Can compound interest interest make us ALL rich?
Time is Money
In theory, a finite amount of money in a bank savings account can draw interest forever. As the interest compounds, the bank balance will grow exponentially. At an infinite time, the original amount of money (no matter how small) will grow to an infinite bank balance.
The problem with this equivalence is that it is very difficult to make predictions. Especially about the future. There is no investment that grows exponentially forever. Bank failures, currency failures, and other financial calamities intervene, rendering the miracle of compound interest something more like a temporary myth. But one of the interesting things that follows from the infinite compounding problem is the realization that a promise to make a perpetual (i.e., forever) stream of payments has a finite present value. That is, if a finite bank balance could hypothetically result in an infinite future stream of payments, then it stands to reason that an infinite stream of future payments might only be worth a finite bank balance.
The equations that govern these relationships are simple. In fact, they are called “simple interest”:
A = P*i
- A ($/yr) is the annual interest payment,
- P ($) is the balance in the account at present, and
- i (%/yr) is the annual interest rate.
With a little rearranging, we find that a perpetual stream of payments is equivalent to a finite present account balance, according to:
Trading in the financial markets are based entirely upon differences of opinion about the future. Maybe you think a certain stock is a bargain and I think it’s over priced because we have different ideas about the future earnings prospects of the corporation. So we agree to make a trade: your cash for my stock.
Given the enormous flows of money thru the financial markets, there must be a great deal of disagreement! But how can we all have different ideas of what things are worth? Even though the financial statements are audited, and the larger public companies receive intense scrutiny from analysts, and earnings estimates are widely available, there are lots of things that people might disagree about when it comes to the worth of a company. But the largest single financial market is US Government bonds. Here, there is no default risk and the payment terms are exactly specified. There is zero uncertainty in how much money will be paid and when, so why is there so much trading?
The answer is in our understanding of time. While trades are made in the present, the traders always base their understanding of value on expectation of future events, and this is where the disagreement lies. People are (generally) not willing to pay a full amount now for promises in the future. In fact, we discount future cash flows when we contemplate what they are worth to us today. And there is a tremendous difference among different people with regard to personal discount rates.
Discounting the Future
An interest rate is an amount paid (or collected) on a loan. A discount rate is used to understand the present value of future cash flows. That is, interest rates are prospective — looking forward in time. But discount rates are used to understand how future cash flows should be valued in the present, which is looking backward (albeit from a future date) in time. This video introduces the term discount rate:
Sal Khan shows how the mathematics of discounting future events is simply the reciprocal of the interest calculations described in previous videos. Expressing all future cash flows in terms of present value allows the analyst to make a comparison among alternatives that is based upon their personal time preferences (i.e., discount rate). Thus the analyst is able to select (or recommend) the alternative most valuable to them.
A different analyst may get a different answer, using a different discount rate (which Khan calls the great “fudge factor” of finance in the previous video) and consequently, the two analysts may be motivated to trade alternatives, each thinking that they got the better of the other. This next video demonstrates how the present value of different cash flow alternatives is sensitive to the discount rate. (Listen closely, and you can hear Khan use the term “interest rate” when he meant “discount rate”. This is incorrect! But because these two rates are used in the same way in the same mathematical formulas, Khan’s mistake is repeated by many experts. Some authors use these terms interchangeably, which only creates confusion).
The choice of discount rate is most important when considering cash flows far into the future. Remember that the relationship between present value and discount rate is non-linear. That is, the amount of time appears in the exponent of the formula for calculating the discount factor. Here’s the formula for converting a future cash payment to PV, where r is the discount rate (per compounding period, such as per month or per year), and n is the number of compounding periods.
PV = F / (1 + r) ^ n
The discount factor is this portion of the formula: (1+r)^n. For small n, the difference between a discount rate of r = 0.01 and r = 0.04 seems pretty small. But over longer periods of time, this gap gets very big!
[Note: A different and more convenient method of discounting is to use the analog of continuously compounding interest, which is called exponential discounting and is given by the formula PV = F exp (-rt) ].
In this next video, Khan makes the case that events that are farther away should be discounted at different rates than those that are close to the present. This is common practice in finance, and it corresponds to the fact that far-future events are understood to be riskier (less certain) than near-term events. Thus, Khan demonstrates that different discount rates can be applied to events that happen at different times, with longer-term events discounted much more heavily — both greater n (more compounding periods) and greater r (higher discount rate). However, the effect of this is to discount future events considerably more than near-term. In environmental decisions, where some consequences might not be felt until decades after a decision, applying financial discounting in this way can result in myopic choices.