Unit Name and Number
Use of Exponential Growth Curves in Predicting Account Balances
Introduction to Exponential Growth
Exponential growth describes a process where the rate of change of a variable increases as the size of the variable becomes bigger and bigger (Agarwal). The rate of the growth is proportional to the present value of the variable. This behavior is observed in the growth of populations where a parent bears an offspring, the offspring too bears its offspring, and so does the offspring of the offspring. As the size of the population increases, the rate of growth increases faster as there happens to be more and more parents each time.
Exponential growth is described by a function that has the independent variable as the exponent (Agarwal). The general formula of the exponential function is shown below
y = ?ap?^x ( 1)
Where a ? 0,b ? 1, and b > 0. The function produces a nonlinear trend in which the dependent variable y is multiplied by a constant factor as the input variable x increases by a constant amount. In time-dependent processes, as in most real world problems, the exponential function describes the increment, I of a quantity after time period, t as shown below (Crauder)
I=x_0 r^t ( 2)
Where x_0 is the initial value of the quantity and r is the rate of change
A more meaningful approach would be to express the value of the final value of the quantity after the time t, rather than the increment.
Final value, x_t=x_0+I ?x_t=x_0 (1-r)^t ( 3)
This is the general format of the exponential function that predicts the value of a variable x that grows exponentially over a time t at a constant rate r.
The Exponential Curve
When plotted in graph, the exponential function assumes a curve that slowly increases at the initial stages, remains nearly flat for a period before taking an almost vertical gradient. A typical exponential curve is shown below (Bronshtei?n).
The curve describes a behavior where a quantity is increasing slowly at the beginning, but as the total number of items becomes bigger, the quantity grows faster and faster.
Modelling using the Exponential function
The exponential function can be used to model any process in which a quantity grows by a fixed rate for a period of time can be modeled by an exponential function (Crauder). The common characteristic of such applications is that the initial value grows steadily over time. In biology, when bacteria is introduced in suitable culture medium, their population increases exponentially Edelstein has provided such curves for different conditions (Edelstein-Keshet). This is so since bacteria can reproduce within hours, and the broods continue to reproduce, and so is the children of the children, not forgetting the original parents too. In nuclear physics, a chain reaction is initiated when a uranium nucleus produces multiple neutrons which when absorbed by other nuclei trigger a release of more neutrons, which attack more nuclei. A self-sustaining chain reaction occurs as each of the parent and subsequent neutrons initiate explosive reactions (Satake). In economics, the average rate of economic growth is usually expressed in percentage form, assuming exponential growth (Crauder). This is so since investment opportunities create more opportunities in different sectors over time. Demography studies have always expressed human population using the exponential curves as noted in world population data sheets (Population Reference Bureau). Exponential curves are subsequently used to predict population in decades to come for the purpose of planning.
In the figure below, ¬the GDP and per capita of Japan, estimated from 1870 to 2008 was shown to take exponential growth (Maddison).
Similarly, the population growth of China, Brazil, India and the US are here below shown to have assumed an exponential curve (Maddison).
In banking and finance, exponential growth is seen in pyramid schemes where a member deposits some amount and recruits other people to join (Artzrouni). The member earns some percentage of the deposits made by those they recruit, and those recruited by their recruits in a chain. However, the most crucial application of exponential growth in banking is in calculation of compound interest for depositors. The growth of capital deposited in a savings account that earns compound interest follows an exponential trend (Larson). The exponential function can therefore be used to predict bank account balances.
The exponential function is used in finance in computing the amount of compound interest (Larson). It is important in describing the continuous accrual of interest in savings. Saving accounts that offer compound interest are the most attractive to investors as they allow accrual of large sums of money with a small initial capital investment. Compound returns mean that, though the interest rate is constant, the amount of interest earned in each period is higher than the previous (Guthrie). This is in contrast to simple interest, where if an initial capital of $1,000 is deposited to an account that earns interest at the rate of 10%, the investor will constantly earn $100 in each interest period, so long as no extra capital is deposited. A compound interest account however attracts interest on the cumulative capital, that is, the principal and the interest already earned. In each period, the interest rate is applied to the sum of the principal and the interest already earned in the previous periods. In such, the interest for each period will be higher than the previous and savings will grow faster than in simple interest accounts. In the case of a $1000 deposit, the principal will earn $100 interest after the first compounding period and the account will be worth $1100. After the second period, the interest rate is applied to the amount $1100, therefore the interest earned will be $110. In each subsequent period, the amount of interest earned increases, giving an accelerated growth of the investment. In 20 periods, for example, the total capital in the account will be $ 6727.50. Compounding refers to the ability of capital, or any other asset to yield earnings that are then reinvested to yield their own earnings (Guthrie).
