Chapter 7: Risk, Diversification & Portfolio Theory

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement not replaced the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

Welcome back to The Deep Dive, the show that takes a huge stack of complex sources, distills the essential insights, and gives you the knowledge you need to be well informed fast.

So we spent a lot of time in previous deep dives talking about valuing cash flows, right?

We've mastered the net present value rule, and we've hammered home that a dollar today is worth more than a dollar tomorrow.

Right, but we've always left this massive hole in the middle of our framework, and that hole is the cost of capital.

Exactly.

We keep using that classic finance phrase, the opportunity cost of capital depends on the risk of the project.

Which sounds great in a textbook, but if you're a manager and you have to decide between two huge multimillion dollar investments, that kind of, you know, hand waving just doesn't work.

You need a number.

You need a way to actually quantify risk and turn it into a discount rate.

And that is our mission today.

We're going to dive deep into portfolio theory to solve this exact problem.

We'll define what risk really is, show how we can measure it with tools like variance and standard deviation, and then show how that magic of diversification actually works.

Right.

And finally, we'll pinpoint the specific type of risk, the undiversifiable risk that actually determines the discount rate for any corporate project.

So for you, the learner,

this conversation is really a fundamental roadmap.

By the end of this, you will get why two companies that look identical on paper, you know, same profits, might get discounted at completely different rates.

And you'll also see why the highest risk assets don't always give you the best expected return.

This is really the bedrock of how we quantify uncertainty in finance.

Okay.

So where do we start?

Let's start with history.

We first need to establish the fundamental fact that in financial markets, risk has to be rewarded.

To really nail down this risk return trade off, we're going to look at some amazing historical data.

We're talking about three classic portfolios that have been tracked in the US for over a century, from 1900 all the way to 2020.

Yeah, that long term view is so important because it lets us see past all the short term noise and get to the real underlying trend.

So what are these three portfolios?

Well, they represent three different levels of risk, which gives us some really useful benchmarks.

The first one, the safest one is treasury bills.

T -bills, that's short term US government debt, right?

Exactly.

They mature in less than a year, they're backed by the government, so they have basically zero risk of default.

They're what people in finance mean when they say risk -free investment.

Okay, so that's our baseline.

What's next on the risk ladder?

Next up, we have long term treasury bonds.

Again, it's US government debt, so still no default risk, but now you've introduced a new kind of uncertainty.

Interest rate risk.

That's the one.

Because these bonds mature over, you know, 10, 20, 30 years,

their prices are sensitive to changes in the economy's interest rates.

If rates go up, the price of your older, lower paying bond goes down.

That makes them riskier than T -bills.

Okay, so safe T -bills, medium risk bonds, I'm guessing the third one is stocks.

You got it, stocks.

They are by far the most volatile.

Their value is tied to the whole economy, of course, but also to how well a specific company is doing.

They carry both broad market risk and that unique company specific risk.

All right, we have our three buckets of risk.

What does the history tell us?

Let's say I put $1 into each of these back in 1900 and just let it ride for 121 years, reinvesting everything.

The result is, frankly, staggering.

Let's look at figure 7 .1, which shows the nominal growth.

That's just the raw dollar amount, not adjusted for inflation.

Your dollar in the super safe T -bills, it grew to $78.

Not bad.

A solid.

Your dollar in the medium risk bonds did much better.

It grew to $282,

but your dollar in the stock market portfolio, it grew to $69 ,754.

Wow, from $1.

That chart is just, it's the perfect illustration of the power of compounding when you're rewarded for taking on risk.

It is, but let's pause there because that number, $69 ,000, it's a little misleading.

Because that's in nominal dollars, right?

Doesn't account for what you can actually buy with that money.

Exactly.

Over those 121 years, we had an average inflation rate of about 3 % and that sounds small, but over a century, it has this just devastating effect on your purchasing power, especially for those safe assets.

Okay, so show us the real numbers, figure 7 .2.

Right.

When we look at the growth in real dollars, so dollars with constant purchasing power, the story flips completely.

That T -bill, which gave you $78 nominally, in real terms, it only grew to $2 .60.

Oh, that is shocking.

You barely made any money.

Barely beat inflation.

The long -term bonds did a bit better.

They managed $12 .50 in real terms.

So after 121 years, your safe investments basically just kept your money from becoming worthless.

That's a huge lesson right there.

It is.

Inflation is not just some minor inconvenience.

It absolutely eats away at your savings.

The only asset that created substantial real wealth was stocks.

That $1 ended up as $2 ,291 of actual purchasing power.

This leads us right to the idea of a risk premium, doesn't it?

It does.

If we just look at the average annual returns in Table 7 .1, T -bills returned about 3 .7 % per year, bonds 5 .4%, and stocks 11 .5%.

So the risk premium is the extra return stocks gave you for all that volatility.

Precisely.

It's the stock return minus the risk -free rate.

So 11 .5 % minus 3 .7%.

That gives you a historical risk premium of 7 .8%.

So in the past, a project with the same risk as the overall stock market needed to promise an extra, what, almost 8 % point to return just to convince investors to take on that uncertainty.

Right.

