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So here, looking at these annual returns from 1927 to 2014.

You see for US Treasury Bills, one month Treasury Bills on

an annual basis, they've averaged about 3.5%.

Ten year bonds about 5.3%, a little higher return reflecting the duration risk.

You're tying up your money for a longer period of time.

US Stock Market averaged about 12% over 1927 to 2014.

Obviously, kind of a lot more volatility to that return than you have from

the return from US Treasury Bills or US Treasury Bonds and then Small US Stocks.

Stocks in the bottom one-tenth in terms of a market capitalization firm size,

they've averaged 19% per year.

So you can see about a seven percentage point difference relative to the stock

market a a whole, but almost double the standard deviation or the volatility.

So you can think of asset pricing models trying to give us a sense of hey,

this seven percentage point difference here of small stocks

versus the market as a whole, what does that account for?

Is that due to small stocks being riskier or does it just mean small stocks have

been under price have just been historically a great investment?

And then of course, you see this equity market premium.

US Stock Market return relative to treasury bills being about 8% per year.

So question, do you know why the higher return on average for

small stocks relative to the stock market as a whole?

Here I just did a graph of the US Stock Market return going from

1927 to 2014, that's indicated by the darker diamond.

US Small Stocks, the bottom decile in terms of size indicated by

the lighter gray squares and let's just put a line in here to indicate zero.

You can see that on average, these gray squares are higher than

the darker triangles indicating a higher average return for

small stock relative to the stock market.

But what's interesting is that you can see when times are bad,

there's a negative stock market return here, these dark diamonds.

Usually in those bad times, small stocks are doing worse.

the light gray squares are always lower.

So maybe that gives us a little clue,

as to why do we see small sizes getting this higher return on average.

Maybe it has something to do with in bad times,

the small stocks seem to do worse in the market.

So, they'll kind of kick us when we're down.

So, let's just kind of go back to basics of return and risk.

And then ultimately, build up to asset pricing models like CAPM,

which will also review in subsequent videos.

Let's just start with a simple example where we have large stocks with

an average return of 8% per year.

Standard deviation at 25% and then Treasury Bills or

Risk Free Asset where they just give us 3% per year, no volatility.

And then by definition, zero correlation between the two.

So, let's look at this capital allocation line or

basically it's the return-risk tradeoff.

If we're 100% in the Risk Free Asset, we have a 3% return with zero volatility.

If we're 100% in large stocks,

we have an average return of 8% with a 25% standard deviation.

The slope of this line represents what's called the famous Sharpe Ratio here.

And remember, this formula for the Sharpe Ratio, the slope of that line,

you could also call it a reward-to-volatility ratio,

if you want is simply the excess return of the asset.

Excess in kind of in excess of the risk free rate,

that's kind of the benchmark, the safe asset.

What's the excess return of the asset?

Divide it by its volatility, its standard deviation.

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So, that was kind of simple example showing the return to risk tradeoffs

where we have just one risky asset and one risk free asset.

We also in module one of the first course and introduced, well, what if we have two

risky assets and this example here really then starts to build the fundamentals that

will go into the development of the capital asset pricing model.

And has some really nice kind of intuitive, hopefully intuitive lessons

that can even guide our own financial portfolio decision making.

So let's say, we have large stocks average return of 8%.

Standard deviation at 25% and then small stocks average return of 15%.

Standard deviation of 50%, correlation between the two is 0.4.

Obviously, this is an example, you can come up with different assumptions and

they'll generate different return to volatility tradeoff, but

the same basic points will apply.

So here, we have a portfolio frontier.

It's giving us all the return to volatility possibilities,

given these assumptions here about large stocks and small stocks.

So this curve is really just tracing out all of the return to volatility

possibilities when we vary our portfolio mixed between large and

small stocks like what if it's 25% small, 75% large or 75% large, 25% small?

What if it's 99% large, 1% small or 99% small, 1% large?

Consider all those possibilities and then graph the expected return and

standard deviation across all of those kind of possibilities here,

you to get this curve.

This curve has several interesting points.

One is what combination of small and large stocks gives the minimum variance?

It actually turns out in this example,

this was 94% large stocks, 6% small stocks.

And again, in module one of my first course, we went into this in depth.

You can solve for what portfolio gives a minimum variance using this

solver optimizer app that you get with data add-ins in Microsoft Excel.

So that minimum variance point is actually very important,

we can draw a line through that here.

Portfolio combinations above this minimum variance portfolio here.

These are called the efficient frontier for

a given level of volatility, a given standard deviation here.

What are the portfolios that give us the highest return?

Given that we only are investing in small and large stocks,

that's given up here in the efficient frontier.

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Assets portfolio combinations below this green line are what's called

the dominated assets.

So these are assets for a given level of volatility,

a given level of standard deviation.

There exist another combination, a small and

large stocks with the same standard deviation that gives a higher return.

Therefore, the original portfolio combination is dominated.

An example, in this case was 100% large stock portfolio.

100% large stock portfolio is close to this minimum variance,

that was 94% large, but a 100% large portfolio

as a standard deviation of 25% and an average return of 8%.

