Ch. 2:  Math of Personal Finance

2.0 Introduction

  As you traverse through life, you're in a constant tug-of-war between the present and the future.  Suppose you need to study for three hours and you’re trying to decide whether to study tonight or tomorrow morning.  If you decide to study tonight, your “future self” benefits at the expense of your “present self”.  Conversely, if you go out for drinks tonight, your present self has a great time while your future self suffers.  Weighing choices through time is one of the most challenging aspects of personal finance (and life in general!) as most people are naturally myopic.  (Note:  I really wish I could get your future self to take my class, I bet they would care a lot more about personal finance.  But alas… anyone reading this is always in the present)

  Juggling the pros and cons of choices on the present and future selves is a key consideration in financial planning.  Spending money today almost always benefits the present self, often at the expense of the future self.  Before we can tackle this philosophical problem, we need to start thinking of money and time as a long-term problem and connect mathematical concepts to our choices.  Perhaps every finance book on earth has the following formulas… it’s that important.

2.1 Future Value Formula

  Imagine you have $100 that you will save in a simple savings account at a local bank.  The local bank offers a 2% interest rate (expressed as “i”) on this account.  Today (in “Year Zero”) your account will have $100 in it.  After one full year (in “Year One”), your account will gain 2% on its initial value, leaving you with $102.  This is expressed mathematically in equation 2.1.

Eqn 2.1. Value in Year1 = Value in Year0(1 + i)

 Value in Year1 = $100(1 + 0.02)

 Value in Year1 = $102

What if you decide to leave the money in the savings account for two years?  It’s tempting to think that you would now have $104 in your account, but this is not the case.[1]  At the beginning of year 1, your account now has $102 in it, so the interest will be paid on both the $100 in principal investment, but also on the $2 in interest.  This absurdly important concept is referred to as compound interest.  This is expressed mathematically in equation 2.2.

Eqn 2.2.  Value in Year2 = Value in Year1(1+i)

 Value in Year2 = $102(1 + 0.02)

 Value in Year2 = $104.04

…or we could think of it this way...

Eqn 2.3.  Value in Year2 = [Value in Year0*(1+i)](1+i)

Eqn 2.4.  Value in Year2 = Value in Year0*(1+i)2

 Value in Year2 = [$100(1 + 0.02)](1+i)

 Value in Year2 = $104.04

Conveniently for our purposes, the superscripted value in equation 2.4 matches the number of years that the investment is allowed to grow (this will continue to be the case).  To comprehend the compounding nature of interest, let’s allow this investment to continue marching along through time

                Value in Year3 ­­= Value in Year0*(1+i)3 = $106.12

                Value in Year4­= Value in Year0*(1+i)4 = $108.24

                Value in Year5 ­­= Value in Year0*(1+i)5 = $110.41

                Value in Year20 ­­= Value in Year0*(1+i)20 = $121.90

                Value in Year100 ­­= Value in Year0*(1+i)100 = $594.31

      Value in Year1000 ­­= Value in Year0*(1+i)1000 = $32,671,572,988.81

The information above shows the relentless power of time.  A measly $100 invested at a paltry 2% grows to nearly $33 billion dollars in 1,000 years.  This is why Albert Einstein once called compounding interest “the most powerful force in the universe.”[2]  Thankfully, you don’t have to be an Einstein to understand this.  Equation 2.5, which will be used hundreds of times in this class, shows the basic calculation for future value.  In this formula, FV is the future value of the investment at a future date, n represents the number of years that an investment is allowed to grow, i is the interest rate, and PV is the initial amount invested (the principal).

Eqn 2.5. FV = PV(1+i)n

From here, we can make a lot of simple, but important calculations.  For example, suppose that I am 22 years old and have $25,000 to invest in the stock market, which typically earns about 10% interest annually (more on this later!).  If I invest this money and don’t touch it until I am 67 (n=45), how much money can I expect to have in my account?

                FV = $25,000(1 + 0.1)45 = $1,822,262.09

It’s not so hard to be a millionaire if you have time!  This is why it is so important to save and invest money when you are young.  Your future self will thank you…    

2.3 Using the Future Value Formula for Other Purposes

  The basic future value formula expressed in equation 2.5 can be used for other purposes aside from solving for future value.  For instance, you might want to know how much you need to save today to achieve a future goal.  Or, you might wonder how many years it will take to become a millionaire.  Such scenarios are discussed below.  In each case, it is key to recognize that we can solve for any of the variables in equation 2.5 (FV, n, PV, i) as long as the other three variables are provided.