And you might be thinking, well, maybe the US is just special, a historical outlier.

That's a fair question.

But that's why figure 7 .3 is so important.

It shows the risk premiums for 20 different countries, and the US number, 7 .8%, is completely normal.

It's right in the middle of the pack.

Spain was on the low end at 5 .4%.

Portugal was high at 10%.

So the takeaway is clear.

Demanding a big long -term reward for bearing stock market risk is a global thing.

It's not just an American story.

Absolutely.

It's a fundamental principle of global finance.

Okay.

So now we have this historical relationship lockdown.

But how does that help a manager today who's trying to set a discount rate for a new project?

If the project has the same risk as, say, the S &P 500, what rate do they use?

That's the key question.

The rate they need is what we call long -o dollars, the expected future return on the market.

Because the alternative for any shareholder is to just take their money and invest it in the broad market.

So that's the opportunity cost.

And here's where we get into a really important and kind of tricky technical point.

You have to use the arithmetic average to estimate that future return, not the geometric average.

Yes.

And this trips a lot of people up because it feels so counterintuitive.

Right.

The 11 .5 % we just used for stocks, that's the arithmetic average.

Just add up all the annual returns and divide by 121.

But the geometric average, the compound return that tells an investor what their money actually grew at, is lower.

It's 9 .7%.

So why on earth would we use the higher number for discounting?

It's a critical distinction.

The geometric average is backward looking.

It tells you the constant rate your money would have had to grow at to get from the start point to the end point.

It describes the past.

But the arithmetic average is the best forecast in any single future year.

And because NPV is all about discounting future cash flows, that single year expectation is the number we need.

Let's use that big pharma stock example to really hammer this home because it's so fundamental.

So say the stock is trading at $100 today.

Right.

And in one year, there are three possible outcomes, each with an equal chance of happening.

The return could be minus 10 % plus 10 % or plus 30%.

So the stock price could be $90, $110, or $130.

Exactly.

So what's the expected value of the stock next year?

You just average those outcomes.

One third of 90 plus 110 plus 130.

That comes out to an expected value of $110.

And the expected return is the arithmetic average of the possible returns.

Minus 10 plus 10 plus 30 all divided by 3.

That gives you an expected return of 10%.

Perfect.

So you have an expected cash flow of $110 one year from now.

How do we find its present value today?

We have to discount it by the expected rate of return, which is that 10 % arithmetic average.

And $110 divided by 1 .10 is $100.

It works.

The calculation gets us back to today's actual stock price.

It has to.

Now, just to prove the point, what if we tried using the geometric average?

For this specific example, it calculates to about 8 .8%.

If we discounted $110 by 1 .088, we'd get a $100.

Which is wrong, because the stock is actually trading for $100 today.

The only way the math works and reflects reality is if you use the arithmetic average to find the opportunity cost of capital for a given period.

Okay, so that's a huge financial principle.

Use the arithmetic mean of historical returns to estimate the opportunity cost.

But that doesn't mean we should just plug in that historical 11 .5 % number for our expected market return today, does it?

No, absolutely not.

That would be a big mistake, because the expected market return or rate, a river dollars, and the expected risk premium.

And that risk -free rate moves around a lot.

A ton.

Think back to 1981.

The T -bill rate, our $3, was around 15%.

If you just use the historical average of 11 .5 % for stocks then, you'd be saying that the risky stock market was expected to return less than a perfectly safe government bill.

Which is just, it's absurd.

No one would ever buy stocks.

Exactly.

So the much more sensible approach for a manager is to take today's risk -free rate and add the historical risk premium to it.

So the formula is like $3 is tall plus the normal risk premium.

That's it.

So for example, if the current risk -free rate is say 2 % and we use our historical 7 .8 % premium, our best guess for today's expected return on the market is 9 .8%.

That becomes our discount rate for a market risk project.

And this is where it stops being pure science and becomes a bit of an art.

A huge amount of art actually, because we're making this massive assumption that the future premium will look like the past one.

And there's a lot of debate around that.

What kind of things do people debate?

Well first there's the question of the time period.

Should we use all 121 years?

Or maybe just the last 50?

A longer history is more stable, but maybe the world was so different a hundred years ago that those returns aren't relevant anymore.

That's a good point.

It's so much easier to diversify your portfolio today than it was in 1920.

You can buy an index fund on your phone in two minutes.

And that increased ease of diversification should, in theory, lower the risk premium that investors demand.

But on the other hand, you could argue there are new massive risks on the horizon today that didn't exist back then.

Like systemic climate change risk or geopolitical instability, things that affect the entire market.

If you believe those risks are bigger now, you'd argue for a higher risk premium going forward.

So there's no formula that tells you which of those arguments wins.

It really is an artistic judgment call.

It is, and it explains why the stock market is so active.

You have one professional investor buying a stock because their model uses a 5 % risk premium, and another selling the exact same stock because their model uses a 9 % premium.

They have different views of the future.

And these small changes in the discount rate have huge effects, right?

Especially for growth stocks or long -term projects.

Enormous effects.

Because that little difference gets compounded over and over again for cash flows that are far in the future.

It can completely change the net present value calculation.