It turns out a portfolio that's 88% large, 12% small has

the same standard deviation at 25, but has an expected return of 8.8%.

So that would suggest, hey, a 100% large stock portfolio's

dominated if that same standard deviation and 88% large,

12% small portfolio gives you a higher return of 0.8 percentage points per year.

So dominated assets, you can refer to them as being inefficient.

A portfolio combination is inefficient., if there's another portfolio

that yields a higher average return with the same standard deviation.

So let's do a real world application of this and this is like the time to start

really paying attention, because this is the tip that could literally save you

thousands and thousands of dollars and you get it for free.

Now, what more could you want?

So literally, this could be a tip that could save you tens of thousands

of dollars by pointing out a real world potential dominated asset and

I brought this up in the first course in module one, worth repeating again.

So let's say, you have an S&P 500 index fund and

it has an annual expense ratio of 0.05%, 5 basis points per year.

There's another S&P 500 index fund.

Again, the S&P 500 is an index of 500 large US companies.

It has an annual expense ratio of 0.4%.

So, each of these mutual funds is following the S&P 500.

So there they have exact same thing,

except one has an expense ratio A 0.4% per year or 40 basis points.

The other 0.05% per year or five basis points.

And by the way, there are mutual funds out there with such low expense ratios for

index funds provided by major fund families.

And just for simplicity,

for the example, let's assume the S&P 500 has a 10% return per year.

So, what's the difference in the wealth that you accumulate by investing in these

two funds?

Funds A and funds B after ten years, after 20 years, after 40 years.

So what's the difference after 10 years and then after 20, after 40?

We did this example before, so no need to write it out again.

If we look at fund A,

it only has this expense ratio of five basis points per year.

So, we're compounding instead of at 10% at 9.95%.

So you aggregate that up a dollar investment after ten years after expenses

is $2.58 where for fund B, where the expense ratio is 0.4% per year.

So aggregated up by 9.6%, as opposed to 10%.

This $1 grows to $2.50.

So, you're 3.2% higher wealth by investing in fund A over fund B.

20 years, this difference is a factor of 6.6% higher wealth.

In fund A, then fund B after 40 years,

this is 1.136 is a ratio of fund A to fund B wealth.

So if you buy investing in fund A, you would have had 13.6% higher balance at

the end of 40 years and they are exactly the same.

They're both investing in the S&P 500 is fund A has a higher expense ratio.

So if your final balance would be a million dollars by investing in fund B,

you would have a million, $36,000 by investing in fund A.

So you may say, this 13.6 is a big deal or it's not a big deal.

But if you don't want the 136,000, you can feel free to send it to me.

I'll be happy to take it off your hands.

Let's consider another example and we'll be getting into examples like this or

talk about the difference between actively managed funds, and index funds later

in the course, and the evidence of the performance of actively managed funds.

So let's assume we have this fund A,

S&P 500 again with this annual expense of 0.05%,

only 5 basis points per year and such funds exist there out in the marketplace.

So now fund B,

instead of being an S&P 500 index fund is an actively managed large-cap fund.

It has annual expense of 1.25% per year.

So this is a mutual fund where the manager is trying to beat the market,

also picking among large stocks.

Therefore, this higher fee.

Maybe this was also purchased through a financial advisor or

broker, which may kind of add fees to the mix as well.

So therefore, we have this 1.25% assumed fee here.

Just an example.

Let's assume both the S&P 500 and

the large-cap fund have a 10% return each year.

So, I'm basically assuming the large-cap fund doesn't beat the S&P 500.

They each have the 10% return per year.

We'll get back to this.

This is a good assumption to make or

not when we talk about mutual funds in module four of this course.

Now, let's see the difference in wealth that emerges after 10 years,

20 years, 40 years.

And again, we did this example in the first course.

So no need to write it again, but

you can see here the differences are quite striking.

After ten years, fund A again, the $1.00 investment goes up to $2.58,

because we're compounding 9.95% per year as opposed to 10.

For fund B on the other hand with the 1.25% expense ratio,

we're now compounding at a return of 8.75%.

That only grows to $2.31.

Fund A has 11.6% wealth at the end of the day than

fund B in your account after a ten-year period.

Once you go to 20 years, the difference in the wealth,

if you have fund A versus fund B grossed almost 25% more if you used fund A.

After 40 years, your retirement account will have

55.1% more wealth in it using fund A, then fund B.

So, there's this wild factor here.

So this is something to take into account and

we'll hit this point again that remember,

expenses you pay are money from mutual funds or to kind of various advisors.

That's money that comes directly out of your pocket, so

you need to be sure you're getting some benefit from that.

In this case, we assume both funds give the same return.

This higher expense basically really reduces the return you

get from the fund and that just compounds over time

leading to these dramatic differences in wealth accumulation.

So now, let's summarize some key points from this video and

module one of my first investments course.

So over the last 80 years, the US stock market has

outperformed Treasury Bills by about 8% per year on average.

That's what we would call the equity market premium.

Also, small stocks have outperformed the overall market by a sizable margin.

Now, the interesting question is do we attribute that difference to small stocks

being riskier or does it just mean small stocks have historically been under-priced

and have been kind of a good investment?