Econ 3016 Lecture 2

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ECON3016: EMPIRICAL FINANCE

WEEK 3/4
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Stock Returns: Annualisations Stock Returns: Continuous Compounding Stock Returns: Stylised Facts Stock Returns: A simple Model

Topics

Annualisations

Continuously Compounded ReturnsStylised Facts

Uncorrelatedness versus Independence

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Annualisations

Consider an investment instrument that returned 10%, 12% and 5%

over three periods. The cumulative gross return will be(1.10)(1.12)(1.05) = 1.29 i.e. if we had invested 1 into the asset,after three period we would end up with 1.29. Note that this is quite

different from simply adding the three sets of returns and obtaining27%.

Here the per annum (annualised) performance is((1 + 0.10)(1 + 0.12)(1 + 0.05))

13 1 = 8.96%. Which question is

this quantity answering? I start with 1 and after 3 years I end up with

1.29. What is the annual interest that leads to this outcome? Checkthat 1(1 + 0.0896)3 = 1.29

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Annualising Returns

Simple scenario: we have a stream of annual returns, say R1 = 0.05,R2 = 0.12, R3 = 0.04 and R4 = 0.10. We know that

(1 + 0.05)(1 + 0.12)(1 + 0.04)(1 0.10) = 1.10074

i.e. if we placed 1 in a fund that returned the above quantities over four

years we would end up with 1.10074.

Can we come up with a composite return measure, say some yearly returnRA, such that if we received the same RA over 4 years we would also end up

with 1.10074? recall FV = 1(1 +RA)n so that what we want to do here is

to find RA such that (1 +RA)4 = (1 +R1)(1 +R2)(1 +R3)(1 +R4). Clearly

RA = [(1 + R1)(1 + R2)(1 + R3)(1 + R4)]1/4 1 = 0.024285

RA is our annualised return (per annum return). Important: We assumed

yearly returns.

S k R A li i S k R C i C di S k R S li d F S k R A i l M d l

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Annualising Returns

Arithmetic average return: R =1TT

t=1 Rt.An alternative measure of average return is the cumulative return over

the period from 1 to T, annualised by taking the Tth root (raising to

power 1/T). This is called the geometric average return

1 + RG T

t=1

(1 + Rt) 1T

Intuition: If we invest $1 in an asset, after one period we have (1 + R1),after two periods we have (1 +R1)(1 +R2) and after T periods we have

T

t=1(1 + Rt). The geometric average RG answers the question: What

constant return RG leads to a $1 investment turning into

Tt=1(1 + Rt)after T periods? Its the RG that solves (1 + RG)

T =T

t=1(1 + Rt).[similar to the APR!].

Think ofRG as the annualised return (p.a).


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Annualising Returns

More formally: Let kbe the number of compounding periods and let n

be the number of compounding periods in a year, so that there are

N = k/n years of data. The annualised return is then defined as thegeometric average of the returns

Rt(k) =k1

j=0

(1 + Rtj)

n/k

1

Note: With yearly compounding we have n = 1 and k is the number ofyears so that the above expression simplifies to

Rt(k) =

k1j=0

(1 + Rtj)

1/k

1


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Annualising Returns

Example: Consider an investment instrument that returned 10%, 12%

and 5% over three periods. The cumulative gross return will be(1.10)(1.12)(1.05) = 1.2936 i.e. if we had invested 1 into the asset,after three period we would end up with 1.29. The annualised return

from this asset is (1.29)13 1 = 8.96%. Note that this different from

the arithmetic average of 9%. Within the previous notation we have

N = 3, n = 1, k = 3.

Meaning of 8.96%: If you start with 1 and the yearly return is 8.96%

you will end up with 1.2936. So the 8.96% figure (annualised return)

has the same interpretation as the APR.


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Annualising Returns

Example: Suppose the compounding interval is monthly (n=12), the

monthly return is 1% and there are two years of data (k=24). The

annualised return is given by ((1.01)24)1224 1 = 0.1268 or 12.68%.

Intuition: I am looking for the annual rate, say RA such that

(1 + RA)2 = (1 + 0.12

12)122. Solve for RA and obtain RA = 12.68%.


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Annualising Returns

Should we use arithmetic averages or geometric averages? Which

average is better? It depends.

In 1999, 2000 and 2001 the SP500 returned R1 = 0.2104,

R2 = 0.0910 and R3 = 0.1189. The two means areR = (0.2104 0.0910 0.1189)/3 0% and

RG = [3

t=1(1 + Rt)]1/3 1 1%.