2.3.1  Solving for Present Value

I would love to buy a beach home when I retire at age 55.  Currently, I am 35.  I expect a beach house to cost $2 million when I buy it in 20 years. If the stock market returns an annual interest rate of 10%, how much do I need to invest today to reach my goal?

In this case, we know all variables aside from PV (FV = $2,000,000; n = 20, i = 10%).  By rearranging the FV formula, we can easily solve for FV...

                FV = PV(1 + i)n

...divide both sides by (1+i)n

                          FV/(1 + i)n = PV

…and voila...

Eqn 2.6. PV = FV/(1 + i)n

                                 …solving for the example question

                PV = 2,000,000/(1 + 0.1)20

                          PV = $297,287.26

So, we need to save $297,287.26 to reach our goals given the parameters provided.  The math isn’t so hard, is it?

2.3.2  Solving for the number of years (n)

I have $47,000 today to invest and don’t plan to save another penny in my lifetime.  I would like to know how many years it will take me to reach my investment goal of $1 million.  I expect the stock market to return 10% annually.

In this case, we know all variables aside from n (FV = $1,000,000; PV = 47,000; i = 10%).  By rearranging the FV formula, we can again solve for n. 

                FV = PV(1 + i)n

…divide both sides by PV

                          FV/PV = (1 + i)n

…take the natural long (ln) of both sides

                ln(FV/PV) = n*ln(1 + i)

…divide both sides by (1 + i) and rearrange

Eqn 2.7. n = [ln(FV/PV)] ÷ [ln(1+i)] 

Arriving at this formula was a bit tricky.  But, I’m not so concerned with your understanding of natural logs.  The goal of my class is to build practical skills, not determine your capacity to explain the meaning of a natural log. But you still need to be able to calculate the answer!  Now, let’s address the question provided above.  Be careful!  Mind your order of operations.  PEMDAS people![3]

 n = [ln(1,000,000/47,000)] ÷ [ln(1+0.10)] 

 n = 32.08 years

So, it will take 32.08 years to become a millionaire if the parameters provided in the question are correct.

2.3.3  Solving for the interest rate (i)

Stock market returns are unpredictable.  I am 52 years old and want to retire at age 60 (in exactly eight years).  I currently have $1,400,000 invested and need $1,750,000 to retire.  I don’t plan to save any more money.  What interest rate do I need to earn on my investment to reach my goal at age 60?

In this case, we know all variables aside from i (FV = $1,750,000; PV = $1,400,000; n = 8).  How can we solve for i? 

 FV = PV(1 + i)n

…divide both sides by PV

 FV/PV = (1 + i)n

…raise both sides by (1/n)

(FV/PV)(1/n) = (1 + i)

…subtract 1 from both sides and rearrange

Eqn 2.8.  i = (FV/PV)(1/n) – 1

Again, if you can solve for this value using just the basic FV formula (eqn. 2.5), this will make your life easier.  But, if that’s too hard, you should be able to at least use equation 2.8 to solve for i—just plug-in the necessary variables and solve!  We can now solve for the question above.

 i = (1,750,000/1,400,000)(1/8) – 1

 i = 2.83%

This person needs a return of at least 2.83% to reach their goal.  Only needing a 2.83% return, this investor may choose a conservative investment allocation.  Thus, this simple exercise helps the investor build a financial strategy!

2.4  Other forms of Compounding Interest

  To this point, all examples have assumed that interest compounds annually.  Annual compounding interest implies that the “interest clock” resets every year.  Suppose I save $100 in a savings account at 12% interest.  If interest compounds annually, the agents handling my account will allow these funds to grow linearly for the first year before re-starting the interest clock after year 1.   In other words, my account will gain $1 each month (1% per month = 12% per year) for the first year.

  But, after the first year, my account will now have $112 and future interest payments will be based on $112.  My account will now gain $1.12 per month for the full year.  This re-assessment of interest will again happen at year 2, year 3, and so forth.  This is annual compounding interest.  If I allow my account to grow for ten years, I can easily solve for it’s future value using equation 2.5.

