Compound Interest and the Power of Saving
Compound Interest and the Power of Saving
The following write up is long, 15 pages, including finely detailed graphs, so you might be able to read it better and more comfortably if you make a high quality print. Please read it carefully, as first, there will be exam questions on it, second it contains instructions for what you are required to turn in, and third and most important, it contains very important material for helping you to build a secure and wealthy future.
As you will see in this assignment it is amazing how powerful saving can be. Even people with very modest incomes can eventually become millionaires – even adjusting for inflation – as long as they save steadily and consistently year in and year out.
As I said in the syllabus, if you save just $100 per week in a balanced diversified stock fund like the Russell 5000, and it ends up earning its historic average return of approximately 12%, then after 29 years you will have $1 million! And after 35 years you will have $2.2 million, but with inflation of 3%/year, this will be worth 1 million today dollars.
Why is compound interest, or return, so powerful? Why has it been called the eighth wonder of the world? Because it causes savings to grow exponentially, and the key to that is that you get interest on interest. For example, suppose that you invest $100 at 12%. What happens?
Year 1
You start with $100, your investment, and at the end of the year you get interest equal to 12% of what you started the year with, that is [pic].
So you get $12 interest, plus you have your initial investment of $100, so at the end of year 1 your account has $112.
Year 2
Now you start the year with your $100 initial investment, plus last year’s interest of $12 – and you will get interest on both! So this year your interest will be:
| |Interest Earned this year (Year 2) |
|On your Initial Investment |[pic] |
|On Last Year’s (Year 1’s) Interest |[pic] |
|Total |$13.44 |
So this year you get an extra $1.44 in interest, which is equal to your interest on last year’s $12 of interest. That’s where the expression “interest on interest” comes from. So now your account is worth your initial investment of $100, plus year 1’s total interest of $12, plus year 2’s total interest of $13.44, which adds up to $125.44.
Year 3
| |Interest Earned this year (Year 3) |
|On your Initial Investment |[pic] |
|On Year 1’s Interest of $12 |[pic] |
|On Year 2’s Interest of $13.44 |[pic] |
|Total |$15.05 |
This year you received 12% interest on the $100 you invested in the first place, plus 12% interest on your $12 of first year interest, which came out to $1.44. In addition, you received 12% interest on your $13.44 of second year interest, which came out to $1.61. It all added up to $15.05 in interest.
Year 4
| |Interest Earned this year (Year 4) |
|On your Initial Investment |[pic] |
|On Year 1’s Interest of $12 |[pic] |
|On Year 2’s Interest of $13.44 |[pic] |
|On Year 3’s Interest of $15.05 |[pic] |
|Total |$16.86 |
As you can see, all you did was invest $100 one time, yet every year your interest payment keeps getting bigger because you keep getting more and more interest on interest.
Year 5
| |Interest Earned this year (Year 5) |
|On your Initial Investment |[pic] |
|On Year 1’s Interest of $12 |[pic] |
|On Year 2’s Interest of $13.44 |[pic] |
|On Year 3’s Interest of $15.05 |[pic] |
|On Year 4’s Interest of $16.86 |[pic] |
|Total |$18.88 |
By year 5, because of all of the interest on interest you are getting, your total interest payment has increased from $12.00 to $18.88, a 57% increase in just 5 years! Notice that your interest on the initial investment portion never changes. That’s always $12.00. But the interest on interest portion always keeps growing.
It grows in part because you keep having more and more years of interest to get interest on, but also the size of the most recent interest on interest payment never stops growing. For example, we have seen it grow from $1.44 to $1.61 to $1.81 to $2.02, and it will continue to grow, and in an exponential way, eventually becoming far bigger than the $12.00 in interest on the initial investment portion. Now let’s see what happens in the ensuing years – over the long run.