Geometric means are very useful for highlighting what happens over a

long period. Note that when returns are constant over time then

R = RGotherwise R is always at least as big as RG.


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p g y p

Annualising Returns

Suppose that half the time an investment earns 60% and half the time it

earns -50%. So if you invest for 10 years, then for 5 years the return is

60% and for 5 years -50%. The arithmetic average is

R = (0.6+ 0.6 +0.6+ 0.6 +0.60.50.50.50.50.5)/10 = 5%.Not too bad!

Suppose however that I hold the asset for 2 years. In the first year it

returns 60% and in the second -50%. How much money do I have at the

end of the second year? I invest 1 dollar. First it goes 1.60 then to

(1.6)(0.5) i.e 80 cents.


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Annualising Returns

Suppose I hold it for 10 years and for five it goes 60% and for 5 it goes

-50%. Then after 10 years my dollar invested is worth

(1.6)5(0.5)5 = $0.33.

If it goes up for 50 of the next hundred years, and down for the other50, then in 100 years I have (1.6)50(0.5)50 0!!!!

If half the time the stock goes up by 60% and half the time it goes down

50% then over the long run I get NOTHING, eventhough the arithmetic

average is 5%. Lesson: Arithmetic returns are unreliable for

understanding long run returns


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Continuous Compounding

One Period Returns: The continuously compounded return rt is defined as

rt = ln(1 + Rt) = ln(Pt/Pt1).

Intuition: Take the exponential of both sides, we get ert = 1 + Rt = Pt/Pt1.Rearranging gives Pt = Pt1e

rt. So that rt is the continuously compounded growth

rate in prices between t-1 and t. Notice: rt = ln Pt ln Pt1 = ptpt1.

To see further where this comes from go back to page 2 of Week 1 slides. We obtain

rt

as Pt

1(1

+r

t)

1

=P

tas

0. Note that in page 2 we have the analogy

1/m. Since lim0(1 + rt)1 = ert we end up with Pt = e

rtPt1.

Economists often prefer to work with continuously compounded returns. The latter

have important technical advantages. Recall that we could not simply add simple

period returns to obtain multi-period versions. Also, the annualisation process

required us to compute geometric averages.

Disadvantage of ccrs: When dealing with portfolio averages we can only use simple

returns in the sense that Rpt =N

i=1 wiRit whereas rpt =

wirit.


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Given a monthly ccr rt it is easy to solve for the corresponding simple net return as

Rt = ert 1. Hence nothing is lost by considering continuously compounded returns

instead of simple returns. CCRs are very similar to simple returns as long as thereturn is relatively small (this is generally the case for daily or monthly returns). Most

of the time we will be working with CCRs.

Idea: ln(1 + x) x for x small. Take ln(Pt/Pt1) = ln(1 + Rt) Rt when Rt small.Multi-Period Returns: Consider the two month ccr defined as

rt(2) = ln(1 + Rt(2)) = ln(Pt/Pt2) = ptpt2. Taking exponentials of both sides

shows that Pt = Pt2ert(2)

so that rt(2) is the continuously compounded growth rateof prices between months t-2 and t. Using Pt/Pt2 = (Pt/Pt1)(Pt1/Pt2) and theproperties of logs it also follows that

rt(2) = ln(Pt/Pt2) = ln(Pt/Pt1) + ln(Pt1/Pt2) = rt + rt1. So the cc twomonth return is simply the sum of the two cc one month returns.


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The cc k-month return is defined as

rt(k) = ln(1 + Rt(k)) = ln(Pt/Ptk). It is now easy to show that

rt(k) = k1

j=0 rtj. This additivity of ccrs to form multiperiod returns

is very convenient for statistical modelling purposes. To annualise

simply divide by k.

Adjusting for inflation: The cc one period real return is

rrealt = ln(1 + Rrealt ). Using our previous analysis we can show that

rrealt = ln((Pt/Pt1)(CPIt1/CPIt)). This simplifies further torrealt = rt t where t = ln(CPIt/CPIt1) (continuously compoundedone period inflation rate).


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Annualising Continuously Compounded Returns: If our investment

horizon is one year for instance then the annual ccr is simply the sum of

the twelve monthly ccrs as in

rA

= rt(12) = r

t+ r

t

1+ . . . + r

t

11= 11

j=0r

t

j.

Define the average continuously compounded monthly return to be

rm =1

12

11j=0 rtj

Notice that 12rm =

11

j=0 rtj so that we may alternatively express rA as

rA = 12rm. That is, the cc annual return is 12 times the average of the

cc monthly returns.


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Stylised Facts

How do time series of stock returns behave?

