticisms of the Dividend Growth ModelThere are two primary criticisms of the dividend growth model. First is the criticism that dividends do not grow at a constant rate. This may be especially true for new companies. They may experience substantial differences in growth rates of dividends over time. The second criticism is that not all companies pay dividends, so the model would not be appropriate to determine their fundamental value. We address each criticism below.

Criticism 1: Non-constant growth of dividends; 2 stage growth model One way to allow for a non-constant growth rate of dividends is to imagine two stages of dividend growth. In stage 1, dividends grow at rate g1 and in stage 2 dividends grow at rate g2. Letting the current period be time 0, let time T be the period in which we switch from stage 1 to stage 2.

To find the present value of dividends for this stock let us find for each stage separately, and let us begin with stage 1.

Stage 1 DividendsThe strategy to follow to find the present value of dividends in stage 1 is as follows:

1. Find PV as if g1 holds for all time. Define this as V 0 , ∞

2. Find PV for g1 starting at T and going till infinity. Define this as V T ,∞

3. Thus the PV of dividends in Stage 1 = V 0 , ∞−V T ,∞

Step 1. The PV of dividends if the growth rate is g1 is constant for all time is just given

by our equation V 0 , ∞=D0 (1+r )

r−g.

Step 2: The PV of dividends starting at T and going to infinity is given by the same equation. However we will have to find the PV at time 0. This gives us

V T ,∞=1

(1+r )TDT (1+r )

r−g1

However, since Dt=Dt−n (1+g1 )n in stage 1, it must be DT=D0 (1+g1 )T

Substituting this into the above equation we have

V T ,∞=1

(1+r )TD 0 (1+g1 )T (1+r )

r−g1

Step 3: Now we find V 0 , ∞−V T ,∞.

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V 0 , ∞−V T ,∞=D0 (1+r )

r−g −1

(1+r )TD0 (1+g1 )T (1+r )

r−g1

Thus, this last equation is the present value of stage 1 dividends.

Stage 2 DividendsFinding stage 2 dividends is simpler. We simply find the PV of dividends from time T to infinity at the growth rate of g2 and find its present value at time 0.Using the expression for V T ,∞, but assuming a growth rate of g2 we have

V T ,∞=1

(1+r )TD 0 (1+g2 )T (1+r )

r−g2

Pricing Formula for Two Stage ModelAdding stage 1 and stage 2 dividends values together gives us the final pricing formula of the stock as

P=D0 (1+r )

r−g −1

(1+r )TD0 (1+g1 )T (1+r )

r−g1+

1(1+r )T

D 0 (1+g1 )T (1+r )r−g2

To understand this equation, the first term is the value of the stock if the growth rate is g1 forever. The second term is subtracting off the term that represents the fact that the stock does not grow at rate g1 after time T, and the third term accounts for the fact that it grows at rate g2 after time T.

NOTE: It would also be possible to have dividends not growing at a constant rate in stage 1. In this case the second stage is the same, but the first stage growth is just the discounted value of dividends in stage 1. Hence the pricing formula is simply

P=D 0+D1

1+r+

D2

(1+r )2+

D3

(1+r )3+…

DT

(1+r )T+ 1

(1+r )TDT (1+r )

r−g2

Criticism 2: Estimating Fundamental Values without Dividends; the Residual Income ModelResidual income refers to the surplus, or extra, value added to a company during a particular period above the required rate of return by the investor.Let r be the required rate of return and let B be the “book” value of the assets of the company. In other words this is the total market value of the company’s assets. It represents the equity in the firm. Given the required rate of return, then the required earnings per share (REPS) in period t is given by

REP St=Bt−1 r

That is, the required earnings per share is the required rate of return multiplied by last period’s book value.

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Residual Income (RI) is the surplus of earnings per share (EPS) over the required earnings per share. That is,

RI=EP St−REP St=EP S t−B t−1r

This residual income now takes the role of dividends in our equation. So the value of the stock is given by

P=B0+EP S1−B0r

1+r+

EP S2−B1 r1+r

+…

If we assume EPS grows at rate g, one can show

P=B0+E P S0(1+g)−B0r

r−g=

EP S1−B0 gr−g

.

Note that you are not required to derive this last equation. The important point is that the fundamental value of the stock can be derived even though there are no dividends. Hence the fact that some stocks do not pay dividends does not imply we cannot determine its fundamental value.

