ECONOMICS # 3: Tracking The M2 Money Supply
Dr. Paul Kutasovic
Mathematics by Dr. Sharon Petrushka
Introduction
The Federal Reserve’s strategy for implementing monetary policy is based on the relation
between the monetary aggregates and nominal GDP. Target growth rates for the monetary
aggregates are set by the Federal Reserve in order to achieve the GDP and price stability
objectives. The target growth rates are based on the historic relationship between the monetary
aggregates and GDP. The Fed then controls the money supply through its control over bank
reserves and the deposit creation multiplier.
Mathematics:
Before you read this section, please refer back to the mathematics in Excel Problem #8 ,
headed % Change for Unequal Time Periods. In that section, we showed how to convert
from a growth rate over a time period of ! years, where ! is greater than one, to an
equivalent annualized rate.
In this section, we will see how to convert from a growth rate over a time period which is
less than a year, to an annualized growth rate.
Suppose that " is the % change , or growth rate, in the quantity # in decimal form
over a period of time equal to !$ th of a year. For example, if " is a monthly growth rate
, then ! % $& '
if " is a quarterly growth rate, then ! % ( ' etc. Let #)*+ denote the value of the
quantity , # ' after * time periods (of length !$ th of a year) . The growth rate is
computed as follows ,
"%
%-./!01 %
#)$+2#)3+
#)3+
%
#)$+
#)3+
2
#)3+
#)3+
%
#)$+
#)3+
2$
Thus,
"4$%
#)$+
#)3+
%
8/791 :; <9/!*6*= /;*1> :!1 *6?1 @1>6:A
6!6*6/7 8/791 :; <9/!*6*=
5
(1)
#)$+ % #)3+ )$ 4 "+ ' and
It follows then, that
#)&+ % #)$+ )$ 4 "+ % #)3+ )$ 4 "+&
#)B+ % #)&+)$ 4 "+ % #)3+)$ 4 "+B
More generally,
#)t+ % #)t 2 $+)$ 4 "+ % #)3+)$ 4 "+t 5
(2)
We would like to convert the growth rate " to an annualized growth rate. To do that,
we assume that for each of the ! time periods of length !$ th of a year, the quantity
grows at the same rate. Then, the value of the quantity ' # ' after one year is
#)!+ % #)3+)$ 4 "+! 5
The % change , > ' after one year is then computed as follows:
>%
Thus
#)!+2#)3+
#)3+
%
#)3+ )$4"+! 2#)3+
#)3+
> % )$ 4 "+! 2 $
%
#)3+ C )$4"+! 2$ D
#)3+
% )$ 4 "+! 2 $
)B+
Example 1:
Suppose that you initially have $1,000 in a bank account, and a month later you have
$1,032 . Then if #)*+ % your balance after * months, then
#)$+
#)3+
%
$'3B&
$'333
% $53B& % )$ 4 53B&+
Thus , the monthly growth rate is " % 353B&
5
Note, that if your money continues to grow at this rate every month, after a year, your
balance will be
#)$&+ % $$' 333)$ 4 353B&+$& % $$(EF5BB
To change this to an /!!9/76G1A 0>:H*. >/*1 , we use formula (3) with ! % $& 5
> % )$ 4 353B&+$& 2 $ % 35(EF33 % (E5F33%
This number is equivalent to having a growth rate of 45.9% compounded once a year.
That is, if your initial balance is $1,000 , and it grows 45.9% every year, at the end of
one year, your balance will be $ 1,000(1 4 35(EF33+ % $$(EF.33
Example 2:
Suppose that your company has a profit of $20,000 initially, and $&8,000 at the end of
one quarter. Then if #)*+ is the profit after * quarters, then
#)$+
#)3+
%
&I'333
&3'333
% $5( % )$ 4 35(+
Thus, the quarterly growth rate is " % 35(
To change this to an /!!9/76G1A 0>:H*. >/*1 , we use formula (3) with ! % (5
> % )$ 4 35(+( 2 $ % &5I($J % &I(5$J%
In the discussion above, the quantity # was growing exponentially.
In the business problem in this section, you will see discussed the notion of monetary
targets. You will be asked to compute lower and upper bounds for the growth range.
This is done using not exponential growth, as above, but rather linear growth.
Suppose that K is your base value, and > is the minimum annual target growth rate,
and L is the maximum target annual growth rate for your quantity # 5
We will find the lower and upper values of the quantity # at the end of each month as
follows: For ! % $ ' & ' 555 ' $&
?:!*.
7:H1> M:9!A ; :> #
9@@1> M:9!A ; :> #
3
K
K
$
>
K 4 KN $&
L
K 4 KN $&
&
&>
K 4 KN $&
K 4 KN &L
$&
!
K 4 KN !>
$&
!L
K 4 KN $&
The reason that the growth rates are divided by 12 is that the rates > and L are yearly
rates and we are converting to a monthly rate. Note that it is not being done here like it
was in the previous discussions assuming exponential functions and compounding, but in
a much simpler manner by dividing the yearly growth rate by the number of time periods
in a year which is 12 .
Example 3:
Suppose that your base value is 3798.3 , > % 53$ and L % 53E . Then for ! % E
which represents five months after your starting point
lower bound for # % BOFI5B 4 BOFI5BN EN53$
$& % BI$(5$&
upper bound for # % BOFI5B 4 BOFI5BN EN53E
$& % BIOO5(B
The reason that this procedure is called 76!1/> 0>:H*. >/*1 is that the function
K>
#)!+ % K 4 KN !>
$& is a linear function with slope $& and = 2 6!*1>-1@* equal to K
5
EXCEL
To be supplied
Problem
The success of the Federal Reserve in achieving the money supply targets is critical for the longterm health of the economy. If the money supply is growing above target, the result may be
higher inflation. In contrast, below target money supply growth may result in an economic
slowdown or recession. Thus, it is critical to closely track the money supply and analyze it in
two ways:
1.
