Numbers and Their Meaning

Numbers and Their Meaning

Use basic math to count, quantify, and understand relationships

John bought all the stock he could since it was priced at only $1.

Something is only cheap or expensive in relation to something else. If an investor refers to a stock as "cheap," what is it cheap in relation to? 100,000 shares of stock priced at $1 have the same market valuation as 1,000 shares priced at $100. The relevant question is: What is the company worth in relation to its price?

TransCorp made a "huge" $1 billion in profits.

Words like "big" or "small" have no meaning in themselves. A number has only a size in relation to another number. $1 billion says nothing about economic performance unless we compare it with how much capital was needed to generate it. What if TransCorp needed $100 billion in equity and debt to run the business? That's only a 1% return.

"Research shows that the new drug reduces the risk of getting the disease by 25%." What benefits can be expected by taking the new drug? Assume statistics show the following: Without the drug 20 people in 1,000 get the disease. By taking the drug, the figure drops to 15 people in 1,000. How efficient is the drug if we look at the absolute number of people saved from the disease? The reduction in absolute risk is 5 people in 1,000 (20-15) or 0.5%. The reduction in relative risk or the relative number of people saved from the disease is 25% (5/20). A 25% relative reduction needs to be read alongside the baseline risk and the absolute benefit.

How many people need to take the drug in order to save one individual from the disease? There are 5 fewer cases per 1,000 people treated (or 1 per 200), so the number needed to treat is 200 over the study’s follow-up period. On average, one case is prevented and the other 199 people receive no additional prevention of this disease. These figures do not tell us in advance who benefits.

How do we know that research findings are true? Surveys of epidemiological studies (studies of factors affecting health and illness) show that many researchers' claims don't replicate in medical trials because of bias and random error. The author points to disputes over claims that aspirin is highly protective against heart disease in both men and women and that vitamin E reduces cardiovascular mortality. Studies by Hygiene and Epidemiology Professor John Ioannidis show that the smaller the studies; the smaller the effect sizes; the hotter the field; the greater the interest; the more opportunities to search large databases; and the greater the flexibility in analyses, the higher the chance that a research claim is false. True replication is the cornerstone of science, and assuming that biases are weeded out, the more researchers reproduce a finding, the better chance it has of being true.

What does it mean that our sun is 93 million miles away?

Often we need to translate a number to something more understandable. Light travels at about 186,281 miles per second in a vacuum. This means that it takes sunbeams about 8 minutes to reach the earth. The star Alpha Centauri is 4.35 light-years away. A light-year is the distance light travels in one year. A light-year measures distance, but it also tells us about light-travel time as measured from Earth. Light from that system takes about 4.35 years to reach us, so we see it as it appeared about 4.35 years ago.

Always look at what numbers mean. For example, ask: Does the magnitude make sense? In relation to what? Also, think about what counts - not how it is counted. There was a sign hanging in the physicist Albert Einstein's office at Princeton that said: "Not everything that counts can be counted, and not everything that can be counted counts."

The effect of exponential growth

John's son David made a proposal: 'I take out the garbage every day for a month, and you only have to pay me a penny today and then every day afterward I want double as much as I got the day before.'

The sequence of numbers 2, 4, 8, 16 grows exponentially. Day 2 his son would have 2 cents, day 3, 4 cents. After 27 days, his cumulative pay would be about $1.3 million. The individual growth is constant - 100% a day- but the sum gets higher faster and faster. This is the power of doubling.

As we have seen there are limits to prolonged growth. Take bacteria as an example. Assume that a certain strain of bacteria divide in one minute. We put the bacteria in a bottle at 11 am and the bottle is full at noon. When was the bottle half full? - 11:59 am. Just one minute earlier.

Even a small, steady growth rate leads eventually to doubling and redoubling. For example, a country whose population grows by 2% a year, doubles in size in 35 years and redoubles in 70 years. A simple formula for doubling time is found by dividing 70 by the percent growth per year.

Compounding refers to "interest on interest." If we invest $1,000 with a return of 6% a year, we receive $60 in the first year. If we reinvest that $60, next year we get another $60 from our original $1,000 investment, plus $3.6 from the $60 we reinvested. If we reinvest all our returns, the approximate total value of our original $1,000 investment after 5 years is: $1,000 x 1.06 x 1.06 x 1.06 x 1.06 x 1.06 = $1,338.

Time is the key to compound interest. Over short periods, compounding produces a little extra return. Over long periods, it has an enormous effect. Investing $2,500 each year for 40 years at a constant 10% annual return would produce more than a million dollars.

The time value of money

"A bird in the hand is worth two in the bush." - Aesop

Why must we reduce the value of money we receive in the future?

At a positive discount rate, money paid in the future has a lower present value than money paid today. If we have a dollar today, we can invest it and earn interest making that dollar worth more than a dollar in the future. This means that money has a price and that price is interest.

How much should we pay today for the right of receiving $1,000 a year from now? Or, how much do we need to invest today in order to have $1,000 a year from now? It's the same question. The answer depends on the interest rate. If the rate is 6%, then the answer is approximately $943. Investing $943 today at 6% produces approximately $1,000 one year from now. Thus, $943 is the rounded present value of $1,000 a year from now. We have discounted or reduced $1,000 to its value today. The further out in time we receive the $1,000 or the higher the interest rate is, the less the present value is.