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Market Efficiency and Behavioral Finance

Market EfficiencyMarket efficiency is the idea that prices of stocks always reflect their long-run fundamental value. This result depends on the assumptions that

1. At least some investors correctly understand the fundamental value of assets2. Those investors have all relevant information to determine the fundamental value

and will buy/sell if the price is different from the fundamental value3. Those investors have sufficient funds invested that the price will change with their

behavior.

Based on the above, we assume all investors have some method of determining stock values that accurately reflect all future earnings from stocks. That is, there exists a known method of determining the long-run fundamental value of a stock. Let this long-run fundamental value of stock j be given by Vj. Hence stock j is undervalued when Pj < Vj, and is overvalued when Pj > Vj.

Now suppose the stock is undervalued. That is, Pj < Vj. In this case all investors will want to buy the stock. But since so many are buying the stock, the market demand rises and the price of the stock will increase. This will continue as long as Pj < Vj. But as the price of the stock rises we eventually will reach a place where Pj = Vj. At this point the stock is correctly valued, hence there is no reason to buy or sell the stock, and thus the price will not change.

Now suppose the stock is overvalued. That is, Pj > Vj. In this case all investors will want to sell the stock. But since so many are selling the stock, the market supply rises and the price of the stock will decrease. This will continue as long as Pj > Vj. But as the price of the stock falls we eventually will reach a place where Pj = Vj. At this point the stock is correctly valued, hence there is no reason to buy or sell the stock, and thus the price will not change.

So we see that regardless of whether we start from a position of undervalued or overvalued stock, we end up with the price correctly valued at Pj = Vj. Moreover, since stock exchanges involve auctions and are generally very active, such adjustments in prices can occur in a matter of minutes. What this means then is that the observed prices on stock markets accurately measure the long-run fundamental value of the stocks. This is known as Market Efficiency.

Price Changes and Market EfficiencyIf Market Efficiency is true then does this mean prices never change? No. Prices of stock will change, but if markets are efficient stock prices will only change when the fundamental value of a stock changes. When such “news” becomes available to the public, investors will realize the stock is incorrectly valued, and the price of the stock will adjust. Hence, if market efficiency is true, apart from news about the stock, there is no reason for the price of the stock to change.

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Implications of Market EfficiencyOne implication of Market Efficiency is that if you think you have found an overvalued or undervalued stock, this means that you think the market has incorrectly valued the stock. Which is to say, you believe all the other investors are wrong and you are right. While this may be true, it also should caution us from betting against the market too much.

Because of the above implication, if markets are efficient in general an investor cannot do better than the market. That is, market efficiency implies that on average you are better off just investing in the stock market rather than trying to find overvalued or undervalued stocks. Clearly then whether or not stock markets are efficient is very important for investors.

Behavioral FinanceRecall, market efficiency is the idea that prices of stocks always reflect their long-run fundamental value. This result depends on the assumptions that

1. At least some investors correctly understand the fundamental value of assets2. Those investors have all relevant information to determine the fundamental value

and will buy/sell if the price is different from the fundamental value3. Those investors have sufficient funds invested that the price will change with their

behavior.

Behavior finance challenges the first of these assumptions and suggests that it may be investors make cognitive errors when determining value. Chapter 8 of the textbook address behavior finance. Please read pages 241 – 249 to understand the types of cognitive errors investors may make.

However, also keep in mind that it is not sufficient to show that some people make errors, but that it must be the case that enough investors make errors so that the above assumptions do not hold.

Key Concepts1. Prospect Theory

An alternative theory to classical, rational economic decision making, which emphasizes, among other things, that investors tend to behave differently when they face prospective gains and losses.

Specific Types Frame Dependence… Whether a bet/investment is framed as a loss or gain

influences the decision.

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Scenario One. Suppose we give you $1,000. You have the following choice:A. You can receive another $500 for sure.B. You can flip a fair coin. If the coin-flip comes up heads, you get another $1,000, but if it comes up tails, you get nothing.Which would you pick?

Scenario Two. Suppose we give you $2,000. You have the following choice:A. You can lose $500 for sure.B. You can flip a fair coin. If the coin-flip comes up heads, you lose $1,000, but if it comes up tails, you lose nothing.

