The Use of the Mathematical Concept of “Average” in Finance: A Pedagogical Note
By
Babu G. Baradwaj*
Department of Finance
College of Business & Economics
Towson University
8000 York Road
Towson, MD 21252
(410)704-5773
and
Moon W. Rhee*
Department of Finance
College of Business & Economics
Towson University
8000 York Road
Towson, MD 21252
(410)704-4075
October 2009
* We are grateful to Rex Dupont and the late Sayeed Kayvan for invaluable comments and
suggestions in earlier versions of this paper. We also thank Seunghyun Woo for research
assistance.
The Use of the Mathematical Concept of “Average” in Finance: A Pedagogical Note
Abstract
Underlying many of the concepts taught in the introductory finance course is the concept
of expressing a key variable as a single number that represents the center of gravity of a range of
possible outcomes. The most common approach to calculating this number is to use a weighted
average. In teaching finance, we can make the concepts clearer and simplify the students’
learning chore by laying out the concept of weighted average in an explicit manner. This article
suggests a focused teaching approach that relates a number of specific formulae to one central
concept, and thus making it easier for the student to understand and remember the material,
rather than blindly following a number of separate formulae. We also have shown how some of
the important issues in finance such as term structure of interest rates and duration can be
explained in the context of a weighted average.
1
1.
INTRODUCTION
Mathematical concepts, both simple and complicated are used in a variety of fields, from science
and technology to humanities and social studies. Many students enter college with a phobia for
all things mathematics. Often this phobia is magnified by complicated looking equations that are
hard to memorize and retain. Since many students survive by memorizing everything that may be
needed to pass a test, they seldom put in the effort to comprehend the underlying concept. Some
instructors being obliged to students’ demands show how to enter numbers into a formula to get
the answer with little explanation about the underlying concept.
Finance is one of the social science disciplines that uses numbers most extensively.
However, it often happens that students in finance use numbers and formulas with no clear
understanding of the true purpose or meaning of them. In Keown et al (1996)'s textbook
promotional brochure, they have listed the top ten reasons students dislike finance courses. One
of them is "there are more formulas in finance than in the baby food aisle at the supermarket."
There is some truth in that. Many introductory finance textbooks have a list of frequently used
formulas and symbols inside the cover of textbooks. To some advanced students these formulas
may not be a major problem, but for many they form a significant barrier to learning.
This paper attempts to develop a framework that can be used to help students understand
any underlying mathematical concept while de-emphasizing the memorization of equations. If
we can get students to understand the basic theme and learn how to match the variations to the
problem, they will retain longer and be more flexible in applying the concepts. The fundamental
questions of "What are we doing now?", "Why do we need this formula", and “What elements do
I need to solve this problem?” can then be made clearer.
2
In this paper, we focus on the concept of “averages”, which is one of the most
fundamental concepts in mathematics and is also widely utilized in finance. Averaging appears
in introductory finance courses in several ways: average historical returns, expected future
returns, portfolio weighting, and weighted average cost of capital. In more advanced courses, it
appears in the form of duration calculations and the term structure of interest rates. As the
student passes each stage, the averaging process becomes more complex. The use of weighted
averages, however, should remain as one of the fundamental techniques in finance and be
considered as such, rather than as a series of formulae. From our discussion on the concept of a
simple average of historical returns to our applications of different types of averages to major
financial concepts and presentations of various examples, we intend to offer a unified concept of
averages for introductory finance courses. This paper may be the first step in showing how
fundamental concepts in other disciplines including mathematics can better explained and
understood by finance students when they take on related finance concepts.
The rest of the paper is organized as follows: The key properties of averages are
presented in section 2. In section 3, we introduce the concept of arithmetic averages and show
some examples in both simple and weighted averages. In each example, we explain how the
specific examples is using a related property of averages. In sections 3 and 4, geometric and
harmonic averages are introduced with their examples. The summary and conclusions follow in
section 5.
2.
PROPERTIES OF AVERAGES
It makes sense to begin this discussion by highlighting some of the properties of averages. If
students can comprehend the underlying properties, they will be better equipped to handle the
3
finance concepts where average is used. This will enable instructors to not spend as much time
on teaching them about averages and focus instead on the finance concepts. Here, then, are the
fundamental properties of averages that students need to understand and assimilate.
