2. Topic 1.1.
LINEAR
FUNCTIONS
• Linear functions represent a
constant relationship between
the dependent and independent
variables.
• A straight line is characterized
by two parameters: intercept
and slope.
• Elasticity is another widely
used sensitivity measure.
1.1.1 Introduction
Linear functions constitute the basis for developing our understanding of
mathematical applications in business and finance. They are the most basic
representation of a function and, as we will see later on, the behavior of
more complicated functions can be reached through them.
Linear functions have various applications in finance, for example when
calculating simple interest or the CAPM. In this session, we will identify the
main characteristics of a linear function, with a special emphasis on the use
of slope and elasticity as sensitivity models.
1.2.1 Linear Functions
Algebraically speaking, a linear function is represented as follows:
y = L(x)
Where there are two variables:
y= dependent variable
x= independent variable
3. Additionally, we have a parameter m= slope, or direction coefficient.
Linear functions have two properties:
1. Linear functions are additive: if the total equals the sum of all
elements:
L(x1+x2) =L(x1)+ L(x2)
m(x1+x2)=mx1+mx2
4(3+2)=4(3)+4(2)
4(5)=12+8=20
2. Linear functions of degree 1. If we multiply all the arguments of a
function by a constant, the function will also be modified in the
same proportion.
L(Kx1) = KL(x1)
m(Kx1)= Km(x1)
4(5*3)=4*5(3)
4(15)=20(3)=60
y = mx
Function: the relationship between an explanatory
independent variable (x) and a dependent variable (y)
4. b0
b1
y
Affine functions are linear functions with an intercept that is equal to the
y
value of the function when x=0. Particularly, affine linear functions do
not pass through the origin.
We must emphasize that a linear function is basically the result of
additions and multiplications of other functions. The no-arbitrage
principle that we will see in session 2.2 is basically an extension of this
principle: the value of a portfolio equals the sum of values of each of its
assets.
Affine functions, which we generally associate with a straight line, are
not linear functions in the strict sense. In practice, however, there is no
distinction between linear and affine functions. In our case, neither will
be necessary , because they both share this fundamental characteristic:
Changes in the dependent variable are proportional to changes in the
independent variable, regardless of whether said changes are big or small.
Moreover, changes are symmetrical. Increases or decreases exert the
same impact.
Said change is measured by the slope. The higher the value of the slope,
the higher the change in y resulting from changes in x. The straight line
looks steeper (m1>m2). A slope’s sign can be negative (m3 <0). This indicates
an inverse relationship between y and x.
y = b1 + mx
y = b0 + mx
x
y = b0 + m1x
y = b0 + m0x
y = b0 + m2x
y = b0 - m3x
x
5. When we observe a straight line, the slope is infinite. A small change in
x causes “infinite” changes in y. If the line is horizontal and the slope
value is zero, variable x does not have an impact on variable y.
1.1.3 Graphic representation
Linear functions are represented on a Cartesian plane where there are
two axes: the X -axis, representing the independent variable, and the
Y-axis, representing the dependent variable. Take the following
function as an example.
y = 5*x or y = 5x
In this case, variable y increases five times the value of variable x. As
we have already mentioned, the coefficient represented by 5 is known
as the slope. See what happens if we increase the value of x by a
constant of 2.
Y=2+ 5x
As we can see, variable x is increased exactly by 2 regardless of the
value of x. The constant 2, in this case, represents the intercept.
.
m3x
y = b0 +m1x
y = b0+
m = 0
b =2
6. 9.4
1.1.4 Simple Interest
Future value represents the monetary value of an initial
investment PV0 at the end of the period t. Assuming
interests are paid at the end of the investment term, value
can be calculated through a straight line:
FVt = PV0(1 + rt)
This formula is referred to as simple interest:
• Dependent variable: future value
• Independent variable: time
• Intercept: Initial value or present value
• Slope: interest rate
Suppose that the interest rate of a treasury certificate,
with a 336 day term (0.93 year), is 5.79%. If the initial
investment is $9.4873, a linear equation allows us to
calculate the way in which the investment’s value will
increase until it reaches $10.00. (See session 4.1 Financial
Mathematics).
Image taken from:
https://www.banxico.org.mx/SieInternet/consultarDirectorioInternetAction.do?
accion=consultarCuadro&idCuadro=CF107§or=22&locale=es
7. 1.1.5 CAPM
The Capital Asset Pricing Model (CAPM) establishes that the
return of an asset over the risk-free interest rate keeps a linear
relationship with the risk premium of the market:
ry = rf + β(rm − rf )
• Dependent variable: asset return y
• Independent variable: risk premium, difference between the
return of a market portfolio and the risk-free asset, (rm − rf)
• Intercept: return of the risk-free asset: f
• Slope: Beta β It measures the systematic risk of an asset
regarding the market
With this information, an analyst is able to determine the
expected return of a stock such as Apple depending on the
expected risk premium.
It is important to understand that CAPM is a linear model so that
we can apply it in different contexts. For instance, if we want to
identify the Beta of a portfolio, we know that the properties of a
linear function allow us to add the Betas of each asset individually.
https://finance.yahoo.com/quote/AAPL/key-statistics/
8. 1.1.6 Slope and Elasticity
To understand the concept of slope in economic terms, take this
example:
5) Qd= 300-1.5*p
This function represents the quantity demanded for a good, Qq,
according to its price, p. Imagine said quantity represents oil barrels. In
this case, if the price increases by one dollar, the demand drops by 1.5
barrels. Suppose that the initial price was $60 per barrel. The demand
would then be 210 barrels. If the price decreased by $10, the quantity
demanded would increase to 225 barrels. If it went up to $70 per
barrel, the quantity demanded would drop to 195 barrels
As it can be observed, the slope measures the sensitivity of the quantity
demanded to the changes in the price of oil barrels since, for every
increase in the price, the quantity demanded goes down by -1.5. Note,
additionally, that the sign of the slope indicates an inverse correlation
between quantity demanded and price.
