1
Value of Gold
Preliminary Version
Floyd Vest, December 2012
From 2000 to 2011. The price of gold has skyrocketed from about $300 per
ounce in 2000 to about $1700 per ounce in 2011. This is a 467% increase.
Example 1. By the Compound Interest Formula, the annual compound rate of
growth in the price of gold from 2000 to 2011 is given by
(1)
300 (1  i )11  1700 . This gives the impressive annual rate of appreciation of
17.1%. (For the general statement and derivation of the Compound Interest Formula, see
the Side Bar Notes.) (This article is based on “Why stocks beat gold and bonds” Fortune,
2-27-2012 by Warren Buffet.) A copy of the three page article in hand would help in
following this article. To get Buffet’s article, go to berkshirehathaway.com, click on
Warren Buffet’s Letters to Berkshire Shareholders, and the 2011 letter. Go down to “The
Basic Choices for Investors and the One We Strongly Prefer.” See the Side Bar Notes
about Warren Buffet’s rating as one of the top investors of our time. )
(In the below You Try Its, and later Exercises, solve with formulas and a scientific
calculator pad. Show your work. Label numbers, answers, and variables. Summarize.)
From 1960 to 2000. (For the history of the price of gold, see Wikipedia.org and
Search, Price of Gold.)
You Try It #1
From 1960 at $50 per ounce, the price of gold increased to about $300 in 2000. (a) What
is the compounded annually rate of increase in price? (b) What is the total percent
increase in price?
The question that is recommended about an investment is how its return compares
to inflation. Does money invested retain its purchasing power? We resort to
usinflationcalculator.com to calculate inflation from 1960 to 2000 and observe an
increase in prices of 481.80% from $20 for an item to $116.35 for an annual inflation rate
of I = 4.5%.
You Try It #2
For the above, put in the numbers and solve the Compound Interest Formula for the
annual inflation rate I.
We see from You Try Its #1 and #2, that from 1960 to 2000, the increase in the
price of gold slightly outperformed inflation. After taxes, the average annual return
would have been less than inflation. Exercise: Verify this at a capital gains rate of 15%
on the profit from a sale.
From 1965 to 2012. Warren Buffet’s article reports that in this period, the S&P
500 stocks grew an investment from $100 to $6072 yielding 9.13% per year.
2
You Try It #3
In the above calculation for the S&P 500, set up the formula with n unknown and
calculate the number of years.
For the 47 years from 1965 to 2012, six-month T-bills grew $100 to $1336.
You Try It #4
(a) Calculate the rate of return on six-month T-bills. (b) Calculate the rate of inflation
for the same period.
Example 2. From You Try Its #3 and #4, in 47 years from 1965 to 2012, inflation
averaged 4.32% and T-bills earned 5.7%. With an income tax rate of 25%, the after tax
accumulation for the T-bills was
47
St  100 1  .057(1  .25)   $715.27 . (See the Side Bar Notes for a
(2)
derivation.)
With inflation and without taxes, the after inflation accumulation was
47
 .0567  .0432 
= $183.00. Using an approximation formula would
S I  100 1 
1  .0432 

give S I 100(1  .0567  .0432) 47 = $187.81. (See the article in this course “Living and
Investing With Inflation.”)
(3)
Warren Buffet’s definition of investing is to gain in purchasing power after
inflation and income taxes. After inflation and income taxes, $100 in T-bills paid
 .0567(1  .25)  .0432 
S I  100 1 

1  .0432

formula would be S I
47
= $97.00 in purchasing power . An approximation
100 1  .0567(1  .25)  .0432
47
. Try it.
You Try It #5
(a) Build and label a table for the above results for T-bills and the time period. Discuss.
(b) Buffet’s article claimed that the effect of inflation is three times the effect of taxes.
Show the calculations to check this statement and explain possible different results in
terms of different kinds of taxes for different investments.
Bubbles. A dramatic increase in the price of gold is suspected to result in a
bubble – where there is a rapid drop in price without an immediate following recovery.
This could happen when investors think prices are not going to continue increasing, likely
because there are no buyers who want to pay more than the current price. This happened
recently in what is called the internet bubble and the housing bubble. In the internet
bubble, the price of Cisco stock dropped from about $80 to about $20. A famous
historical example was investments in tulips.
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Example 3. A drop from $80 to $20 is a 75% drop. What percent gain is required
for $20 to recover the former high of $80. Consider
20(1+x) = 80. 1 + x = 80/20 = 4. x = 3. It takes a 300% gain to recover the former high.
You Try It #6
In 1980, the bubble burst in gold and the price dropped 60%. (a) For such a drop from a
price of $1700, what would be the new price? (b) What percent gain is required to
achieve the former high. (c) For the current $9.6 trillion for the world’s gold, with a
drop in price of 60%, how much money would be lost? What would be the new value of
the world’s gold.
Example 4. For a decline of y%, what percent gain, x%, is required to recover the
former high? We calculate
 y 
y 
x 