Are there any commonalities across the observed returns around the

world?Why important? We want to think about simple models for describing

the dynamics of stock returns. Reasoning: Look at how returns behave

and tailor your model in a way to capture the observed stylised facts.

Example: rt = + et.


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Stylised Facts

Is the normal distribution a reasonably good approximation to thedistribution of stock returns?

Approach: Think of important features of the normal distribution: Tails,

Skewness, Kurtosis etc. Do the data match these features?

Sk[Y] = E[(YE[Y])3/3] (Normalised third moment of randomvariable Y with mean E[Y] and variance 2)

Ku[Y] = E[(YE[Y])4/4] (Normalised fourth moment of randomvariable Y with mean E[Y] and variance 2)

Sample Counterparts: Sk = 1TS3

Tt=1(yt y)

3 and

Ku = 1TS4

Tt=1(yt y)

4 where S2 is the sample variance. Excess

Kurtosis: Ku 3.


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Stylised Facts

The notion of skewness captures any bias (or lack of symmetry) in thedispersion of the random variable. Measure of asymmetry. There are 3possibilities:

(i) The distribution ofY exhibits no skewness. In this case the distribution is

symmetric around its mean. For a symmetric distribution the mean,

median and mode are all equal.

(ii) Positive Skewness: Long Tail to the Right (mean > median > mode).

(iii) Negative Skewness: Long Tail to the Left (mean < median < mode).


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Stylised Facts

A distribution with positive kurtosis is called leptokurtic. In terms of

shape, a leptokurtic distribution has a more acute peak around the

mean (that is, a higher probability than a normally distributed variable

of values near the mean) and fat tails (that is, a higher probability than a

normally distributed variable of extreme values). Think about marketcrashes.

A distribution with negative kurtosis is called platykurtic. In terms of

shape, a platykurtic distribution has a smaller peakaround the mean

(that is, a lower probability than a normally distributed variable of

values near the mean) and thin tails (that is, a lower probability than a

normally distributed variable of extreme values).


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Stylised Facts

How to formally test for symmetry, tail thickedness, normality?

Symmetry: use Sk/

6/T N(0, 1).

Tail thickedness: use (Ku 3)/

24/T N(0, 1).

Practical Implementation: Construct the test statistics and reject if the

numerical value falls beyond the cutoffs from N(0,1). If you are

conducting the test at 5% for instance (2-tails) use 1.96 as the cutoffs

i.e. reject if|Sk/

6/T|1.96 and |(Ku 3)/

24/T| > 1.96


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Stylised Facts

Jarque Bera test for normality

JB = T6

Sk2 + (Ku3)

2

4

Under the null of normality JB 22. Reject H0 when JB > 2(2)%.


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Stylised Facts

Is it reasonable to assume that simple net returns are normally

distributed? Empirical Evidence: Observed returns have typically much

fatter tails than the Normal. Also some skewness. Common Belief:

Normality is not a suitable assumption. Note: At this stage we are

thinking ofRt = (Pt Pt1)/Pt1.

Since many financial assets exhibit limited liability (the largest possible

loss is the total investment) the normal distribution (which assumes that

y varies from minus to plus infinity) is not appropriate. Think that a

lower bound on returns is 1.

If single period returns are assumed to be normal, then multiperiod

returns cannot be normal (since they are the products - not sums - of

single period returns). Example: (1 + Rt(2)) = (1 + Rt)(1 + Rt1). Ifdaily returns were normal then multiperiod returns would be the product

of normals! Also looking at empirical data suggests that returns show

greater kurtosis (fatter tails) than expected with a normal distribution.


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Stylised Facts

Why normality is not appropriate: Let Rt denote the simple annual

return on an asset, and suppose that Rt = N(0.05, (0.50)2). Because

asset prices must be non-negative Rt must always be larger than -1

(indeed note that Rt < 1 meansPt

Pt1< 0). However, based on the

assumed normal distribution Pr(Rt < 1) = 0.018. That is, there is a

1.8% chance that Rt is smaller than -1. This implies that there is a 1.8%chance that the asset price will be negative. This is why the normal

distribution is not appropriate for simple returns.

A Way out? What about considering continuously compounded (log)

returns. Recall we defined the ccr at time tas rt = ln(Pt/Pt1). Instead

of the simple returns what if we assume that rt is normal, sayrt N(,

2). What does this imply for simple returns and why is thisnot a bad idea?


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Stylised Facts

Recall rt = ln(1 + Rt) and suppose rt N(0.05, (0.50)2

). Unlike thesimple net return the ccr can take values less than -1. For example

suppose that rt = 2. From 2 = ln(1 + Rt) we have thatRt = e

2 1 = 0.865. ThenP(rt < 2) = P(Rt < 0.865) = 0.00002.