9. Conversely, suppose we do not know the slope, and we can only rely
on the data previously observed. We know that when the price of oil
changes from $60 to $50, the quantity increases to 225 barrels. We can
determine the slope based on this formula:
m =
y2 − y1
x2 −x1
=
Δy
Δx
At times, we will have the value of the slope as our primary
information. At others, we will only have the data observed that will
help us calculate the slope.
In this example, we observe that not all the bonds are equally liquid but
there are tenors to give us an idea (usually 1.5, 10, and 30 years). Based
on these values, we use interpolation to determine the yield of other
bonds, for example 3 or 7 years. While there are different interpolation
techniques, linear interpolation continues to be the most popular.
10. Slope is, consequently, one of the most important concepts because
it helps us estimate the sensitivity or the change of a variable based
on changes of another variable. Nevertheless, we must take two
considerations into account. First, the causality between two
variables is not always easy to calculate. For example, if we observe
that the exchange rate increases and inflation increases, too… which
is the causal variable?
Another problem is that the slope relies on the units in which variables
X and Y are measured. If instead of measuring quantities in oil barrels
we did so in gallons or liters, sensitivity would be different. To get rid
of such problem, the concept of elasticity is commonly used. Elasticity
measures the percent change in a variable that results from the percent
change in another.
By using the percent change in each variable, we eliminate the problem
caused by units of measurement. As for elasticity, results are generally
classified into three categories: a) elastic, b) inelastic, or c) unit elastic.
11. When changes in variable x cause a bigger percent change in another
variable, we say this variable is elastic. If the price of oil increases by
one percent, and the quantity demanded decreases in less than one
percent, the demand is inelastic. If there is a change of 1% in oil price,
and the quantity demanded goes down exactly by 1%, then this
quantity is unit elastic. As we can see in the graph, although slope is
constant in a linear function, elasticity changes at different points.
The concept of elasticity has wide implications. For example, it is
extremely important for a company to calculate elasticity so that it can
determine what the best price strategy is. Revenue is determined by
the product and the amount thereof that a company sells. At the same
time, the function of demand establishes that there is an inverse
correlation between price and quantity demanded. If price goes up, the
quantity demanded goes up. If price goes down, so does the quantity
demanded. What is the dominating effect in revenue? Is it price or is it
quantity?
∣ ϵ ∣ =
1
• Inelastic: ∣ ϵ ∣ <
1
• Unit elastic:
∣ ϵ ∣ >
1
• Elastic:
Slope is constant, but elasticity varies
from to 0. There are three cases:
12. The quantity effect dominates over elastic demand. This means that a
1% drop in price will result in an increase of over 1% in quantity
demanded. The net effect is an increase in revenue. The company, in
this case, must reduce its price.
If the company is in inelasticity of demand, a 1% increase in price will
cause a decrease of less than 1% in the quantity demanded. Therefore,
the price effect will prevail and the revenue will go up. The company
must increase its price.
We can then anticipate that the price against which a company faces
demand in a unit elastic situation will increase its revenue. An increase
or drop in price will be exactly offset by a change in the quantity
demanded. In such case, the company does not have any incentives to
change the price.
• If demand is inelastic, a higher price will increase revenue.
• A change in price is not offset by a percent change in a drop of
demand
• If demand is elastic, a lower price will increase revenue.
• The price change does not compensate for an increase apart from
that which is proportional to the quantity demanded.
13. Another case in which calculating elasticity proves useful is when we
want to compare the effects of different independent variables.
Supposetheoildemanddoesnotdependonpricealone,butonthegrowth
of world economy (g) as well:
6) Qq= 300-1.5*p +.1.2 g
The demanded quantity depends on two variables. Graphically, the
equivalent of a straight line in two dimensions equals a tridimensional
plane.
We can additionally observe that, in the direction of the price axis, the
relationship is negative whereas in the direction of growth, demand is
positive.
In this case, if we wish to understand the impact on the quantity
demanded, comparing elasticity is more natural than comparing slopes
as the units for price and growth are different.
14. 1.1.7 Elasticity Applied to Marketing
If you had to explain these ideas to a colleague, it would be easier to do so in
terms of elasticity (percent changes) or slope (unitary changes).
Source: https://hbr.org/2015/08/a-refresher-on-price-elasticity
15. Conclusion
• A linear equation is characterized by two parameters:
intercept and slope.
• Linear functions can anticipate changes in a variable in
relation to another in a very simple way. Said relationship
is constant and it is measured by the slope.
• Linear models have a direct application on models like
CAPM to calculate the value of money over time.
• It is frequently more convenient to use elasticity as a
measure of sensitivity.
• It measures percent changes, not unitary changes.
• Elasticity, unlike slope, is not constant in linear
functions.
References
• Chiang, A. & Wainwright, K. (2005). Fundamental Methods of
Mathematical Economics.4 ed. USA: McGraw-Hill.
• Treviño, R. (2008). PreMBA Analytical Premier. USA:
Macmillan.
• Gallo, A. (2015). A Refresher on Price Elasticity.Harvard
Business Review. Recuperado de https://hbr.org/2015/08/a-
refresher-on-price-elasticity.