1 
 1 
  1 . Solving for x gives x  100 
 . For example if
 100   100 
 100  y 
 75 
y = 75%, x = 100 
= 300%.
 100  75 
You Try It #7 (Graph by hand.)
Build a table for values of x for y = 25%, 50%, 75%, 90% and graph x against y. What
happens for y = 0% and y = 100%? Graph y against x. On the graph, what happens at
y = 100%?
Example 5. What fraction increase F comes from a compound interest rate i for
n=11 years (from 2000 to 2011)? We calculate
F = (1  i )11  1 . Using 6%, one dollar increased by the fraction
F = (1+.06 )11 -1=.898 on 89.8%.
You Try It #8.
From Jan 2000 to Jan 2011, the price of gold increased by about 390% or about 15.5%
per year. (a) Build a table for the fractional increase F with i = 0, .05, .1, .155, .5 for 11
years. Express F both as a fraction and a percent. (b) What i gives an increase of 100%?
(c) Use a graphing calculator to graph an approximation to the fractional increase in the
price of gold from Jan. 2000 to Jan. 2011.
Example 6. Buffet’s article reported that the dollar fell 86% in purchasing value
from 1965 to 2012. The rate of inflation was 4.32%. Assume that a pound costs $1 in
1965. To show this 86% drop, we calculate 1 (1  .0432) 47 = $7.30. A pound in 2012
1
costs $7.30. One dollar buys a pound in 1965 and in 2012 it buys
= .137 of a
7.30
pound. The loss is 1 - .137 = .863 pound, and 86.3% of what $1 bought in 1965. The
value of $1 in 1965 declined to $.137 in 2012.
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Example 7. If you want to compound a negative rate for the inflation rate I to
calculate the new purchasing value of a dollar after n years, don’t use (1-I ) n , use
n
I 

10
1  1  I  . Consider the example 1(1+I ) =2, so that the new purchasing value of one


1
1
dollar in terms of year one is
or $.50 or
= (1+I ) 10 . For this example
10
2
(1  I )
10
.0717735 

I = .0717735 = 7.17735%. Try 1 
 = .5 .
 1  .0717734 
n
I 
1

Proof: Set 1 
and simplify and then reverse the steps. Using (1-I ) n

n

1

I


1  I 
will only give you an approximation.
Example 8. The negative of a rate of depreciation can be used to calculate the
depreciated value. For example, a machine cost $1000 and depreciates at 7.4% per year.
Its depreciated value in five years is 1000(1 - .074 ) 5 = $680.86 . More precisely,
5
.074 