|Year |Total Interest Payment for that Year |Amount of Money in the Account at the End of |
| | |the Year |
|0 (Initial Investment) |$0.00 |$100.00 |
|1 |$13.44 |$112.00 |
|2 |$15.05 |$125.44 |
|3 |$16.86 |$140.49 |
|4 |$18.88 |$157.35 |
|5 |$21.15 |$176.23 |
|6 |$23.69 |$197.38 |
|7 |$26.53 |$221.07 |
|8 |$29.71 |$247.60 |
|9 |$33.28 |$277.31 |
|10 |$37.27 |$310.58 |
|11 |$41.74 |$347.85 |
|12 |$46.75 |$389.60 |
|13 |$52.36 |$436.35 |
|14 |$58.65 |$488.71 |
|15 |$65.68 |$547.36 |
|16 |$73.56 |$613.04 |
|17 |$82.39 |$686.60 |
|18 |$92.28 |$769.00 |
|19 |$103.35 |$861.28 |
|20 |$115.76 |$964.63 |
|21 |$129.65 |$1,080.38 |
|22 |$145.20 |$1,210.03 |
|23 |$162.63 |$1,355.23 |
|24 |$182.14 |$1,517.86 |
|25 |$204.00 |$1,700.01 |
|26 |$228.48 |$1,904.01 |
|Year |Total Interest Payment for that Year |Amount of Money in the Account at the End of |
| | |the Year |
|28 |$286.61 |$2,388.39 |
|29 |$321.00 |$2,674.99 |
|30 |$359.52 |$2,995.99 |
|31 |$402.66 |$3,355.51 |
|32 |$450.98 |$3,758.17 |
|33 |$505.10 |$4,209.15 |
|34 |$565.71 |$4,714.25 |
|35 |$633.60 |$5,279.96 |
|36 |$709.63 |$5,913.56 |
|37 |$794.78 |$6,623.18 |
|38 |$890.16 |$7,417.97 |
|39 |$996.97 |$8,308.12 |
|40 |$1,116.61 |$9,305.10 |
|41 |$1,250.61 |$10,421.71 |
|42 |$1,400.68 |$11,672.31 |
|43 |$1,568.76 |$13,072.99 |
|44 |$1,757.01 |$14,641.75 |
|45 |$1,967.85 |$16,398.76 |
|46 |$2,203.99 |$18,366.61 |
|47 |$2,468.47 |$20,570.61 |
|48 |$2,764.69 |$23,039.08 |
|49 |$3,096.45 |$25,803.77 |
|50 |$3,468.03 |$28,900.22 |
As you can see, the annual interest payment, and the total amount in your account, really take off over a long period of time as the interest on interest grows exponentially. Graphically it looks like this:
[pic]
This is what I mean when I keep mentioning exponential growth, growth with a ski slope shape, that over a long period of time takes off.
I’ve gotten at a lot of the intuition why in talking about interest on interest, but another way to see why compound interest causes your investment to take off is to consider this:
In year 1 you get 12% of $100.00 – so your wealth grows by $12.
In year 2 you get 12% of a bigger amount, $112.00 – so your wealth grows by $13.44.
In year 3 you get 12% of a still bigger amount, $125.44 – so your wealth grows by $15.05.
And so on; you keep getting 12% of a bigger and bigger amount. Look at the graph above. By year 28 your account has grown to about $2,000, so in year 28, you’re getting 12% of $2000 in interest! Or about $240, as opposed to the only $12 in interest you got in year 1, so now your money is growing at a really fast rate. It’s growing at a rate of $240/year, as opposed to the $12/year rate it was growing at when you started out.
By year 50 your account has grown to about $29,000, so now you’re getting 12% of $29,000 as opposed to the 12% of only $100 you were getting when you started out. Now your money is growing at a rate of 12% x $29,000 = $3,480/year as opposed to the $12/year rate it was growing at when you started out!
In any case, whatever the mathematical intuition, the bottom line is the exponential growth from compound interest means that if you’re willing to save and invest for the long run, you can become very wealthy, and I encourage all of you to take advantage of this.
Let’s look at one more example. Instead of saving $100 just one time, as I mentioned in the syllabus let’s look at what would happen if we saved about $100 per week, that is $400 per month. The 12% interest rate, or return, we will be using, remember, is the historical average return on a balanced diversified stock portfolio like the Russell 5000. Here are the results:
|Year |Net Worth |
|1 |$5,107 |
|2 |$10,826 |
|3 |$17,232 |
|4 |$24,406 |
|5 |$32,441 |
|6 |$41,441 |
|7 |$51,521 |
|8 |$62,810 |
|9 |$75,453 |
|10 |$89,614 |
|11 |$105,475 |
|12 |$123,238 |
|13 |$143,133 |
|14 |$165,416 |
|15 |$190,373 |
|16 |$218,324 |
|17 |$249,629 |
|18 |$284,691 |
|19 |$323,961 |
|20 |$367,943 |
|21 |$417,203 |
|22 |$472,374 |
|Year |Net Worth |
|23 |$534,165 |
|24 |$603,371 |
|25 |$680,883 |
|26 |$767,695 |
|27 |$864,925 |
|28 |$973,823 |
|29 |$1,095,788 |
|30 |$1,232,389 |
|31 |$1,385,383 |
|32 |$1,556,735 |
|33 |$1,748,650 |
|34 |$1,963,595 |
|35 |$2,204,332 |
|36 |$2,473,959 |
|37 |$2,775,941 |
|38 |$3,114,160 |
|39 |$3,492,966 |
|40 |$3,917,228 |
|41 |$4,392,402 |
|42 |$4,924,597 |
|43 |$5,520,656 |
|44 |$6,188,241 |
|45 |$6,935,936 |
|46 |$7,773,355 |
|47 |$8,711,265 |
|48 |$9,761,723 |
|49 |$10,938,236 |
|50 |$12,255,931 |
Graphically it looks like this:
[pic]
Again, notice the exponential, ski slope, growth, how it starts out growing slowly and moderately, but then starts to take off. It takes about 20 years to save the first $500,000, but then it’s only about 9 more years to add another $500,000. At that point you have $1,000,000, so with the 12% return, your money’s growing at $120,000 per year, and the growth rate will just keep accelerating; now it takes only about 3 ½ years to make the next $500,000. By year 40 you have almost $4 million, from saving just $400 per month! But, of course, 40 years from now $4 million won’t be worth what it is today, as inflation will erode its purchasing power, so let’s look at how much your real wealth, adjusting for inflation, will grow when consistently investing $400/month.
