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Mathematical Induction in Financial Mathematics
Floyd Vest (Preliminary Version)
Versions of most of the formulas in this financial mathematics course can be proven with
Mathematical Induction.
Mathematical Induction Principle:
Assume a statement is claimed to be true for a finite set of positive integers 1, 2, …, t-1, t.
To do a proof by Mathematical Induction, complete
Step 1: Show that the statement is true for t = 1.
Step 2: Show that if the statement is true for t = k, then it is true for t = k+1. By completing Step
1 and Step 2, you have proven that the statement is true for all positive integers 1, 2, …, t.
Notice that this set of positive integers is finite.
Example 1. Basic to financial mathematics is the well known formula for the sum of the first t
terms of a geometric sequence which is usually expressed as
a(1  r n )
S t  a + a r + ar  ar  ...  ar 
(1)
for positive integers t = 1, 2, …,n. The t
1 r
is the index and can be different from the exponent. This gives , S1  a, S2  a  ar , … .
2
3
n 1
Step 1. When t = 1, S1 means the sum of the first term which is S1  a . Formula (1)
a(1  r )
gives S1 
 a . Thus the statement is true for t = 1.
1 r
Step 2. To complete Step 2, we will show that if St is true then St 1 is true. Our aim is
to derive from St the formula St 1  a  ar  ...  ar n 
(1)
S t  a + a r + ar 2  ar 3  ...  ar n 1 
a(1  r n 1 )
.
1 r
We will assume Formula 1
a(1  r n )
and add ar n to both sides giving
1 r
a(1  r n )
 ar n . Calculating we have
1 r
a(1  r n )
a(1  r n )  ar n (1  r )
St 1  a  ar  ar 2  ...  ar n 1  ar n 
 ar n =
=
1 r
1 r
a  ar n  ar n  ar n 1
a(1  r n 1 )
a  ar n 1 a (1  r n 1 )
=
=
. Thus St 1  a  ar  ...  ar n 
1 r
1 r
1 r
1 r
n 1
a(1  r )
St 1  a  ar  ...  ar n 
as was to be shown. By completing Step 1 and Step 2, we
1 r
have proven by Mathematical Induction the formula for the sum of a geometric series.
St 1  (a  ar  ar 2  ...  ar n 1 )  ar n 
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Example 2. The Compound Interest Formula is
(2)
S = P(1+i ) n where P dollars is deposited at the beginning of year one. The compounded
annual interest rate is i . The sum or future value at the end of year n is S n . We consider a
proof by Mathematical Induction.
Step 1. By the above, S1  P  iP  P(1  i). This is the formula for t = 1.
Step 2. We wish to prove if St  P(1  i)t then St 1  P(1  i)t 1 . By the above, the
amount St and the end of year t earns interest for the next year as
St  iSt  St (1  i)  P(1  i)t (1  i)  P(1  i)t 1 . Thus we have completed Step 1 and Step 2 and
have a proof by Mathematical Induction.
Example 3. A formula for the sum of an ordinary annuity is
 (1  i)n  1
Sn  R 
 where R dollars is deposited at the end of each year for n years and
i


the sum earns interest at the rate i each year. We will do a proof by Mathematical Induction.
(3)
For Step 1: By the above, the amount at the end of year one is S1  R. In the above
 (1  i)1  1
i 
formula we let n = 1 to get S1  R 
 R    R. This completes Step 1.

i
i 


 (1  i )t  1 
 (1  i)t 1  1
Step 2. We need to show that if St  R 
then
S

R

t 1

 .
i
i




t

 (1  i )t  1 
 (1  i)t  1
i
  (1  i)  1

S

R
(1

i
)

R

R
(1

i
)

Given St  R 
,


t 1





t
i
i
i








 (1  i)t 1  (1  i)  i 
 (1  i)t 1  1
= R
  R
 as was to be proven. So we have proven a formula
i
i