1000 1 
= $699.81 .
 1  .074 
Side Bar Notes:
For a nice graph of the price of gold go to MeritGold.com. Click on Gold. You
should get a graph from about 2000 to your date. Beginning in 2012, the price of gold at
$1816 per ounce dropped to $1,388 by April 15, 2013.
The history of Warren Buffet’s investment success. For this history, see
Wikipedia.org/Warren_Buffet. Write a summary of the growth of his wealth and his
methods of investing. When did he first become a billionaire? When was he caught in a
bubble and how much did it cost?
Warren Buffet as a philanthropist. In 2006, his largest contribution was to the
Bill and Melinda Gates Foundation which funded the STEM Foundation for the
improvement of education in science, technology, engineering, and mathematics. See
Consortium, Spring/Summer 2012, page 7.
Seven derivations of the Compound Interest Formula. For a derivation where
students use recursive equations to build the mathematical model, see “Tim and Tom’s
Financial Adventure” in Tech Math at COMAP. See algebra books where they use
general geometric sequences. Try a proof by mathematical induction. See Kasting,
Martha, “Concepts of Mathematics for Business: The Mathematics of Finance,” p. 29, or
Luttman, Frederick W., “Selected Applications of Mathematics to Finance and
Investment”. For the most direct derivations of the basic mathematics of finance
formulas, see Vest, Floyd, “A Master Time Value of Money Formula,” which derives the
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formulas in the TI83/84 manuals. These articles are in the free download course in
financial mathematics at COMAP.com.
For the most direct derivation, consider the following table to get S = P(1 + i ) n :
Balance Beginning of Year
Interest Paid
Balance End of Year
Year 1
P
iP
P + iP=P(1+i)
Year 2 P(1+i)
i  P(1  i) 
P(1+i) + i  P(1  i) 
= P(1 + i ) 2
Year 3 P(1 + i ) 2
.
.
Year n P(1 + i ) n1
iP(1 + i ) 2
P(1 + i ) 3
S=P(1 + i ) n
The general Compound Interest Formula.
S = P(1 + i ) n where P is the principal (example, the amount invested), S is the
final value called Sum, and n is the number of compounding periods (such as compound
monthly or yearly). The periodic interest rate i = (the annual nominal rate)/(the number
of compounding periods per year). Exercise: Solve for each of the variables.
The production assets that $9.6 trillion in gold could buy. We summarize from
Warren buffet’s article that $9.6 trillion could buy all US cropland and several Exxon
Mobiles. For the value of Exxon Mobile, see Wikipedia.org and Search Exxon Mobile.
Then go down to Financial. What is the value of US cropland? The production is what
percent of the value? Explain how a company producing consumer goods could preserve
its value in the face of inflation.
S&P 500 Stock Index and Current Price of Gold. For the current price of gold and
history of gold prices since 2000 and other economic data, see MeritGold.com and check
Market Data. How does the US national debt compare to the value of the world’s gold?
Trace the history of percent declines in the value of the S&P 500 Index of stocks. For
more history of the S&P 500 stocks, see Wikipedia.org and Search S&P 500. See
measuringworth.com and click on Savings Growth-US $. Report on the bubbles. This
site gives the yearly increase in price and total return (price change and dividends
reinvested) for the years 2000 through 2011. Plot the volatility and calculate the twelve
year average total return to get .54% per year. This may be one reason people are buying
gold. Note: On 12-22-12, the S&P 500 was 1443.67. It entered 2012 at 1243 with a
year to date total return of 16.28%. For recent performance do an internet search for S&P
500 total return and select S&P500 Index(SPX)-Charts & Returns-Morningstar. Add
2012 to the above 2000 to 2011 calculation of S&P 500 total return and get an estimate of
a 13 year average return of 1.67% per year.
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Exercises: Show your work. Use formulas and a scientific calculator pad. Label
numbers, variables, and answers.
#1.
Buffet’s article reported that from 1965 to 2012, an investment of $100 in gold
grew to $4455. (a) Calculate the average annual compounded rate of return on gold. (b)
What was the simple interest rate of return? (c) What was the total percent increase in
the price of gold?
#2.
According to Buffet’s article, the total value of the world’s gold in 2012 is $9.6
trillion and current production of new gold is $160 billion per year which purchasers
would have to absorb to maintain current prices. (a) The production is what percent of
the current supply. Apply these figures to purchasers. Buffet says that production is
more than that consumed by jewelry businesses and industry. (b) How many ounces are
produced each year at $1700 per ounce? How many pounds? (c) How many ounces and
pounds are in the $9.6 trillion world supply at $1700 per ounce?
#3.
If an investment makes 5% per year and inflation is 3.2% and income taxes are
25% for ten years, (a) what is the annual gain in purchasing power of $1 before taxes?
(b) After ten years of these earnings and inflation, how much would $1 buy? This is
called the “real value of one dollar.” (c) Without inflation, how much would it buy?
The gain is what percent of a dollar? (d) With inflation and no investment of the $1,
after ten years, what is the purchasing value of a dollar? (e) After ten years of inflation
and income taxes, what is the sum for the investment of $1.
