When calculating your capital asset pricing model, one problem we have is trying to find the risk premium. Our risk premium here could be your equity return minus your T bond or treasury bond return. That's your equity risk premium. It's the premium you hope for, for taking on the extra risk for investing in items other than T bonds. So risky corporate bonds. So where can we get this premium from? Well, one option is to look historically. We could look at the arithmetic average of returns over a period of time. So at the top row here, we've got the S&P 500 giving us return of 11.41% per annum. We can see the 10 year T bond or treasury bond yielding 5.23. So the difference between them is your risk premium of 6.18. However, we notice that if you choose the time period carefully, that can dramatically change your risk premium. If we instead go for 2005 to 2015, that gets us a risk premium of 3.88% as opposed to the 6.18%, which we saw earlier.

In addition, investment returns this year are not independent of the investment returns you made last year. If you lost almost all of your money last year, then even if you make 100% return this year, it's the 100% return on a tiny, tiny starting figure. So an alternative to the arithmetic average of returns is you find the geometric average of returns, but still we have the same problems here. Changing your historical period changes the risk premium.

Another option is, instead of looking historically to look forward to look for implied returns in the future, in order to do this, we start with the aggregate value of equities formula. This is where you take the dividend from time period one i.e, next year's forecast dividend, and you divide it through by your cost of equity minus your long-term growth rates, and that gets you the present value of equities. We can rearrange that to find the cost of equity you take dividend next year, divide by the value of equities, plus your long-term growth rates. What you've now got, instead of having to use cap M to get to your cost of equity, you've now got it via an implied formula.