for the sum of an ordinary annuity. You will notice that the letter S has been used for different
meanings in Examples 1, 2, 3.
We note that the Well Ording Axiom implies that any non-empty set of consecutive
positive integers contains a least element. We use it to start the proof by Mathematical
Induction. It actually says that any set of positive integers has a least element.
We will apply Mathematical Induction to a finite value for n. If there is an infinite
number of terms, the sum may or may not be a real number.
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Exercises: State and prove by Mathematical Induction the indicated formulas. Give or describe
a time-line for each formula and define the variables. For the other problems, do the usual.
#1.
State and prove a formula for the present value of an ordinary annuity.
#2.
State and prove a formula for the present value of an ordinary annuity with rents R
increasing each year at a constant rate I and earning interest at the rate r. See the article in this
course “The Mathematics of Financial and Social Responsibility.”
#3.
From #2, state and prove a formula for the future value.
#4.
Prove that the sum of the first n positive integers is 1+2+…+n =
n
(n  1) .
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#5.
State and prove a formula for the sum of an arithmetic sequence. In what article in this
course was an arithmetic sequence used. Also, prove this formula by a method other than
Mathematical Induction.
#6.
Prove that the sum of the squares of the first n positive integers is
1
12  22  32  ...  n2  (n)(n  1)(2n  1).
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#7.
Assuming that  x1  x2  ...  xn   xn1  x1  x2  ...  xn  xn1 prove a generalized Left
Hand Distributive Property of Multiplication over Addition
a(b1  b2  ...  bn )  ab1  ab2  ...  abn .
#8.
Discuss how you would describe in general the set of positive integers by using the
concepts associated with the Mathematical Induction Principle. See Wikipedia.
#9.
Once you have the positive integers, discuss how would you characterize the negative
integers? What number plays a key role? How would you characterize it?
#10. Once you have the integers, how would you tell the difference between the positive and
negative integers?
#11. Prove by Mathematical Induction that for any positive integer n, log xn = n log x. Prove
it by another method.
#12. “A solar power system costs $10,000 to $30,000 after the federal tax credit-depending in
part of the size of system required. That investment yields monthly electric savings of $100 to
$200, which is most of the average household’s electric bill. So you should recoup your
investment in five to ten years.” (Money, June 2015, page 32) Do the math to check these
figures. Check the figures for some household you know. In Denton, Texas, electricity costs
range from 3.9 cents to 5.86 cents per kilowatt hour. See Google for averages.
#13. In 2013, the average net worth of the top 1% for ages 60-69 was $11,657,000. For the
top 5%, it was $3,527,000. For the top 10% it was $1,732,000. (Denton Record Chronicle, May
17, 2015) For retirement security, how much is required in 2013? How much will be required
when you age 60-69?
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#14.
What is wrong with the following so-called proof by Mathematical Induction?
“Theorem”: Every positive integer is both even and odd.
“Proof”: Assume k is both even and odd. Since k is even, it can be written as 2s, where s is a
positive integer. Since k is odd, it can be written as 2a+1 where a is a positive integer. So
k+1 = 2s+1, k+1 is odd. Since k+1 = (2a+1) +1 = 2(a+1), k+1 is even. Therefore k+1 is both
even and odd.
#15.
Show that a  0 is undefined. There are two cases.
#16. Median wealth after 30 years of retirement withdrawals from a $500,000 portfolio of
20% stocks, 80% bonds, at retirement starting with 4% withdrawal increasing each year at 3% is
$93,594. What is a constant average rate of return on the portfolio? For 80%/20% median
wealth is $467,761. What is a constant average return. How does this relate to the sequencing
effect? (T. Rowe Price Report, 3/15)
#17. Taxation on Social Security goes up to taxation of 85% of Social Security. Assume
Social Security is $18,000 and other income is $18,000. According to the IRS Social Security
Tax Worksheet, your tax bill is $733 .
Increase the other income by $6000, giving a total other income of $24,000. According
to IRS, the tax is $2031. The tax increase is from $733 to $2031, an increase of $1298 caused by
the $6000 additional income. Thus the $1298 gives a 21.6% tax rate on the additional income.
Step this up by another $6000 in additional income and the tax rate on it is 27.4%. Step
this up by another $6000 and the tax rate on it is 30.4%. Without the Social Security, the 28%
rate doesn’t apply until taxable income exceeds $90,750. (Scott Burns, Denton Record
Chronicle, April 26, 2015) See the article in this course on taxation of Social Security. Do the
math to explain the 27.4% and 30.4% .
For traditional IRAs, it was thought: invest tax-deferred today, expect to withdraw the
money later at an equal or lower rate. But from the above examples, these IRA withdrawals can
be taxed at rates such as 27.4% and 30.4%. What about a Roth IRA? See the article in this
course “Does T M Need a Roth?” How do the arguments in these examples affect people whose
other income has already brought them to full taxation of 85% of their Social Security, before the
IRA withdrawal?