#4.
(a) Assume for 50 years of saving for retirement, bonds pay 6%, inflation
averages 3.2%, and income taxes are 25%. Not counting inflation, but income tax, what
is an investment of $3000 worth? (b) In a traditional IRA, not counting inflation and
annual taxes, what is it worth? (c) If taxes are paid at 25% on the total value of the IRA
at withdrawal, what is the amount of the withdrawal after taxes? (d) How does the after
tax value of the IRA compare with the investment which is taxed annually? (e) What is
the after inflation and taxes value of the taxable bond investment? What is the after
inflation and taxes value of the IRA bond investment? (f) If eggs are one dollar per
dozen at time zero, what is the cost of a dozen eggs after 50 years of inflation? If after 50
years, the $3000 investment after taxes is $27,096.91, how many dozen eggs would it
buy?
#5.
(a) How much P should be invested in 2000 at 17.1% to amount to $10,000 in
11 years. (b) On a graphing calculator, graph your S=P(1 + .171 ) x for x = 0 at 2000 and
x = 11 at 2011. Compare this with the graphs of gold prices given on the internet such as
Roselandcapital.com.
#6.
Derive a general formula for Formula (2) for St which is the after yearly tax,
where t is the annual tax rate on the interest payments, i is the interest rate per year, P is
invested for n years, and St is the final after tax sum.
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#7.
(a) How much would have been invested in gold in 2000 at 17.1% to have
$200,000 in 2011? (b) What percent increase is this?
#8.
(a) Derive a formula giving the percent increase N in principal P from a
compound interest rate i over n years. (b) Solve your formula for i and n.
#9. (a) Derive a formula giving for an inflation rate I, the half-life n years of a dollar.
(b) Build a table of examples. (c) Graph I against n.
#10. (a) Warren Buffet said that in the 1980s cash was king. Twelve percent per year
was common. Compare 12% to inflation for the 1980s and give examples of preservation
of purchasing power. (See uninflationcalculator.com.) (b) Current rates in 2012 are a
disaster. See bankrate.com and compare to current inflation rates and calculate examples
of the change of purchasing power.
#11. (a) How long would it take money to double at 12%? (b) Derive a formula for the
number of years n for principal P to double at the interest rate i. Solve for n. (c) Solve
for i. (d) Build a table of examples. (e) Graph I against n. Graph n against I. (f) Add
72
to your table a test of the rule of 72: Years n to double
.
100i
#12. Some think the price of gasoline is too high? In 1965 gasoline cost about $.25 a
gallon, In 1990, about $1. In 2012, about $3. Calculate the average rate of inflation in
the price of gasoline from 1965 to 1990 and from 1965 to 2012. Build a table of the
average inflation in the price of gasoline from 1965 to 1990 and from 1965 to 2012 and
for the average rate of general inflation for these periods. What do you conclude?
#13. Use the usinflationcalculator.com to show that the inflation rate from the end of
1977 through 1987 averaged 6.5%. As in Example 7, use a negative rate compounded to
calculate the reduced purchasing power of one dollar. Try using (1-.065 )10 and compare.
#14. Use a graphing calculator to graph on the same screen the growth of $100 over 40
years at 6%, 8%, 10%, and 12% to an S. Calculate some percent differences in S at 40
years. Develop a formula for the fractional form of percent difference D in S for r and i.
#15.
If the purchasing value of a dollar dropped from $100.98 to $79.40 in 12
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12
years, what was the average inflation rate I?
#16. (a) If from the beginning of year 2000 to the end of 2012 a stock index averaged
$1440 (without reinvestment of dividends) but paid 2% dividends, estimate the annual
compounded rate of return on the investment. (b) What was the 2012 value of an
investment of $10,000 made in 2000? Compare this to inflation during that time period.
#17. Roselandcapital.com/product/history will give you the 25 year history of the price
of gold. (a) What was the average rate of appreciation of gold from 1987 to 2005?
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What was the loss in purchasing power? (b) What was the average rate from 1987 to
2012? (c) What was the average rate from 2000 to 2012?
#18. Derive the approximation formula used for Formula 3 where S I is the after
inflation sum, I is the annual inflation rate, i is the annual interest rate, for one year, and
where S I P (1  i  I ) .
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Answers to You Try Its:
You Try It #1. (a) i = 4.58% . (b) 500% increase
You Try It #2. I = 4.5% per year.
You Try It #3. 47 years.
You Try It #4. (a) i = 5.67% . (b) I = 4.3% .
You Try It #6. (a) $680 (b) 150% increase (c) The world lost $5.76 trillion. New
value of gold $3.84 trillion.
You Try It #8. (b) i = 6.5% gives 100% increase.
Answers to Exercises:
1.
(a) 100(1+r ) 47 =4455. r = .084 = 8.4% . (b) 100(47)i=4455, i = 94.8%.
(c) 4,355%
2.
(a) Production is 1.67% of current supply. Purchasers would have to absorb
$160 billion in gold, or about 1.67% of current supply, in new gold for the year.
(b) 94.118 million ounces, 5.882 million pounds (c) 5.647 billion ounces.
352.9 million pounds.
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3.
 .05  .032 
 $1.19 .
(a) Approximately .05-.032 = .018 = 1.8% . (b) 1 
1  .032 