 FV = PV(1 + i)n

 FV = 112(1 + 0.12)10

 FV = $347.85

  Often times, savings accounts are not compounded annually.  Instead, interest compounds each month.  Using the same example ($100 invested at 12%), let’s consider how this is handed when interest compounds monthly.  In the first month of investment, my funds are not compounded at all.  So, after one month of investing, I will have gained $1 in interest, leaving me with $101.  With monthly compounding interest, bank agents will not re-start my interest clock.  During month two, I will receive interest on both my principal ($100) and the first month of interest ($1).  Every month, my account’s interest growth will be reassessed based on the interest that has already accrued.  Since interest compounds faster, my account will grow more quickly.  Thankfully, there is a formula that will allow us quickly solve for important variables when interest compounds at various levels.  To do so, we need to understand a new variable—t signifies the number of times that interest compounds in a year.

Eqn. 2.9.  FV = PV(1 + [i/t])nt

Equation 2.9 provides another key formula that is used incessantly in personal finance.  Note that equation 2.9 and equation 2.5 are equivalent when t = 1; when interest compounds annually, these two formulas are identical.  Using 2.9, we can now quickly solve for simple, but important questions.  Earlier, we showed that $100 invested at ten years using annual compounding interest of 12% would result in a future value of $347.85.  Now, let’s see what our account would equal in ten years if interest compounds monthly and at other time intervals.

Semi-annually (t=2)     FV = 100(1 + [0.12/2])(10*2)         = $265.33

Monthly (t=12)               FV = 100(1 + [0.12/12])(10*12)          = $270.70

Weekly  (t=52)         FV = 100(1 + [0.12/52])(10*52)          = $271.57

Daily (t=365))           FV = 100(1 + [0.12/365])(10*365)        = $271.79

As indicated by these calculations, the more times interest compounds, the faster the money will grow.  When you are saving and investing, more compounding periods are better, but when you are the borrower, fewer compounding periods will be preferred.

2.5  Conclusions

  Chapter 2 provides our first taste of the key mathematical concepts that you will use throughout the semester (and hopefully, your life).  Comparing the relative merit of choices is difficult.  If I buy a Kanye West concert ticket today for $250, the cost and benefits attributed to my present self are immediately obvious, but it’s difficult to reconcile this choice against the costs that such spending will have on my future self.  By using the future value formulas provided herein, we can more easily assess the “damage” to my future.  For example, imagine that I could have invested this $250 in the stock market to save for retirement, expecting to ultimately spend this money in 40 years.  Using an annual compounding interest rate of 10%, we can find that this $250 cost today robs me of $11,314.81.  Should you go to the concert?  I’m not sure.  Kanye is f*cking nuts these days.  But at least we have created a better method for assessing the costs and benefits of the spending!  Much more analysis is needed (e.g. what about taxes?!), but this is a good start.

Endnotes

[1] If this were the case, the account would be paying simple interest rather than compounding interest.  Examples of simple interest in personal finance are extremely rare.

[2] Or maybe he didn’t say it.  Who knows?!  Source:  https://www.snopes.com/fact-check/compound-interest/

[3] PEMDAS refers to the order of operations when solving.  Always solve for the parentheses first, followed by exponents, multiplication, division, addition, and finally subtraction.  Having a good calculator makes this easier.

Key Terms: 

Annual compounding interest:  A practical (and common) scenario where interest is allowed to compound in 1-year increments. 

Compound Interest:  The concept that invested money grows super fast because the dollar-value builds not only on the principal but also on any interest that has already accrued.

Myopia: The tendency to overvalue the present relative to the future.  People who are myopic (most of us) often put their future selves in quite the pickle!

Interest Rate:  Financial compensation offered by a borrower to a saver.  When you borrow $10,000 to buy a car, you will need to fork over much more than $10,000 to repay the borrower.  Likewise, if you save $10,000, the interest rate determines how much these funds will grow over time.

Principal:  An initial amount invested before any interest accrues. Or, in the case of borrowing, the amount owed, ignoring interest.

PRACTICE PROBLEMS

For questions 1-7, assume that interest compounds annually.