1.
The sum of all the weights is equal to 1.
n
w
i 1
2.
i
1
An average is greater than or equal to the smallest number and smaller than or equal to
the largest number. Assuming wi ≥ 0 for all i,
X min  X  X max
3.
The order for the computation of a “grand” average is irrelevant. In addition, an overall
average is equal to a weighted average of group sub-averages1.
4.
The average of a series of number does not have to belong to the set.
X  { X 1 , X 2 ,..........X n }
5.
The average could be considered as the center of gravity. Thus if a leverage is placed
under the average value, then one can achieve a balance.
6.
As wi increases, X → Xi. The greater the value of the weight, the closer the average
tends to the number in the series associated with the weight.
7.
Among the three Pythagorean averages, the harmonic mean is the least and the arithmetic
mean is the greatest. The geometric mean is in the middle. These averages are equal if
and only if all members of the data set are equal.
4
3.
SIMPLE AND WEIGHTED AVERAGES
It is quite common to use simple average and weighted average in many fields. Business
students routinely take an introductory mathematics and statistics courses that introduce them to
these concepts even if they somehow have managed to miss it during their middle and high
school classes. What is not made clear is the relationship between the simple arithmetic average
and the weighted average. Both are drawn from the basic equation that is used to compute the
simple average of a series of numbers.
x
x1  x 2 ...  x n
n
(1)
For the simple average we use the assumption of equal weight to each number of the set,
while for the weighted average we allow for the possibility of the weights varying from each
other. Thus, it is important to point out to the students that the simple average is a special case of
the weighted average with the weight being 1/n, and we can use the weight average equation
described below to cover the simple average equation in (1) as well.
N
X   wi X i
(2)
i 1
 w 1 X 1  w 2 X 2    w N X N
With an introduction of these average equations, we may need to explain to students that an
average is just a stylized number which exists in the abstract4.
3.1
Calculating Historical and Expected Returns in Finance
In the introductory finance course, we talk about historical mean returns and expected
returns for securities. Without emphasizing the implicit assumptions that we make with each
computation, the students are often left in the dark as to why we use tow different equations for
5
computing these mean returns. The historical mean return is introduced to students as the sum
of all the period returns divided by the number of periods. An expected return is computed as a
probability-weighted average using the contingency table (or probability distribution), which is
basically describing the returns from different states of nature, and the corresponding
probabilities.
In using the above approaches to compute the historical mean return, we make an implicit
assumption that each period’s return has equal bearing or weight towards the average return. In
other words, we assign a weight of 1/n to each period’s return. What is not apparent to the
students is that this weight of 1/n is justified only when there is no structural change and each
historical period is drawn from an independent and identical distribution. The fact that we
calculate the simple arithmetic average of 5-year stock return despite having 10-year return data
can be interpreted as assigning zero weights to the first five year returns and giving equal
weights to the recent 5-year returns. If we feel differently about the appropriate choice of
"average" to characterize the set of return numbers, then we could have a different average. For
example, if we think that more recent returns are more important than more distant returns, then
we change the weights to reflect our assessment of relative importance of returns at various
periods to determine the appropriate historical returns.
In computing the expected return for a security, we assign probabilities for each state and
then use these probabilities as weights in calculating the weighted average return.
E ( R )  p1 R1  p2 R2 .........  pn Rn
(3)
n
  pi Ri
i 1
In computing the expected return it is made clear to the student that the probabilities can
vary, and hence the expected return will vary depending on which state(s) have the higher
6
probabilities assigned to them. We can do the same with historical mean returns by indicating
that each period’s return is assigned a particular weight and that while we often assume equal
weight for each period, that it does not have to be the case every time.
By using just one approach to explain the calculation of two different types of returns, we
can minimize the use of a multitude of equations that seems prevalent in texts and make it less
daunting for our students.
After obtaining an average return, either a historical return or an expected return, one can
ask students what is the meaning of computing an average and ask them whether the average
return is one of the realized or possible returns we have in the data or in the probability
distribution. At this moment, students may wonder why we are trying to determine the average
return. The historical average or an expected return is just a stylized number in the abstract and
determining an average is just a way for us to simplify and characterize a series of returns2.
