The coin toss also introduces the difference between the average and the annualized return of an asset. However, these are extremely unlikely events, and the above formulation and graph is a accurate representation of the odds in your favor.) If you toss 16 straight heads then 19 straight tails you will still come out behind, but if you toss 27 straight tails followed by 8 straight heads you will actually come out ahead. The order of the coin tosses matters a great deal. (Actually, this is a bit of an oversimplification. What are the odds that you will flip less than 13 heads, and come out behind? Less than 5%. You visit your former college statistics professor, who chides you for forgetting that you could have easily calculated the odds of any combination of coin flips with the so-called "binomial distribution function." The blank look elicits a sigh from him, he heads over to his computer, pulls up a spreadsheet program, and after a few keystrokes hands you the graph in Figure 1-1: However, with some trial and error on your calculator you discover that you would have to get 12 heads and 23 tails before you come out worse than the first option, and you decide that the odds of this are quite low. Of course, you could have a string of bad luck and get tails more than half of the time. This is clearly superior to the 3% return of the first option. You again get out your calculator and find that a 17% return at the end of 2 years is the same as an annual return of 8.17%. For each dollar you had at the beginning of the 2-year period, you now have $1.17.) (The first year return of 30% results in your account being multiplied by 1.3, while a 10% loss multiplies your sum by 0.9. If you represent this with an alternating series of heads and tails, then your return in each 2-year period is represented by: Over a long enough period, you will get exactly half heads and half tails. Let’s look a bit more closely at the second choice. On the other hand, if you get heads all 35 years you know that you will bankrupt poor Uncle Fred with your gains-he will owe you $162,000,000! What if you have a string of losing years? If you get tails all 35 years, you could be left with only a pittance for your retirement. The thought of losing 10% of your hard-earned retirement money with the toss of a coin is too much to bear. In fact, if inflation is also 3%, you will be left with only $107,436 of current spending power. You realize that inflation will diminish the future value of this princely sum. You are adept with a financial calculator, and in a few seconds determine that this option will yield a sum of $302,310 with which to support your golden years. The first choice gives you a fixed rate of return, and in fact an absolutely certain lump sum at the end of your 35 years. This will be hereafter referred to as "Uncle Fred’s coin toss," or simply, the "coin toss." Heads you receive a 30% investment return for that year, tails a minus 10% (loss) for the year. A most peculiar option: At the end of each year Uncle Fred flips a coin. Certificates of deposit with a 3% annualized rate of return, or,Ģ. Further, you must pick ahead of time one of two investment choices for the duration of your employment:ġ. Each year he will contribute $5000 to your retirement account. You are 30 years old and will work for your uncle until you retire in 35 years at age 65. He is a conscientious and kind employer, and after some years in his service he decides to let you in on the company pension plan. Imagine that you work for your rich but eccentric Uncle Fred.
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