#18. (a) Buying a car or leasing? Scott Burns explains why buying is better than leasing.
(Denton Record Chronicle, March 29, 2015) Do a paper on this comparison and send it to us.
(b) Work out the following example. In 2013, a 2013 Ford Fusion had a national average price
of $20,962. Money earns 1.25% on a high yield two year CD. To lease costs $345 a month for
36 months plus $345 due at the signing, and $12,597 residual due after 36 months. Specify your
assumptions. Explain your calculations. Give the financial comparison and your conclusion.
Assume the car is driven ten years. Who pays the maintenance? Leasing may require higher
insurance costs, etc.
#19. In 2013, starting teacher’s salaries were up 5.3% to $44,928. Petroleum engineers started
at an average $93,500. With 3% step raises, how long would it take a teacher to catch up to the
beginning salary of engineers? If the petroleum engineer was laid off, he could always teach.
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What about merit raises and inflation raises? When do automatic step raises end? What is a
master’s degree worth? Liberal arts graduates averaged $37,058. (The Kiplinger Letter, April
2013)
#20.
What are the advantages and disadvantages of proof by Mathematical Induction?
#21.
How do you define a < b for integers.
#22.
Define the additive identity for integers. Why is it unique?
#23.
Define a – b for integers.
#24. Is the current bull market over- priced and getting old? In May, 2015: The market has
returned 23% a year on average since the bull market began on March 2009. It is in its seventh
year. A remarkable feat accomplished only three times in the past 85 years. Many experts say
we are about to see the bear.
Most evaluation is done by the Shiller-price-earnings ratio which is based on average
inflation and adjusted earnings over the past ten years. It is currently at 27.1 with a long term
average of 16.6, 63% over valued. Other measures: Price to trailing earnings, Price to
estimated earnings, Price to book value, are all at or above their averages. (Kiplinger’s Personal
Finance, 7/2015, Pages 44 to 50.)
(a)
Write a discussion of each of these ratios. See references such as finance.yahoo,
Wikipedia, Investopedia, and other sources such as Google.
(b)
Write a discussion of the history S&P prices and S&P total return back to the year 2000
from bottom to peak and peak to peak. See Wikipedia, finance.yahoo, and the article in this
course on the S&P 500 Index of Stocks.
(c)
Study the history of bull and bear markets.
#25. Take a typical 50 year old who earns $70,000 a year, saves at a steady clip of 10%, and
has $350,000 socked away in his 401k-the target for that age. Assuming standard 2% yearly
raises and average annual returns of 5%, he’ll amass $915,500 by 65.
If instead, he lowers his savings rate to 5% until his kids are raised and are out of college,
bumps up to 15% savings at 55, and to 25% at 60, he’ll have $980,000 by the time he retires at
age 65. Do the mathematics to check these results. (Money, August 2013, page 63)
#26. See bankrate.com, Payroll deduction calculator. Do an example. Total taxes are what
percent of your gross income? If your employer didn’t have to match, he could pay the match to
you. On this basis, taxes are what percent of your income?
#27. Use Formula 1 for the sum of a geometric series to prove Formula 3 for the sum of an
ordinary annuity. In high school algebra, Formula 1 is considered more general and is presented
before Formula 3. Give the time line for Formula 3 with the variables and the substitutions into
Formula 1 to arrive at Formula 3.
#28.
For Formula 1, discuss the possible meanings of the left and right side, if r =1or r = 0.
For Formula 3, if i = 0 what is the right side? If i = 0 what is the sum or an ordinary annuity?
Could i be negative? Discuss. In finance, what are the customary values for i?
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Side Bar Notes:
Thirty six percent of parents of graduating college students expect to support them. Fifty
percent of graduates expect help. (Fox News, 5-26-15)
The National Institute for Retirement Security found that 84 percent were unprepared
financially for retirement. (Denton Record Chronicle, May 24, 2015)
For Millennials, 18% say high earnings potential is important in a partner. Thirty two
percent of Boomers say it is important. (Money, June 2015, page 53) There is an average
difference of $28,000 in household incomes.
Americans are losing $24 Billion each year by not participating in 401k match employer
programs. (Fox News, 5-12-15) Participants in 401k programs were given the option of
increasing their savings by 3% each raise. (Money, June 2015, page 68) Their savings increased
from 3.5% to 13.6% after four raises. Check these figures.
By 2020, 16.8% of the population will be 65 and older. In 2010 the number was 13%. In
1970 it was 9.9%. Workers over 65 were the only group not to suffer employment declines
during the recent recession. (The Kiplinger Letter, April 2015). What will be the future trend?
How does this affect the economy and why? What do you recommend? Do you recommend
importing young people from Mexico?
Buy federal bonds direct from the Fed, thus avoiding the nasty spreads. Explain. Check
this idea with your bond dealer.
The world is your friend, investment diversification advice for the layman: Currently
52% US, 23 % Europe, 25% everything else. For sectors, 22% financials, 14% tech, 13%
consumer discretionary, 12% health care, 10% staples, 10% industrials, 8% energy, and 11%