Compute monthly, quarterly and annual growth rates for the money supply.
Determine shifts in money supply trends by comparing growth rates computed for
different periods.
2.
Compute the money supply growth targets and compare them to the actual money
supply numbers released weekly by the Federal Reserve.
Monetary Targets and Money Supply Data
Thee Federal Reserve computes weekly and monthly estimates of the monetary aggregates. It
sets the money supply targets in terms of percentage annual rates of change.
The main target of policy is the M2 measure of the money supply. For 1997, the Fed established
a target growth range for M2 of 1% to 5% measured from the 4th quarter of 1996 to the 4th
quarter of 1997. The target range of 1% to 5% is also applied to 1998.
Table 1
Target Growth Rate for M2
(Annual Rate 4th Quarter Over 4th Quarter)
1997
1998
Growth Rate
1%
to
5%
1%
to
5%
The M2 money supply data is presented in Table 2.
Table 2
M2 Money Supply
(Billions of $)
1995
January
February
March
April
May
June
July
August
September
October
November
December
3639.3
3655.1
1996
3669.9
3685.0
3713.9
3724.5
3725.4
3741.6
3749.1
3759.7
3769.7
3779.6
3798.3
3819.3
1997
3834.7
3850.0
3865.5
3883.3
3880.9
3894.4
3905.3
3940.5
3960.0
3975.8
3998.3
4040.1
1998
Current
Event
Project
From the current event tracking assignment discussed in class, find the values for M2 for
1998.
Growth Rates in M2
For each month in 1997 and for the first three months of 1998, calculate the following for M2:
1.
The annual year-over-year growth rate.
2.
The quarterly growth rate at an annualized rate.
3.
The monthly growth rate at an annualized rate.
These growth rates provide an assessment of how quickly or slowly the money supply is
growing. Relations between growth rates estimated for different periods are used to determine if
there has been a shift in Fed policy or the growth trend in the money supply.
Spreadsheet Construction
The data used to calculate the growth rates is provided in Table 2. Enter the data from Table 2
into an Excel spreadsheet. Label each row by its month. Once the data is input, solve the
problem using the following steps:
Part One
The monthly growth rates are calculated in a separate column. The formula used to compute the
monthly growth rate at an annual rate is:
(((current M2/ M2 lagged by one month)^12)-1)
The formulas are entered into the spreadsheet beginning with an =, with the current and previous
year M2 values (computed in step 1) entered as spreadsheet addresses.
Part Two
The quarterly growth rates are computed in a separate column. The formula used to compute the
quarterly growth rate at an annual rate is:
(((current M2/ M2 lagged by three months)^4)-1)
Once again, the formulas are entered into the spreadsheet beginning with an =, with the current
and previous year M2 values (computed in step 1) entered as spreadsheet addresses.
Growth rates are computed for each month in 1997 and 1998 and are formatted as percents.
Part Three
Finally, the annual growth rates are computed in a separate column. The formula used to
calculate the annual growth rate is:
(((current year M2 / previous year M2)-1)
Growth rates are computed for each month for 1997 and the first three months of 1998 and are
formatted as percents.
The results of the calculations for the first three months of 1997 are provided in Table 3.
Discussion question: What do the growth rates suggest about the direction in monetary
policy and the trend rate of growth in M2 ?
Table 3
Growth Rates in M2 at Annual Rates
(Annual Rates)
1997 January
February
March
April
May
June
July
August
September
October
November
December
1998 January
February
March
M2
3834.7
3850.0
3865.5
3883.3
3880.9
3894.4
3905.3
3940.5
3960.0
3975.8
3998.3
4040.1
Monthly
4.9%
4.9%
4.9%
Quarterly
6.0%
5.6%
4.9%
Annual
4.5%
4.5%
4.1%
Compute Monetary Targets
To monitor the success or failure of the fed in hitting its monetary target, the actual money
supply is compared to the target growth range. For this problem, target growth range is
estimated for 1997 and 1998 using Excel. This is accomplished as follows
For 1997
1.
Use November 1996 as the base for 1997.
2.
Calculate the target range for each month from December 1996 to November 1997.
3.
Use the following formulas to compute the growth range.
Lower Range
3798.3 + (3798.3*0.01*(n/12))
Upper Range
3798.3 + (3798.3*0.05*(n/12))
Where n = month of year with December equal to 1.
For 1998
1.
Use November 1997 as the base for 1998.
2.
Calculate the target range for each month from December 1997 to November 1998.
3.
Use the following formulas to compute the growth range.
Lower Range
3998.3 + (3998.3*0.01*(n/12))
Upper Range
3998.3 + (3998.3*0.05*(n/12))
Where n = month of year with December equal to 1.
The target range for the first three months of 1997 are provided in Table 4. Complete the
following:
1.
Compute the range for the remaining months of 1997 and all the months for
1998.
2.
Using the charting features in Excel, graph the target range for 1997 and
compare it with the actual values of M2.
3.
Using the charting features in Excel, graph the target range for 1998 and
compare it with the actual values of M2 through the month of March.
4.
How is the Federal Reserve doing in hitting the targets so far in 1998?
Table 4
Monetary Growth Targets
1996 November
December
1997 January
February
March
April
May
June
July
August
September
October
November
December
1998 January
February
March
April
May
June
July
August
September
October
November
Lower Range
3798.3
3801.5
3804.6
3807.8
3811.0
Additional problems to be supplied
Research problem to be supplied
Upper Range
3798.3
3814.1
3830.0
3845.8
3861.6