1.9: Understanding the Time Value of Money

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The Miracle of Compound Interest

No one who understands the miracle of compound interest better than Warren Buffett, a multi-billionaire and Chair of Berkshire Hathaway, best known as the Sage of Omaha. [Nebraska] and Chair of Berkshire Hathaway. An article in The Wall Street Journal by Jason Zweig (8/28/20) details Buffett’s thinking about time and the value of money. Patience and endurance are the “investing superpowers” that helped him achieve his $82 billion of personal wealth:

From the earliest age, Mr. Buffett has understood that building wealth depends not only on how much your money grows, but also on how long it grows. Around the age of 10, he read a book about how to make $1,000 and intuitively grasped the importance of time. In five years, $1,000 earning 10% would be worth more than $1,600; 10 years of 10% growth would turn it into nearly $2,600; in 25 years, it would amount to more than $10,800; in 50 years, it would compound to almost $117,400 (2020).

Because we will be discussing the time value of money, we will inevitably be discussing math in this chapter. However, it is not advanced math, so you should find it easy to understand.

The Time Value of Money

Receiving a payment today is better than receiving one a year from now, in part because the general rate of inflation (e.g., two percent) makes the money worth less next year due to decreased purchasing power (that is, two percent less). It is also better if you are going to save or invest that money, putting it into a savings account at 3% interest (for example) means that in a year you will have an additional earning of 3% on top of the original amount. We can illustrate it in this equation:

$\begin{align*} \text{P}_{year2} = \text{P}_{year1} \times 1.03 \end{align*}$

P_year1 is the principal amount you received at the beginning of year one, and P_year2 is the principal amount you will have at the beginning of year two, including the interest you earned. However, interest or dividends money in savings or an investment is compounded. That is, if you leave the interest or dividends you earned over year one in savings for year two, you will again receive 3% interest on the principal plus 3% interest on the interest you already earned in year one. We can represent this mathematically as follows:

$\begin{align*} \text{If P}_{year2} = \text{P}_{year1} \times 1.03 \end{align*}$

$\begin{align*} \text{Then P}_{year3} = \text{[P}_{year2} \times 1.03] \times 1.03 \end{align*}$

$\begin{align*} \text{Substituting, we have} \end{align*}$

$\begin{align*} \text{P}_{year3} = \text{[P}_{year1} \times 1.03] \times 1.03 \end{align*}$

Let’s say that you put $1,000 in a savings account at 3% interest and leave it to compound. Below, you can see the amounts you will have at the beginning of each year:

$\begin{align*} \text{Beginning of year 1: \$1,000.00} \end{align*}$

$\begin{align*} \text{Beginning of year 2: \$1,030.00} \text{[\$1,000.00} \times 1.03] \end{align*}$

$\begin{align*} \text{Beginning of year 3: \$1,060.90} \text{[\$1,030.00} \times 1.03] \end{align*}$

$\begin{align*} \text{Beginning of year 4: \$1,092.72} \text{[\$1,060.90} \times 1.03] \end{align*}$

The compounding of the interest may not seem like a lot here, but it makes a huge difference when you are saving for retirement. For another example, let’s say you start working at 21 and retire at 68, spending 47 years in the labor force. As we will discuss in more detail later, one investment in a mutual fund with a widely diversified portfolio saw a return of an average of 10.1% per year for ninety-four years. If you were to invest $1,000 in this diversified portfolio and did not touch it for 47 years, you would have a retirement nest egg that looked like this:

Original Amount: $1,000.00 at beginning of year 1
Interest Rate: 10.1% compounded
Time Period: 47 years
Amount at end of 47 years: $92,045.80

Furthermore, you will most likely deposit more into your retirement account each year, rather than just $1,000 once at the beginning of your career. If you invested $1,000 per year each year in this diversified stock portfolio, at the end of your career your nest egg would look like this:

Principal Amount: $1,000.00 each year invested at the beginning of each year
Interest Rate: 10.1% compounded
Time Period: 47 years
Amount at end of 47 years: $1,084,535.20

Most of the time, if you work for a good employer, they will sponsor a 401(k) retirement plan and match your contributions. The most common plan is that you contribute 3% of your salary, and your employer matches. Let’s say that together you contribute $4,000 per year for 47 years and put it all in a diversified stock portfolio. In that case, here is your retirement nest egg:

Principal Amount: $4,000.00 each year invested at the beginning of each year
Interest Rate: 10.1% compounded
Time Period: 47 years
Amount at end of 47 years: $4,338,140.81

This is also tax free until you retire and withdraw money to live on.