3.2
Portfolio Return Calculations
Portfolio returns and the weighted average cost of capital provide a chance to expand the
averaging concept to show how unequal weights come about in “real world” problems. These
are good examples that demonstrate the essence of weighted averages. We have seen, so far two
types of weights (frequencies and probabilistic) and one of the properties of weights - the sum of
all the weights being one. When we move to portfolio weights, we show yet another type of
weight, one based on value.
The expected return (or historical return) of a portfolio of securities can be computed in
two different orders.
1.
Compute the portfolio return for each state (or for each period) and then the
expected (or historical average) return of the portfolio.
7
2.
Calculate the expected (or historical average) return for each security in the
portfolio, and then the expected (or historical average) portfolio return.
Most introductory text books only highlight the second approach, completely ignoring the
third property of averages listed above that the order for the computation of an average is
irrelevant. This property, however, is certainly not well understood by students unless
demonstrated to them. When we require students to compute the portfolio returns either way, it
gives students not only the assurance that both approaches are valid, but also a chance to think
about the relationships between the two approaches. Note that portfolio mean returns are like
overall means and thus the order of computation for them, either from mean returns of individual
assets first or portfolio's return for each period/each state first, is irrelevant.
It may also be noted that this argument does not directly apply to the computation of the
portfolio variance as the linearity property of averaging is lost in higher moments. However, it is
worthwhile to demonstrate that one can get the variance or the portfolio return from computing
the portfolio return for each period (or each state) instead of just presenting the portfolio variance
formula in terms of individual security variances and various covariances.
A good example of average computation for portfolio returns can be found in the
determination of various (stock) market indexes. While Dow Jones Indexes are price weighted
averages of price changes of the component stocks, S&P Indexes are market-value weighted
averages factoring into the size of the company. The original Value Line index is an equally
weighted index using a geometric average. In order to compensate for the effects of stock splits,
stock dividends, and other adjustments, the price weights used for the Dow are altered by a so
called divisor in order to generate a consistent value for the index. As a result, the sum of the all
the “adjusted weights” does not equal one in this case. However, even in this case, the weights
8
can be proportionally altered to restore this property of the sum of all weights being one, as often
done in a set-theoretic treatment of probability.
3.3
The Use of Weighted Average in Computing Duration
For the more advanced student, the concept of duration represents one more application
of the weighted average. Our definition of duration is that it is the present value weighted
average of payment periods of the cash flows, where the weights are the normalized size of the
present value of the cash flows. Here, the weights are more complex than the simple, market
weighted average, and some groundwork must be laid to explain why these particular weights
are useful.
Here we think of the weighting scheme as a way of finding the center of gravity (property
5) of the present values of the components of the cash flow stream for a bond, for example. This
measure is particularly useful since it also provides the basis for calculating the sensitivity of the
price of the bond to changes in the interest rate. If one thinks of applying that interest rate
change to the center of gravity of the payments, one can visualize the effect.
0
1
2
PV1
PV2
T
D
PVT
We may represent duration as the weighted average of a series of time periods as in the
equation below.
D = w1*1 + w2*2 + …………..+ wT*T
(4)
9
In this case, the weights have a complex appearing equation, but one that is relatively
easy to explain. In this case the individual components of the weights are made up of the
formula:
wt 
PVt
PV
(5)
where,
wt is weight to be given to period t, and
PVi is the present value of the tth cash flow, and
PV is the sum of present values of all the cash flows.
The weight for each cash flow, wt, is obtained by dividing the present value of the ith cash
flow by the present value of the sum of all the cash flows.
As we examine the properties of duration, we can see the impact of the weights on the
measure of duration.
1.
Let us start with the first property that states that as T (time to maturity) increases,
duration also increases. That is self evident from equations 3 and 4.
2.
1  D  T . This is also pretty obvious since duration, as an average, should be
greater than or equal to the smallest number (1) and smaller than or equal to the
largest number (T).
3.
For a zero coupon bond there is only a single payment at maturity. Hence, wi=0
for I = 1 to T-1, wT=1 and D=T.