The graph below shows how your real wealth grows in inflation adjusted today dollars assuming a common long term forecast for inflation of 3% per year.
[pic]
The growth is less when adjusting for inflation, but still you reach $1 million real today dollars in 35 years from saving just $400/month! That’s not even $100 per week! So everyone in this class, utilizing the power of compound interest can become a millionaire, and in real inflation adjusted dollars.
But what if you want to save $500 per month, starting after you graduate, and are earning college degree level compensation? Or what if you want to start out saving $400 per month, but then increase it by 20% every 4 years, to account for the fact that throughout one’s career compensation typically increases over time, with regular raises and promotions? Then, how will your wealth plan play out?
That’s what we are going to work on next. I have designed an easy to use wealth calculation Excel work book for you to help answer these kinds of questions, and do this kind of planning. You have full permission to keep this spreadsheet after the course is over for your future financial analysis and planning, and please feel free to let your friends and relatives copy it.
Even for someone who knows little, or nothing, about Excel, it shouldn’t be too hard to learn how to use the spreadsheets in this work book. Once you look at them you will quickly see what I mean.
Instructions are included at the bottom of each of the 3 spreadsheets, however if you really do know very little or nothing about Excel, I strongly recommend purchasing a copy of the widely available book “Teach Yourself Visually Excel 2003”.
This book is absolutely outstanding for beginners and intermediates. You will learn Excel far faster, easier, and more enjoyably from this book than the typical ones out there. It contains clear beautiful full color step by step illustrations of all of the most important tasks in Excel. It’s a great investment. Excel is a very widely used and important program in the professions that most of you will go into. Most of you will also have to use it in future courses. So, reading this book now, and having it as a reference, can make things much easier and more enjoyable in the future.
Personal Financial Forecasting, Analysis, and Planning
Please first take a look through the wealth calculation Excel work book; read the instructions at the bottom of each of the 3 spread sheets, and try entering different numbers in the yellow boxes and observing the results. Basically, just familiarize yourself with the work book.
Next, please perform the following personal financial forecasting, analysis, and planning, which includes filling out the key findings table below:
Step 1: Estimate what your monthly take home pay will be when you graduate and start your first job. As you will see, this is not as hard as it may sound to some of you, and even a ball park estimate will allow for some very useful and instructive analysis and planning. First, estimate your before tax pay. A great place to start is the University of Arizona Annual Starting Salary Survey. This is a self reported survey of the starting salaries of newly minted U-A graduates. Note that even though, unfortunately, they weren’t designed to look like links, you can click on the major areas to get a listing of sub areas. For example, if you click on Eller College of Management, you will get listings for 11 sub areas from Accounting ($41,618) to Public Management ($36,745).
These averages do come from people who have gone through the effort to self report their starting salaries to the University of Arizona Career Services Center, so the figures are probably bigger than the true averages, as people who received poor offers are going to be less enthused about self reporting them to Career Services, while those who received large ones tend to be happy to report the good news. As a result, you may want to adjust these estimates down by 10 – 20%.
At the same time, if you have an unusually high GPA, or an otherwise unusually strong resume, then you may want to adjust them back up 10 – 30%, plus the survey includes only salary, and not other forms of cash compensation like bonuses, so if your career is known for paying substantial bonuses you should probably add an estimate of this on. Basically, though, as I said, a ball park estimate will still allow for some very useful and instructive analysis and planning, so don’t worry about getting it perfect.
Another survey you can look at, which examines newly minted college grads nationwide, is The National Association of Colleges and Employers (NACE) Salary Survey, also available from U-A Career Services.
Basically, just use your best judgment – after studying the above surveys – and otherwise researching your career thoroughly in coming up with an estimate of what your after graduation starting compensation will be.
For those non-traditional students; you already know your current compensation, but still, please get the U-A starting average from the Career Services Survey and enter it into the appropriate space in the key findings table below. First, it can be useful to know what the current going wages are like in your area, and second, for this exercise I’d like you to use the salary that you estimate you will be earning once you complete your degree, which should be significantly higher, and the Career Services Survey should still provide some clues even though it includes predominantly traditional students.