Intertemporal Consumption and Savings

Saving and borrowing allow intertemporal consumption. Basically, you move your consumption from one time period to another. If you do not spend all your income in year one, your savings can increase your consumption in later years. On the other hand, if you spend more than your income in year one (by using credit cards or taking out a personal loan), you must consume less than your income in subsequent years to pay back your debt. As the prime example of this, saving money for retirement each year means you are consuming less currently in order to have money for retirement. However, you are also earning interest or dividends that will allow you to consume even more than the original amount when you reach retirement.

Dr. Franco Modigliani, Nobel Prize winner in economics, explains our consumption and saving decisions over a lifetime with the life cycle hypothesis. Until college graduation, when we have student loans, we are dissaving; that is, we are consuming more than our income and financing it with student loans. After we begin our career, we consume less than we earn because we are saving for retirement. Finally, when we retire, we are once again dissaving by spending the retirement savings we have built up.

The Future Value of Dollars Received Today

The mathematical formula for the amount of principal at the end of n periods is:

$\begin{align*} \text{A} = \text{P}(1 + \frac{\text{r}}{\text{n}})^\text{nt} \end{align*}$

$\begin{align*} \text{P} &= \text{Principal amount} \end{align*}$

$\begin{align*} \text{r} &= \text{annual nominal interest rate (written as decimal)} \end{align*}$

$\begin{align*} \text{n} &= \text{number of times the interest is compounded per} \end{align*}$

$\begin{align*} \text{t} &= \text{number of years} \end{align*}$

However, you do not need to work out the future dollar value by hand. The internet offers lots of calculators that will do this for you.

The Present Value of Dollars Received in the Future

The present value of future dollars is called the Net Present Value (NPV), and it involves the economic principle of Opportunity Cost. The opportunity cost is the next best use for your money instead of your current purchase, or it can be the next best use of your time instead of what you are using it for now. For example, the opportunity cost of paying college tuition could be giving up on buying a new car. The opportunity cost of going to class could be getting a few more hours of sleep. The opportunity cost of not having a specific amount of money this year instead of next year is the interest or dividend you earn through investment.

This concept is important in business because the principal way a business can value an investment is the stream of income the investment throws off, discounted to the present. This is called Net Present Value of Discounted Cash Flow.

What interest rate (or discount rate) should you use to discount future streams of income? As a student, your opportunity cost would most likely be the 2% interest you would earn in a savings account. For business, the discount rate used is most often 8% or 10% per year because this is the return they would get by investing in their business if they had it now instead of later.

If a company buys an investment that generates $100,000 per year for ten years. The discounted cash flow or net present value of this cash flow stream is

Amount Per Year: $100,000.00
Discount Rate: 8%
Time Period: 10 years
Discounted Cash Flow: $724,688.78

The mathematical formula for Net Present value is:

$\begin{align*} \text{NPV} = \sum_{t=1}^{n} \frac{\text{R}_{t}}{(1+i)^{t}} \end{align*}$

$\begin{align*} R_{t} &= \text{Net cash inflow during a single period } t \end{align*}$

$\begin{align*} i &= \text{Discount rate or return that could be earned in alternative investments} \end{align*}$

$\begin{align*} t &= \text{Number of time periods} \end{align*}$

Note that we are dividing the cash flow or income from each period by 1 plus the discount rate, so this is reducing the cash flow by the discount rate. There are many online calculators that you can use to calculate the Net Present Value of future cash flows.

The Present Value or Future Value of an Annuity

An annuity is simply a stream of money paid periodically. It could be interest from a savings account or dividends from a stock investment. The present or future value of an annuity can be calculated using the present or future value calculators presented above.

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