4.
As the coupon rate increases, all PVi increases and PV also increases, but wT
decreases since the face value is fixed. Hence, duration decreases.
10
5.
As the discount rate increases, all PVt decreases and PV also decreases. Further
term PVt decreases relatively more due to a compounding effect than the near
term PVt. As a result further term w t decreases, while wt increases (Note that the
sum of all the weights has to be equal to one). Therefore, duration decreases to
reflect the shift in weights.
6.
Being considered as the center of gravity as shown in the graph above, duration is
where any impact on cash flows prior to and after the duration payment period
may be balanced. This is why an immunization can be achieved with duration
being equal to an investment horizon.
4.
GEOMETRIC AVERAGE
Geometric mean may be defined as the mean of n numbers expressed as the nth root of their
product. In other words the geometric mean of X1, X2,..., Xn can be expressed as follows:
Geometric Mean  (X 1 * X 2 * ......X n ) n
1
(6)
The geometric mean is useful to determine "average factors". For example, if a stock rose
8% in the first year, 15% in the second year and fell 10% in the third year, then we compute the
geometric mean of the factors 1.08, 1.15 and 0.90 as (1.08 × 1.15 × 0.90)1/3 = 1.0378, and we
conclude that the stock rose on average 3.78 percent per year.
In finance, the geometric mean is used whenever we deal with interest rates and trying to
find future values. The geometric average recognizes the compounding effect involved. Using
the geometric average to calculate the average of multi-period returns on a stock or a portfolio of
11
stocks would be a better approach than using the arithmetic average because the geometric mean
is exact about what the actual rate of return was over the periods being considered. The
geometric average is lower than the arithmetic average as discussed earlier (property 7), unless
the returns are logarithmic scaled.
One example to differentiate between arithmetic and geometric means is when the price
changes from $100 to $110 (or $90) to $99. The percentage changes in the price are +10% (10%) followed by -10% (+10%). An arithmetic average of the percentage price changes is 0%,
while the geometric average is (1.1*0.9) ^ ½ - 1 = -0.5%, showing a slight decline in the price
from $100 to the final price of $99.
4.1
Term Structure of Interest rates3
Under the pure expectations hypothesis, we explain that for a given holding period the
total projected return should be the same regardless of the combination of securities to be held.
In other words, the total return expected to be earned over a two year period, for example, by
holding a 2-year security should be equal to the return expected by holding a one year security
and rolling it over for another year, or by investing in a 5 year security and selling it at the end of
two years.
Therefore, the long-term rate of interest should be a geometric average of the current shortterm interest rate and a series of expected short-term rates.
(1 t Rn )  (1 t R1 )(1  t 1 E 1 )(1 t  2 E 1 ).......(1 t n1 E 1 )n
1
where,
R

Observed market rate
E

Expected future short-term rate
12
(7)
t

Time period when rate is applicable
n

Maturity of the bond
The behavior of this kind of average obeys the same rules as the arithmetic average, and using
the second property, the movement of the long term rate can be predicted. If it is expected that
(short-term) interest rates rise (i.e., E >tR1), then the long-term rate (tRn, the average return)
should be greater than tR1. On the other hand, if interest rates are expected to fall, then tRn
should be smaller than tR1 to yield the same overall expected returns regardless of the long-term
versus short-term investment strategies. The opposite side of the coin is that if tRn is greater
(smaller) than tR1, one can make an inference that the market expects an increase (decrease) in
the interest rates in the future, according to the expectations hypothesis.
One can also approximate the long-term rate as a simple average of current and projected
future short-term interest rates. This long-term rate estimated from this approximation is often
very close to the exact equation from equation (7), but always lower as one can anticipate from
the relationship between the arithmetic and geometric averages. Given the compounding interest
(interest on interest), the “true” average can be lower than the simple average rate, yet still yield
the same overall total return.
13
5.
HARMONIC AVERAGE
Harmonic mean may be defined as an inverse of the mean of reciprocals of n given numbers. In
case of a more general weighted harmonic mean, it is defined by
Harmonic Mean 
5.1
1
(8)
n
1
 (wi * )
i 1
xi
Dollar Cost Averaging
Financial planners often advice the clients to consider dollar cost averaging as a disciplined
strategy to accumulate wealth for retirement. Dollar cost averaging is investing equal amounts of
money on a regular schedule over time in a particular investment. By doing so, more shares are
purchased when prices are low and fewer shares are purchased when prices are high.