An important aside is we have an extremely useful career services center, and web site. You’ve already seen that it contains some very nice compensation surveys, but there’s also a wealth of other information you can use in your career research and planning, both for your assignments, and for your life in general. I strongly encourage you to explore their site, and to stop by for a visit with a counselor. In their words:
Career Services is charged with assisting all UA students in making career plans, obtaining career-related experience and in finding permanent employment upon graduation. This assistance is provided through individual counseling, workshop and classroom presentations, special events that bring together employers and students and the vast resources contained within our office.
Step 2: Use your starting compensation estimate along with the wealth calculation excel work book to complete the key findings table below. As I said, the work book is largely self explanatory, but still, there are instructions available at the bottom of each of the 3 included spread sheets. Again, if you have little knowledge of Excel, I recommend purchasing the book “Teach Yourself Visually Excel 2003”.
Once you have filled in the key findings table, please copy it into a separate Word document and submit it. Although this is all you will turn in, it is still important that you learn the material in this write up, and in the Excel work book, thoroughly, as there will be related questions on the final exam.
One other Excel spread sheet I’d like to give you is a financial calculator. Although you will not need this for the requirements of this course, you may find it very useful in the future. If you go on and take the finance departments first term course, FIN 311, you will learn to use a financial calculator, most likely the Hewlett Packard 10b II.
Financial calculators are widely used in the business world, but they have some substantial draw backs, and I consider them largely obsolete. The biggest problem is that they are relatively difficult to learn how to use, and to remember how to use if you don’t work with one regularly. The financial calculator I designed in Excel is far easier to learn and remember how to use.
An analogy is the old MS-DOS software used on PCs in the 80’s as opposed to the Windows software of today. MS-DOS required rote memorizing code like commands. These commands were painstaking to learn, and if you didn’t use them regularly, you would tend to forget them rather quickly. With the advance of Windows, however, software was far more self explanatory and intuitive. You could learn how to do things much more easily, and remember, because they were a lot more self explanatory, and more obviously made sense than a bunch of gibberish code like:
@echo off
prompt $p$g
comspec=c:\dos\
path=c:\;c:\dos;c:\bin;
cls
ver
Note also that this kind of thing is much more picky and error prone, as if you forget a colon or slash you can get very wrong results.
The same things are true when comparing a financial calculator to my financial calculator spread sheet. The calculator requires rote memorizing some relatively code like commands that are picky. Little deviations can result in a wrong, or very wrong, answer. With my spread sheet, once you know what an annuity or a present value is, it’s obvious how to calculate them. There’s really nothing to memorize. And with Excel you can easily save or print your results, not so with a calculator.
The only real benefit to a calculator is its portability, and it’s a bit faster for calculating the simplest things, but important financial analysis is usually done sitting at a computer anyway, and laptops are already pretty portable, with super portable tablet computers soon to come, so Excel really is the future.
My Excel calculator currently has the basic financial functions, but in the near future I should be upgrading it to do all of the financial functions of a business calculator, but in addition, it will generate for you very nice tables and graphs. At that time I will post it on my web site: , where you are free to copy it.
I encourage you to learn more about its functions, as well as much other useful finance, by taking FIN 311. There currently is not a second term personal finance course, but I think there definitely should be. If I can find the time, and get university approval, I’d like to teach one in the near future. Such a course would include learning these financial functions.
Below is the key findings form you are required to turn in.
Good luck!
|Personal Financial Analysis and Planning: Key Findings |
|Section 1: Calculation of your forecasted monthly after tax starting pay |
|Average annual starting salary from The University of Arizona Annual Starting Salary Survey. | |
|Average annual starting salary from another source such as The National Association of Colleges and Employers | |
|(NACE) Salary Survey (optional) | |
|Your forecasted annual starting pay, including all forms of cash compensation such as bonuses and commissions | |
|Your forecasted tax rate including all taxes, federal, state, local, etc. (if you are unsure, just use 35%. That| |
|should be fine for a ball park estimate) | |
|Your forecasted annual after tax starting pay | |
|Your forecasted monthly after tax starting pay | |
|Section 2: Future Wealth |
|A. Saving 20% of your forecasted starting after tax pay every month |
|Value of savings after: |5 years | |
| |10 years | |
| |20 years | |
| |30 years | |
| |40 years | |
| |50 years | |
|B. Saving 20% of your forecasted starting after tax pay initially, but then increasing your monthly savings by 20% every 4 years. |
|Value of savings after: |5 years | |
| |10 years | |
| |20 years | |
| |30 years | |
| |40 years | |
| |50 years | |
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