FIN 19 Calculating the Rate of Return

Let's examine problem three, five, a rates of return. So we know the formula that we've been using this whole chapter, present value times one plus the rate of return raised to the power of the number of periods, uh, equals the future value. And so we're doing lump sums here, right? This is just one chunk of money in the present, one chunk of money in the future. In future questions, this chapter, we'll look at frequent cash flows and annuities they're called, but here it's just one chunk. So, so one of the most famous baseball cards is the T206 Honus Wagoner card. I urge you to Google it. It's a very famous baseball card. A mint condition copy was sold in 1933 for $50. By the way, $50 in 1933, it's like a ton of money, uh, and was the most expensive baseball card in the world at the time in 1991, a group, including professional hockey legend, Wayne Gretzky, uh, if you're Canadian, you definitely know who Wayne Gretzky is. Very famous ice hockey player, bought a mint condition copy of the famous card for $451,000, four years later in 1995, they sold the group to a, uh, sold the card to a group led by Walmart for $500,000. So these are famous sports cards bought and sold by famous people. The same card sold to another collector in 2006 for $2.6 million. So the question says, what was the rate of return on the card from 1933 to 1991? If you bought the card for 50 bucks in 1933, and then sold it to Gretzky in 1991 for $451,000, what was your return? So we're solving for R. Okay. Let's punch in what we have today is 1933. We buy the card for $50. Times one plus R raised to the power of T. Well, how many years is it? 1933 to 1991, 70, 60, 58 years, 58 years doing the math in my head. Always hazardous to a guy's health. 1991 minus 1933. Yeah. 58 years. Okay. Raised to the power of 58 equals. So we bought it for 50, 58 years go by and we sell it for $451,000. What is our annualized rate of return or compounding rate of return here? Divide 451 divided by 50. So we get one plus R raised to the power of 58 equals 451 divided by 50, 90, 20. Okay. So one plus R to the 58th equals 90, 20. We have to raise these to the power of one over 58. You have to take the 58th root of these. So to the power of one over 58 to, uh, uh, get that working. So again, we're just getting rid of this exponent. So we raised both sides to the power of one 58, one divided by 58. I need it as a decimal is 0.01724. So I get 90, 20 to the power of 0.01724 equals one plus R. 90, 20 to the power of 0.01724. I get 1.17, almost exactly. It's unusual to have a number so close to an exact number. R equals 0.17, R equals 17%. So in answer to a, what's the rate of return of the card from 1933 to 1991? It's a crazy example, right? This is an unusual type of investment, but there you go. You would have earned an annualized return of 17%, way outperforming any stock market over the same time period. I'm sure. Um, what did the Gretzky group get on the card? Uh, okay. The Gretzky group is more human terms, buys the card for 4.51 in 1991. And then four years later, they sell it for 500 and they didn't get a great return. So, uh, let's do the math here. So the PV 4.51 times one plus R to the power of how many years? Four. So four years later, um, they sell it for 500. Okay. Divide both sides by 4.51. So 500 divided by 4.51, 1.1086, 1.10865. I'll throw that in. So one plus R to the power of four equals 1.10865. One plus R equals 1.10865 to the power of one fourth, sort of same thing we did here doing one 58th. Here we do one fourth. So, uh, 1.10865 to the power of 0.25, right? One fourth is 0.25. So I get 1.02612. So one plus R equals 1.02612. R equals 0.02612. R equals 2.612%. Okay. So, uh, the Gretzky group did not earn a great return 2.612%. It would have been better buying, uh, an S&P 500 index or something like that. Let's just, I want to show this in the financial calculator, just in case you were curious about that. So, uh, let's do the Gretzky group. We won't do them all in the financial calculator. We'll do the last one by hand, but let's just do this one in our financial calculator. Make sure we know how to do that. Um, make sure everything's clear. Our, uh, present value here is 451,000 negative. We make a 451,000 investment to get $500,000 in the future. That's our F fee. We enter the N as four. Our payment is zero. And then we compute the IY and you can see 2.612. And that's the same answer. So I just wanted to show how you would do that in the calculator. And we do produce the same answer. Let's do the last piece. It says, what's the total rate of return from 1933 to 2006? Okay. 1933, the PV of the card was 50 times one plus R raised the power of the number of years. 1933 to 2006, 73 years. 73 years raised the power of 73. And in that year, it sells for $2.6 million. So we get a rate of return here. We'll solve for our, so divide both sides by, uh, 50, 52,000, this group by 52,000 times, right? The value didn't double, didn't triple at 52,000 times itself. That's what long time periods and high growth will do for something. Uh, so one plus R to the 73 equals 52,000. We have to take each side and raise it to the power of one 73rd. One divided by 73 is 0.0137. We're going to call that 0.0137. One plus R equals 52,000 to the power of 0.0137. So 52,000 to the power of 0.0137, 1.1604. So R equals 0.1604, R equals 16.04%. There we have it. We've solved part A, 17% is the return, uh, from 1933 to 91. The Gretzky group did very poorly on the card, only earning 2.6% over their short ownership, but over the life of the card, well, from 1933, it's an older card than that, uh, from 1933 to, uh, uh, 2006, if you had owned it the whole time, you would have got 16% return on your money. That's our answer to part C. All right. Kind of a long one. An interesting one though, sports cards, at least they interest me. And if they interested you, hit me with a thumbs up. Thanks for watching. Have a great day. See you in the next video. Bye bye.