Discrete compound interest
If money is deposited in a savings bank account, it earns an interest at the end of a certain time period, like a day, a month or a year depending on the agreement. If for example the interest is earned in each year, then the bank is said to offer annual compounding. Monthly compounding will therefore mean the process where the interest is compounded after a month. Quarterly and biannual compounding means that the interest is compounded once each 3 months and 6 months respectively. The process of compounding is here below explained considering biannual compounding. A case is considered where $1000 is deposited in a bank account that offers an annual interest rate of 4%. Let the amount of money in the bank account after a time period t, in years be represented by the function P(t). P(t) is a function of the time t and the initial condition in our knowledge is that at t=o, the amount in the account is the principal, i.e. $1000. This can be represented as P(0) = 1000
Biannual compounding means that interest is earned every 6 months, or conveniently, in half a year. The equivalent biannual rate is therefore 4/2 %. In decimal form this can be written as 0.04/2.
Let the account balance after the first compounding period (6months) be denoted by the function P(1). Similarly, let P(2),P(3),P(4) … represent the bank balance in the 2nd, 3rd, 4th … periods, referring to 12, 18 and 24 months respectively.
At the end of the first compounding period, the interest earned on the principal at the stipulated rate is computed as
I_1=(0.04)/2 x1000=$20 ( 4)
The account balance after that period will be the sum of the initial amount deposited and the interest earned.
This computation can be expressed in a single equation as
P(1)=1000+(0.04)/2 x1000 ( 5)
This can be simplified in to P(1)=1000(1+(0.04)/2) ( 6)
The later equation shows that the account balance at the time t_1=6months can actually be represented as a product of the principal and a term defined by the interest rate.
In line with the principle of compounding, the interest after the second compounding period will be computed from the total amount in the account P(1). The interest earned after the second period (12 months), is thus calculated as
I_2=1020×(0.04)/2=$20.4 ( 7)
The account balance after the second period, P(1) will be the sum of the interest, I_2 and the amount P(1)
i.e. P(2)= P(1)+I_2
Substituting for I_2=P(1)×0.04/2
Substituting for P(1), P(2)=1000(1+0.04/2)+1000(1+0.04/2)×0.04/2
The term 1000(1+0.04/2) is common and can be factorized to obtain the following equation
Further simplification yields the equation, P(2)=1000(1+0.04/2)^2 ( 8)
The above equation gives a simplified expression of the account balance at the end of the second compounding period in terms of the principal $1000 and the rate of interest 4/2%. The similarity between the two equations, eqn. (6) and eqn. (8) is noted.
A third iteration is here below investigated.
At the end of the third period, i.e. after 18months, the interest will be computed on the amount P(2) as per the equation below
Substituting for P(2), I_3=0.04/2×1000(1+0.04/2)^2 ( 9)
The account balance after the end of the third period is the sum of the interest, I_3 and the amount in the account at the end of the period 2, i.e. P(2)
i.e. P(3 )= P(2)+I_3
Substituting for I_3
P(3 )= P(2)+0.04/2×1000(1+0.04/2)^2
Substituting for P(2)
Factorizing the term 1000(1+0.04/2)^2
P(3 )=1000(1+0.04/2)^2 (1+0.04/2)
After further simplification, the following equation is obtained
P(3 )=1000(1+(0.04)/2)^3 ( 10)
From the three equations obtained above, we can generate a general formula by noting the following trend.