everything else. (Forbes May 25, 2015) See finance.yahoo, investopedia, or Wikipedia for the
meanings of these terms.
Home ownership deductions of mortgage interest and property taxes are not a middleclass tax benefit. In a list of 175 metropolitan areas, 152 of them had estimated home ownership
deductions which were less than the standard deduction. In the 23 areas with home owner
deductions greater than the standard deduction, in some of these areas the median price of a
home is $855,000, thus large property taxes and mortgage interest. But this is not for the middle
class. By the time we are down to the bottom median home price for the 23 areas, the median
price is $265,000. (Scott Burns, Denton Record Chronicle)
Conflicted professional financial advice is costing investors $17 billion a year, said
President Oboma’s Council of Economic Advisors (Kiplinger’s Personal Finance, 5/2015, page
11). The Fiduciaries Principle: Investment fiducials are legally bound to act in your best
interest, even at the cost of their own, when giving advice or recommendations. Not all salesman
of financial products are legally held to this principle.
A young person and afraid to invest in stocks and are keeping your long term savings in
cash. For young people, consider that over periods of 30 years, stocks have never lost money,
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they and outperformed cash (short term Treasuries) by an annualized average of 7 percentage
points. (Money, April 2014, page 17)
Will kids be better off, or worse off than their parents? For parents, 45% say the kids will
be better off, 18% say kids will be worse off than parents. For kids, 54% say they will be better
off, 14% say they will be worse off. (Money April 2014, page71)
Will spoiled kids be able to support their parents? It is well known that a percentage of
young people are still depending financially on parents. Parents should do the math. Are they
developing adequate retirement funds? Spending too much on kids could mean that the kids will
be challenged to support their parents. A president of a major financial company said that
parents could be creating a generation of financially disabled people.
Nearly two-thirds of people 50 and older are helping family members, mostly
their adult children. Also about one-half of those 50 and older are not saving adequately for
retirement and cannot afford the average amount extended to family members. Do the math.
(Kiplinger’s Personal Finance, 12/2014, page 50)
What about the prospect of financially and mathematically educated children who
can be financially affluent and socially productive?
Small companies often offer an expensive 401k. If you are working for such a company
and don’t have a strong match, you’ll benefit by moving to an inexpensive IRA. Some 401k’s
charge as much as 1.98 percent expense ratio. Some IRAs charge as little and .05 percent
expense ratio. (Scott Burns, Denton Record Chronical, July 17, 2014) As Scott did, do the
research and the mathematics.
$4.5 billion in overdraft charges in big banks are expected by the end of the year. (CNN,
5-27-15)
Most residents in 9 out of 11 major cities were renters, that’s up from 5 cities. (Fox
News, 5-28-15)
Scott’s advice: Question: I am 62 years old. I have $400,000 in a bank wealth
management firm which charges 1% per year and earns less than index funds. But to sell and
reinvest will cost $9400 in taxes. I don’t need to spend the money. Does this sound like a good
way to go? Scott says that the tax will be recouped in three years by the lower expense ratio of
index funds. By Scott’s figures, what is the expense ratio on the new investment? Could it be
recouped in less than three years? (Scott Burns, Denton Record Chronicle, Sept. 14, 2014).
Happy spending. After reading “Happy Money: The Science of Happier Spending,”
Scott says what is important in happy spending is that none of it requires that you be rich.
Beyond a certain amount, more spending adds little to happiness.
Comparing Social Security to a life annuity. Research has shown that people who buy
life annuities get back 85 cents on the dollar. Life annuity costs varied by as much as 20%. Do
the research and the math. What about inflation increases in benefits? Social Security is similar
to a life annuity in that it pays as long as you live. A very-low-income female worker who
expects to retire at age 65, on average gets back $1.51 on the dollar. A maximum income worker
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will get back 55 cents on the dollar. (Scott Burns, Denton Record Chronicle, July 7, 2013) . See
the articles in this course on Social Security.
References:
For more formulas, see the articles in the following course. What kind of proof was
given for those formulas? See Google or Wikipedia for Mathematical Induction.
For a good approach to studying or teaching Mathematical Induction, see the course on
discrete mathematics at COMAP.
For a free course in financial mathematics, with emphasis on personal finance, for upper
high school and undergraduate college, see COMAP.com. Register and they will e-mail you a
password. Simply click on an article in the annotated bibliography, download it, and teach it or
study 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: Solving Financial Formulas for Interest Rate, For about thirteen more
advance or technical articles, see the UMAP Journal at COMAP. The last section is Additional
Articles on Financial Mathematics or Related to Personal Finance. In all, there are about eighty
articles.