The “true” price (or cost) per share of this investment strategy, determined by a harmonic average or
the total amount invested divided by the total number of shares purchased, is then lower than the
average price determined by a geometric or arithmetic average.
5.2
P/E Ratios
Another example in finance about the harmonic average is the price/earnings ratio, share
price divided by earnings per share. The true P/E ratio of a portfolio such as an industry P/E, can
be obtained by the total aggregate market capitalization of all the companies included in the
portfolio divided by the total aggregate earnings of the entire companies. This portfolio P/E ratio,
however, can also be determined by a weighted harmonic average of all individual company P/E
ratios with the weights being the market capitalizations. It is very interesting to note that a
change in a P/E ratio for any particular stock can have an impact on the portfolio P/E ratio
14
differently depending on whether the increase (decrease) in the stock P/E is due to a higher
(lower) price or a lower (higher) EPS. If a higher (lower) stock price contributes to an increase
(decrease) in the higher (lower) stock P/E ratio, then the stock’s market capitalization also
increases (decreases), resulting in an increase (decrease) in the weight associated with this
company stock P/E ratio. In short, a change in a stock price has a compounding effect on the
portfolio P/E ratio via a change in the individual stock’s P/E and the change in the weight given
to that P/E ratio.
Ceteris paribus, a chance in P/E due to a change in EPS does not cause any change in the
weight associated with the particular stock’s P/E ratio. This means that there is no additional
impact on the portfolio P/E due to a change in the portfolio weights.
6. SUMMARY AND CONCLUSIONS
Underlying many of the concepts taught in beginning and intermediate finance is the concept of
expressing a key variable as a single number that represents the center of gravity of a range of
possible outcomes. The most common approach to calculating this number is to use a weighted
average. In teaching finance, we can make the concepts clearer and simplify the students’
learning chore by laying out the concept of weighted average in a clear and explicit manner as
we teach the courses. This article suggests an explanation of specific formulae in the context of
one central concept, and thus should help the student to better understand and remember the
material better, rather than blindly following a number of separate formulae.
More can be done for other mathematical and statistical concepts used in finance. This is just the
first step in showing examples of how some concepts can be better explained in finance classes
15
and how to avoid mechanical application of finance equations. Students need to understand the
concept and should be able to relate the material to the prerequisites they took before.
16
REFERENCES
Bodie, Z, A. Kane and A. J. Marcus, Essentials of Investments, 5th ed., McGraw-Hill Irwin, New
York, NY. 2010
Keown, A. J. , D. F. Scott, Jr., J. D. Martin, and J. W. Petty, Basic Financial Management, 7th
ed., Prentice Hall, Upper Saddle River, NJ 1996.
17
ENDNOTES
1.
For example, when you compute the average height of all the students in class, you could calculate the subaverages of students' heights based on sex for example, and then take a weighted average of these averages with
weights being the relative sizes of male and female students. The overall averages computed from sub-averages
from different criteria (e.g., race, income) would be the same, indicating that the grouping based on different criteria
is irrelevant to the computation of the overall average.
2.
Actually, the entire number system exists in the abstract. It is just that we use fingers and toes to teach children
how to relate the number system in concrete terms. Frequently reminding students of the abstract nature of the
number system is as important as making the number system concrete and visual by drawing an analogy between it
and real life examples.
3.
Being confused and trying to blindly use a formula for expected returns for historical average returns, some
students ask for probabilities for historical returns. We also have seen cases in which a student will pick four as the
expected mean number from a fair die simply because a die does not have a “3.5” side.
4.
Some of the discussion here is taken from Bodie Kane and Marcus. The term structure of interest rates and the
yield curves are often thought to be identical, although they are, in fact, rooted in different definitions. The term
structure is based on zero-coupon bond, while the yield curve is an IRR obtained from a coupon paying bond.
Therefore, the term structure is a very special type of a yield curve, where the coupon rate is zero.
18