Assuming that the ccr are normally distributed implies that singleperiod gross simple returns are distributed as LogNormal random

variables since rt = ln(1 + Rt).

LogNormal Distribution: The log-normal distribution is the

probability distribution of any random variable whose logarithm isnormally distributed. If X is a random variable with a normal

distribution, then exp(X) has a log-normal distribution; likewise, if Y is

lognormally distributed, then log(Y) is normally distributed.


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Stylised Facts

X = N(X, 2

X). If I write Y = eX it is clear that the logarithm ofY is

normally distributed so that Y is lognormally distributed. We write

Y = Lognormal(X, 2

X) with 0 < Y < . Similarly, if we take the logofY we have X which is normally distributed. Due to the exponential

transformation, Y is only defined for nonnegative values. It can beshown that E[Y] = eX+

122X and V[Y] = e2X+

2X(e

2X 1).

Recall rt = ln(1 + Rt) (ccr). If we assume that rt is NormallyDistributed, it means that our simple return 1 + Rt will follow a

lognormal distribution. If ccrs are normally distributed then the stockprice is lognormally distributed.


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Details: rt = ln(1 + Rt) = ln(Pt/Pt1). Suppose rt = N(0.05, (0.5)2).

That is r = 0.05 and r = 0.5. Let Rt = (Pt Pt1)/Pt1. Clearly(1 + Rt) = e

rt. Since rt is normally distributed, (1 + Rt) is lognormallydistributed. Notice that the distribution of(1 + R

t) is only defined for

positive values of(1 + Rt). This is appropriate since the smallest valuethat Rt can take is -1. Using the above formulae we have

1+R = e0.05+ 1

20.52 = 1.191 and 21+R = 0.563.


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Stylised Facts

Assuming that ccrs are normally distributed is also handy since the

sum of a finite number of normal random variables is normal, the

conceptual problem with multiperiod returns is eliminated. Still,

empirical data suggest that returns show greater kurtosis (fatter tails)

than expected with a lognormal distribution.Summary: Stock returns exhibit greater kurtosis than the normal or

lognormal routines would suggest. This means that extreme events

(both positive and negative) are observed more often than predicted by

these distributions. Stock returns also exhibit a certain amount of

skewness. Certainly extreme events are more likely to be crashes thanexplosions.


A Si l M d l

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A Simple Model

rt = + et with et = N(0, 2). Equivalently: rt = N(,

2).

We need to impose more structure. The above is only stating that the ets

are normally distributed. We are not saying anything about the

dependence structure of the e

ts (or r

ts).What if we assume et = IID(0,

2). Here IID stands for independentlyand identically distributed. We could do away with the identically

distributed part (since we assume normality) and write NID (i.e

normally independently distributed) i.e rt = NID(, 2).

Is the model rt = + et with et = NID(0, 2) realistic?


A Si l M d l

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A Simple Model

rt = + et with et = NID(0, 2) realistic?

Issues: Normality? Independence? Constant variance?

Independence: For any two functions m1(.) and m2(.) it holds thatCov(m1(x), m2(y)) = 0.

Note that Independence of X and Y implies Cov(X, Y) = 0(uncorrelatedness) but Cov(X, Y) = 0 does not imply independencebecause we could have something like Cov(X2, Y2) = 0 while

Cov(X, Y) = 0. CAUTION! Two variables may be uncorrelated whilebeing dependent. The same holds for a single variable at two different

time periods.


A Si l M d l

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A Simple Model

rt = + et with et = NID(0, 2)

IID also implies that returns are not predictable

Can we relax independence?

A more realistic specification: rt = + et with et a zero meanuncorrelated random variable.

Note that we havent said anything about the volatility of the ets.Imposing uncorrelatedness leaves the door open for phenomena such as

time varying volatility i.e. 2t = f(2t1).


A Simple Model

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A Simple Model

Why is the above model important?

We will try to relate economic theories to simple econometric

specifications so as to be able to test theories.

A model such as rt = + et with et uncorrelated rules out

predictability of the rts with their past values, say rt1, rt2 etc. At thesame time it does not rule out dependence in returns in the sense that r2tmay still be related to r2t1. This is very close to what we often observe

when analysing the time series properties of developed market returns.

Particular versions of the Efficient Markets Hypothesis (EMH) may

translate into the above statement.

Useful Remark: rt = ln(Pt/Pt1) pt = + pt1 + et with et asabove.

Econ 3016 Lecture 2

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