APPENDIX F - U.S. Federal Budget Overview - Technical Index (fb-90.html) (252)

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Appendix F - Table Of Contents

[A] Interest Rate Used In Present Worth Calculations (253)

[B] Current Dollars, Constant Dollars, Inflation-Adjusted Dollars (254)

[C] Formula For Calculating Average Growth Rate (255)

[A] Interest Rate Used In Present Worth Calculations (253)

According to Table II.D1 in The March 1999 Social Security Trustees Report (TR99), page 58 {74}, they project interest rates on a mix of treasury securities to be as follows:
====================================================

         Interest
   Year  Rate          
   ----  --------
   2000  4.9%
   2001  5.2%
   2002  5.4%
   2003  5.6%
   2004  5.8%
   2005  6.0%
   2006  6.1%
   2007 and beyond:  6.3%

To keep things simple, when present - worthing the $7809 Billion that is incurred from 2014 to 2034 back to year 2000, I used a 6.3% interest rate for the entire period, thus obtaining the equivalent lump-sum $1500 Billion number. If I had used the lower actual interest rates during the 2000-2006 period, I would have come up with a little bit higher equivalent lump-sum number. So the $1500 Billion is a slight understatement.  

[B] Current Dollars, Constant Dollars, Inflation-Adjusted Dollars (254)

If gasoline is $1.00 a gallon in 1999 and increases at the rate of 5% a year for the next few years, then its price will be as in row A below. So "current dollars" or "nominal dollars" means the actual quoted dollar prices, i.e. how many dollar bills or "greenbacks" one must actually pay.

Suppose that the general rate of inflation (the CPI) is 4% a year. Then the price of gas in "constant" dollars or "real" dollars or "inflation-adjusted" dollars is the price of gas after general inflation has been removed. So, if we are talking about the price of gasoline in "constant 1999" dollars, then the "current dollar" $1.050 price of gasoline in the year 2000 becomes the "constant 1999 dollar" price of $1.050/1.04 = $1.010. And the "current dollar" $1.102 price of gasoline in the year 2001 becomes the "constant 1999 dollar" price of $1.102/(1.04^2) = $1.019. And so on.
====================================================

                                1999    2000   2001   2002   2003  
Gas Prices:
(A) Current Dollars             1.000  1.050  1.102  1.158  1.216 
(B) Constant 1999 Dollars       1.000  1.010  1.019  1.029  1.039  
  

Current Dollars Are Also Known as Nominal Dollars

Constant Dollars Are Also Known as Real Dollars and Inflation-Adjusted Dollars.  

[C] Formula For Calculating Average Growth Rate (255)

Given the below formula:
           n
     y = x

i.e. y = x to the n power, then

           LN(y)
           ----
            n 
     x = e

In Excel, x = exp( ln(y) / n )

In the above, e is 2.7182818   and LN is the natural log.  (  LN(e) = 1  ).

Suppose I want to find the average growth rate for some quantity 
for a two - year period 1920-1922, where the growth rates are as 
given in row B.  Then in row C, I calculate the quantity indicated 
from the column B growth rates:

 A)   Year:                              1920   1921
 B)   Growth Rate, %/Year                 2.0    8.0

 C)   1 + (growth rate percent)/100     1.020  1.080    

To calculate the 2-year average growth rate, I first calculate an 
intermediate quantity which I call "f":

        LN(1.020 * 1.080)
    f = -----------------   =  0.0483818
               2

(The "2" in the denominator of f is the number of years in the period.  
"f" is an intermediate quantity.  LN is the natural logarithm).

                                  f                   0.0483818
The average % increase = 100 * ( e  - 1 ) =  100 * ( e          -  1 )  
    = 4.957134 %.

Check: 1.020 * 1.080           = 1.1016
       1.04957134 * 1.04957134 = 1.1016     

In the above, e is 2.7182818    (  LN(e) = 1  ).

Now, to find the growth rate for 1980-1990 (the 1980s).  From Table II.D1, 
the growth rates are as in column B below.  (I'm showing the first 3 years 
and the last 2 years as this is just a short illustration of the formula, 
but of course all ten values are used): 

          Real GDP Growth   
    Year  Rate, %/Year      1 + B/100 
      A         B              C 
    ----  ---------------   -------  
    1980       -0.3          0.997
    1981        2.3          1.023
    1982       -2.1          0.979
    ...         ...           ...
    1988        3.8          0.038
    1989        3.4          0.034 

In the above, the column C values = 1 + B/100,  where "B" is the column B 
values.

To calculate the 10-year average growth rate, I first calculate an 
intermediate quantity which I call "f":

       LN( 0.997 * 1.023 * 0.979 * ... * 0.038 * 0.034)
  f =  ------------------------------------------------ 
                            10

       LN(1.310757)      0.270605
    =  -----------   =   --------   =   0.0270605
            10              10

Then the growth rate for the two year period is calculated as:

(The "10" in the denominator of f is the number of years in the period.  
"f" is an intermediate quantity).

                                  f                   0.0270605
The average % increase = 100 * ( e  - 1 ) =  100 * ( e          -  1 )  

    = 2.74300 %.

Back to "Real GDP Growth Rate - More On", section non35