1.      I save $4,000 in a savings account today.  The account earns 4% interest.  In four years, how much money will be in the account? 

interest rate = 4.0%. 

FV = PV(1+i)n

FV = $4,000(1 + 0.04)4

FV = $4,679.43

2. I need $50,000 to start a business in 10 years.  If I earn an interest rate of 6%, how much will I need to save today in order to reach this goal in 10 years?

PV = FV/(1 + i)n

PV = $50,000/(1 + 0.06)10

PV = $27,919.74

3. I need to have $250,000 saved in order to buy a rental property.  If I currently have $18,500 set aside, how many years will I need to wait in order to buy a rental property if I earn an interest rate of 10%?

n = [ln(FV/PV)] ÷ [ln(1+i)] 

n = [ln($250,000/$18,500)] ÷ [ln(1+0.10)] 

n = 27.32 years

4. I need to have $250,000 saved in order to buy a rental property.  If I currently have $18,500 set aside, how many years will I need to wait in order to buy a rental property if the general interest rate is 5%?

n = [ln(FV/PV)] ÷ [ln(1+i)] 

n = [ln($250,000/$18,500)] ÷ [ln(1+0.05)] 

n = 53.365

5. I need to have $1,750,000 to retire.  Currently I have $110,000 saved for retirement.  If I don’t save another penny, in how many years will I be able to retire if the interest rate is 6.2%?

n = [ln(FV/PV)] ÷ [ln(1+i)] 

n = [ln($1,750,000/$110,000)] ÷ [ln(1+0.062)] 

n = 46.00 years

6. I have $60,000 in a savings account, earning a general interest rate of 2%.  How long will it take for this money to double?

n = [ln(FV/PV)] ÷ [ln(1+i)] 

n = [ln($120,000/$60,000)] ÷ [ln(1+0.02)] 

n = 35.00 years

7. I have $1,000 in an interest-earning account.  I need to have $10,000 in the account in 10 years in order to travel to Europe.  What interest rate will I need to achieve this goal?

i = (FV/PV)1/n – 1

i = (10,000/1,000)1/10 – 1

I = 25.89%

...now, let's consider changes in the rate of compounding

8. I have $3,000 in credit card debt.  The current interest rate on this debt is 18.5%, compounded annually.  How much will I owe in three years if I never make any payments?

FV = PV[1 + (i/t)]nt

....this is annual compounding interest so t = 1, which explains why we can use the following formula for annualized interest!

FV = PV(1 + i)n

FV = $3,000(1 + 0.185)3

FV = $4,992.02 

9. I have $3,000 in credit card debt.  The current interest rate on this debt is 18.5%, compounded monthly.  How much will I owe in three years if I never make any payments?

FV = PV(1 + i/t)nt

FV = $3,000*(1 + 0.185/12)3*12

FV = $5,203.74

10.   I have $3,000 in credit card debt.  The current interest rate on this debt is 18.5%, compounded daily (365 times per year).  How much will I owe in three years if I never make any payments?

FV = PV(1 + i/t)nt

FV = $3,000*(1 + 0.185/365)3*365

FV = $5,225.09

11. I have $3,000 in credit card debt.  The current interest rate on this debt is 18.5%, compounded 1,000 times per year.  How much will I owe in three years if I never make any payments?

FV = PV(1 + i/t)nt

FV = $3,000*(1 + 0.185/1000)3*1000

FV = $5,225.55

12. You are opening a credit card and expect to carry a significant balance on the card.  One credit card offers a 15.5% interest rate, compounded bi-annually.  The other offers a 15.0% interest rate that compounds 360 times per year.  Ceteris paribus, which credit card is better?

This question doesn’t ask for specifics, but it’s pretty easy, really!  Choose a given amount of debt.  Let’s say $100.  Then, choose a certain amount of time… let’s say 10 years.  We can then compare…

FV of 15.5% bi-annually compounded = $444.99

FV of 15.0% interest rate compounded 360 times = $448.03

So, the 15.5% interest rate is better since the debt accumulates slower!

13. I need to have $1.3 million to retire in 40 years.  How much do I need to save now in order to his this goal exactly if interest compounds 6 times per year at 5.9% interest?

PV = FV/(1 + i/t)nt

PV = $1,300,000/(1 + 0.059/6)40*6

PV = $124169.48