Period, n Account balance, P(t) Comments
1 1000(1+0.04/2) P(t) is given by the product of the principal $1000 and the term (1+0.04/2)
2 1000(1+0.04/2)^2 P(t) is given by the product of the principal $1000 and the term (1+0.04/2), raised to power 2 (the period of compounding)
3 1000(1+0.04/2)^3 P(t) is given by the product of the principal $1000 and the term (1+0.04/2) to power 3, (the period of compounding)
The account balance at the end of compounding period t can be written in a general format
P(t)=1000(1+(0.04)/2)^n ( 11)
But $1000 and 0.04/2 represent the principal and the rate of interest respectively. The equation above can be re-written in a more general term as
P(t)=P(1+r)^n ( 12)
Where P is the principal, r the rate of interest per compounding period, in decimal form and n the compounding period in which the account balance is been calculated.
The validity of this expression can be checked by computing the account balance at the start of period 1, which is undoubtedly known to be the principle, P=$1000
In that case, n=0,P=1000,r=0.04/2
Placing these conditions in the general equation
Suppose we apply the same equation to calculate the account balance at the end of the first period (6months), which we calculated from first principles to be $1020
In this case, n=1, P=1000,r=0.04/2,n=1
We therefore conclude that a valid general formula has been developed which calculates the account balance for a given principal, interest rate and period time.
Compound interest formula as an Exponential Function
The general formula developed above for computing the bank account balances is in the format of the exponential growth function presented earlier. It was shown that, for a quantity that assumes an exponential growth, its value after time t will be given by the equation
This is synonymous to the equation (12)
The term x_t corresponds to the account balance, x_0 the principal, r the rate of interest and t the period of compounding. It is thus inferred that the growth of capital in a compound interest bank account is described by the exponential function and can be accurately predicted using the exponential curve.
Graphical representation of Account Balances
Having modelled the compound interest using the exponential function, the amount of money that accrues in a bank account after a given time can be predicted from the exponential curves if the rate of interest and the principal are known. For example, the Barclays Bank under the international banking portfolio, lists its interest rates as below (Wealth.barclays.com).
Suppose it’s required to calculate the account balance after 12months (1 year) in the case where $1,000 is deposited in to the bank; the interest rate as shown in the chart would be 0.45% p.a. and the account balance is calculated as below
Or by using the monthly equivalent interest rate,
To recheck the answer, we make reference to Wealth.barclays.com, where the bank advises clients that a capital investment of $1000 deposited in an account yields $1,004.51 after 12months. The computation above is therefore correct.
The above function can be plotted to give an exponential curve that can be used to predict the account balances at different time periods and principal amounts. The figure below shows the exponential growth curve corresponding to the current Barclays Bank international banking interest rates. Three curves are shown corresponding to different principal amounts which also have different interest rates. The graphs shows the account balances for principal amounts $30,000, $80,000 and 200,000 for a period of up to a period of 160 months.
Primarily, it is noted that the three plots assume an exponential curve as expected. However, the higher the principal amount, the faster the rate of growth increases, as indicated by the time period the curve takes before getting steeper. The account balances for the different principal amounts can be read off easily from the curves. For example, at the end of the tenth year, i.e. 12 months, the account balances corresponding to the principal amounts $30,000, $80,000 and $200,000 will be $51417.9, $137114.4 and $660077.4 respectively. The exponential curve above can be used to predict bank account balances for any period under the three principals.
Exponential curves however can be more meaningful when they are used to predict and compare different rates, either from different banks or from different account types from the same bank. In order to satisfy this need, an exponential curve for the bank balances for an account from another bank (Standard Bank) is here below presented. The data was obtained from the banks website (International.Standardbank.Com).
As expected, the account balances show an exponential growth over time, starting quite gradually but increasing more rapidly in subsequent periods. From the curve, it is seen that at the end of the tenth year, the account balances corresponding to the principal amounts $30,000, $80,000 and $200,000 will be $99,011.6, $264,031 and $836,934.5 respectively.
It has been shown that money deposited in a bank account that earns compound interest increases exponentially. A general exponential function for computing the bank balances after a given period time was developed and verified. By using interest rates from two different banks, it was shown how the bank balances can be predicted from the exponential growth curves. Though similar modelling is used in reality to investigate various financial parameters most cases are often more complicated. The above work has assumed that the interest rates are constant for the period investigated. In real financial markets however, interest rates are subject to many factors and changes more often. Stock market returns for example shows erratic fluctuation of interest rates.