10
 1 
(c) (1+.05 ) = $1.63 . (d) 1 
 1  .032 
10
 .05(1  .25)  .032 
(e) 1 

1  .032

= $.730 = 73 cents .
10
= $1.055 .
4.
(a) 3000 1  .06(1  .25)  = $27,097.91 . (b) 3000(1+.06 ) 50 = $55,260.46 .
(c) 55,260.46(1-.25) = $41,445.35 after taxes for the IRA. (d) 52.9% more.
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(e) Approximately 3000 1  .06(1  .25)  .032  = $5722.60. For the IRA
50
.032 

41,445.32 1 

 1  .032 
50
= $8580.07. Note: The correct after inflation value for the
50
 .06(1  .25)  .032 
annually taxed investment is 3000 1 
 = $5609.85. Notice that the
1  .032


approximation overestimates by a total 2%. (f) After 50 years, eggs would cost $4.83
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per dozen. The after tax $27,096.91 would buy 5609.64 dozen where at the beginning
the original $3000 would buy 3000 dozen.
5.
(a) P(1+.171 )11 = 10,000 .
P = $1761.47 .
SP
= (1  i ) n  1 . To
P
1
log( N  1)
change to a percent, multiply by 100. (b) i = ( N  1) n 1 , n =
.
log(1  i)
log 2
9.
Derivation: 1(1+I ) n = 2, n =
.
log(1  I )
8.
(a) Derivation: S  P (1  i ) n , N = fraction increase =
(b) Derivation: P (1  i ) n  2 P , 1+i = n 2 , i =
log 2
.
n  log(1 i ) =log2, n =
log(1  i)
11.
n
2  1 . Solving for n,
14.
Fraction D for the percentage difference between rate r and rate i is
(1  r ) 40  (1  i ) 40
(1  r ) 40
 1 . To change to a percent, multiply by 100.
D=
=
(1  i ) 40
(1  i ) 40
15.
I = 1.88%.
(a) 1440 + 13(.02)(1440) = $1814.40, 1440(1+i )13 =1814.40, i = 1.79%.
(b) 10,000(1+.0179 )13 = $12,594,08 . (See the article in this course entitled “High
Dividend Yields on Stocks, and Low Interest Rates on CD’s and Bonds” for dividends
reinvested.)
16.
17.
(a) From 1987 to 2005, gold had no appreciation for the total period. (b) From
$500 per ounce in 1987 to $1700 in 2012, in 25 years 500(1+i ) 25 = 1700. i = 5%.
I 

Derivation: S I  P(1  i ) 1 
from the article “Living and Investing with
 1  I 
Inflation” in this course. Change 1+I to 1 since I is small.
S I P (1  i )(1  I )  P(1  I  i  iI ) . Discarding the iI, since it is small, gives
S I P (1  i  I ) . How do each of the two discarding’s affect the outcome?
18.
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References:
Vest, Floyd, “Living and Investing with Inflation, Fisher’s Effect,” in the free download
course in financial mathematics at COMAP.com. Derives the correct formula for
after inflation returns.
COMAP.com offers a free download course in financial mathematics with over 40
articles, for upper high school and undergraduate college, with emphasis on personal
finance. See if your school will give you tests and credit. See
http://www.comap.com/FloydVest/Course/index.html. Simply register, click on an
article in the annotated bibliography, download it, and study it or teach it.
Unit 1: The Basics of Mathematics of Finance
Unit 2: Managing Your Money
Unit 3: Long-Term Financial Planning
Unit 4: Investing in Bonds and Stocks
Unit 5: Investing in Real Estate
Unit 6: More Advanced or Technical