Which would you pick?If you look closely at the two scenarios, you will see that they are actually

identical. Youend up with $1,500 for sure if you pick option A, or else you end up with a 50-50

chanceof either $1,000 or $2,000 if you pick option B. So you should pick the same option in both scenarios.

About 85 percent of the people who are presented with the first scenario choose option A, and about 70 percent of the people who are presented with the second scenario choose option B.

Mental accounting and Loss Aversion…Basing decisions to buy or sell a stock off the purchase price, even though it is a sunk cost. Researchers have noticed people tend to hold stocks too long to avoid losses.

Playing with House Money… Being more risk taking with winnings than with money you started with.

2. Overconfidence…a typical investor exhibits behavior suggesting they are irrationally overconfident in their abilities. This is seen in

Trading Frequency Lack of Diversification (which leads to almost half of investors outperforming the

market, fueling overconfidence).

3. Not Understanding Randomness and Chance Events

Representativeness heuristic: Concluding that causal factors areat work behind random sequences.

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Consider the two following random flips of a coin. Do you see a pattern?

First 20: T T T H T T T H T T H H H T H H T H H HSecond 20: T H T H H T T H T H T H T T H T H T H H

Clustering illusion: Human belief that random eventsthat occur in clusters are not really random.

Gambler’s Fallacy: People commit the gambler’s fallacy when they assume that a departure from what occurs on average, or in the long run, will be corrected in the short run. Another way to think about the gambler’s fallacy is that because an event has not happened recently, it has become “overdue” and is more likely

Speculative BubblesMarket efficiency provides a very powerful result; prices do not change unless the fundamental value of a stock changes. However, there is significant evidence that at times stocks are overvalued. In particular, there is evidence that an entire stock market (or sector of stock market) is overvalued. When this happens it is called a Speculative Bubble. The reason it is called a bubble is because the value is high, but it is not based on long-run fundamentals. So, like a bubble, it can be “popped” and reduced back to its fundamental value. It is called speculative because it is thought to be driven by speculators in search of short-term capital gains, rather than long-run fundamental value.

How do speculative bubbles work? Remember that one way to make money on the stock market is to invest in a stock for which the price will rise. That is, buy a stock at a low price and sell it a high price. The money made in this transaction is called a capital gain. Hence if an investor truly believes the price of a stock will rise, then he will want to buy the stock regardless of whether he believes the fundamental value of the stock has risen. So if many, or all, investors believe the stock of the price will rise, they will all buy the stock. However, when they all buy the stock, the demand for the stock will rise, and the price will rise. Thus the resulting price increase confirms their prior belief that the price will rise. This is called a self-fulfilling prophecy. It is their belief that the price will rise that ultimately causes the price to rise. Furthermore, now that the price has risen giving confirmation that the price of this stock is rising, other investors will buy the stock, thus causing the price to rise even further. And, remember, these price increases are taking place even though the fundamental value of the stock has not changed. In this sense, the stock is overvalued, or there is a speculative bubble.

Since the stock is overvalued it will eventually return to its fundamental value. And, while the price increase may be gradual and take some time to increase, the subsequent fall in price will be rapid, thus providing the image of the bursting bubble. One might wonder why the bubble must burst. Is it possible that the price simply keep rising on the basis of people’s belief? The answer is no. The reason is that investors will soon realize the stock is overvalued and that the future stream of dividends/earnings do not justify such a high price. In other words, people will no longer be willing to pay such a high

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price for the stock. However, just because one individual realizes it is overvalued does not mean he will immediately sell. He may way to see if the price continues to rise, and thus increase his capital gain when he does sell. But as the price continues to rise, more and more investors will come to the conclusion it is overvalued. Eventually, all of them will decide its overvalued. At this point as soon as there is any small decline in the price all investors will sell off their shares to cash in on their capital gains. And since they are all selling their shares, the supply increases dramatically, and the price crashes down.

While there is evidence of speculative bubbles for specific stocks and stock markets, such speculative bubbles are not completely understood. For example, it is not clear why they begin. That is, it is not clear why so many investors develop the belief the price of a stock will rise. Second, it is not clear why it takes so long for many investors to sell off their stock once it is obvious the stock is overvalued.

While such questions remain, the existence of speculative bubbles make clear that stock prices can change apart from a change in their fundamentals, as would be true if markets are efficient.

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