So given all that uncertainty, what's the standard practice?

The standard practice is to use the historical data as a guidepost and work within an accepted range.

Most practitioners use a market risk premium, somewhere between 5 % and 8%.

We often just use 7 % for examples to keep it simple.

Okay, so now we have our anchor points.

We have $3 for safe projects and an estimated $3 for market -level risk.

But most projects fall somewhere in between.

Exactly.

And to figure out those in -between cases, we need a much more precise way to measure risk itself.

We need a ruler for risk.

So the price you pay for those high historical stock returns is just massive volatility.

If you look at figure 7 .4, which plots out all 121 annual returns, it's a wild ride.

It really is.

You have years like 1931, where the market lost almost 44%.

But then just two years later, in 1933, it roared back with a gain of over 57%.

And when you take all those data points and plot them as a histogram, as you see in figure 7 .5, that volatility shows up as a very wide spread, a very wide distribution of outcomes.

Right.

And if we're going to quantify risk, what we're really trying to do is put a single number on the width of that spread.

And this is where we turn to statistics, and specifically to the normal distribution, that classic bell curve.

Yes.

If you look at the daily returns of a stock like IBM, as in figure 7 .6, they tend to cluster around the average in a shape that looks a lot like a normal distribution.

And the really elegant thing about a normal distribution is that you can describe the entire curve with just two numbers.

Just two.

The first is its center, the average or expected return, and the second is it spreads the risk.

And the two statistical measures we use for that spread are variance and standard deviation.

So what is variance?

Variance, or sigma squared, is the expected square deviation from the mean.

Okay, hold on.

Why do we square the deviation?

If a stock is 10 % above its average, the deviation is plus 10.

If it's 10 % below, it's negative 10.

Why square that?

Two really important reasons.

First, squaring makes every deviation positive.

So a big positive surprise and a big negative surprise both count as risk.

They don't cancel each other out.

Okay, that makes sense.

But more importantly, squaring gives much more weight to the big outliers.

A deviation of 2 becomes 4, but a deviation of 10 becomes 100.

And since it's those big, painful surprises that we're most concerned about when we think about risk, this weighting is exactly what we want.

Got it.

So variance measures the weighted average of these squared deviations, and standard deviation is just?

The square root of the variance.

We usually prefer using standard deviation, sigma, because by taking the square root, we get back to the original units, in this case percentages.

It makes it much easier to compare the risk directly to the return.

Let's make this intuitive with those hypothetical investments in figure 7 .7.

So we have investments A and B.

They both have an expected return of 10%.

Right, but A has a standard deviation of 15%, while B is only 7 .5%.

So same reward, but B has half the uncertainty.

Any rational person would prefer B.

Absolutely.

Now compare B and C.

This time, they have the same risk, the same standard deviation.

But C has an expected return of 20%, which is double B's 10%.

Again, an easy choice.

Same risk, higher reward.

You pick C every time.

This shows how we use expected return and standard deviation together to evaluate investments.

And when we calculate the actual historical standard deviations for our three portfolios, the logic holds up perfectly.

T -bills had the lowest risk, a standard deviation of just 2 .8%.

Bonds were in the middle at 8 .9%.

And stocks were way out there with a standard deviation of 19 .5%.

Higher risk, higher return.

The theory matches the data.

Okay, I think it's time to actually get our hands dirty and calculate one of these.

Let's walk through that coin tossing game from table 7 .2.

Great idea.

So you invest $100.

We flip two coins.

For every head, you gain 20%.

For every tail, you lose 10%.

This gives us four possible outcomes, and they're all equally likely, so they each have a 25 % probability.

Right.

You can get two heads, which is a plus 40 % return.

A head and a tail, or a tail and a head.

Both of those give you a plus 10 % return.

And finally, two tails, which is a negative 20 % return.

Okay, step one is to calculate the expected return.

We just take the weighted average.

So it's 0 .25 times 40, plus 0 .25 times 10, plus 0 .25 times negative 20.

You run the numbers, and it comes out to a nice, clean, expected return Now for the tricky part.

The risk.

The variance.

We take each outcome, measure how far it is from that 10 % average, square that distance, and then weight it by its probability.

Let's do it.

For the plus 40 % outcome, the deviation is plus 30%.

A 30 squared is 900.

And we multiply that by its 25 % probability.

So that's a contribution of 225.

Okay, now for the plus 10 % outcomes, the deviation from the average of 10 % is 0.

0 squared is 0.

So that contributes nothing to the variance.

Right.

And for the negative 20 % outcome, the deviation is negative 30%.

Minus 30 squared is also 900.

We multiply that by its 25 % probability.

And get another contribution of 225.

So the total variance, sigma squared, is the sum of those parts.

225 plus 0 plus 225 equals 450.

And to get the risk number, we can actually use the standard deviation.

We take the square root of 450, which is about 21%.

So this game has a 10 % expected return and a 21 % standard deviation.

Exactly.

And the power of this method is that if we imagine a much riskier game, say with outcomes of plus 70 % and narrative 50%, the expected return could still be 10%.

But the deviations would be huge.

And when you square them, the variance would be massive.

The standard deviation would jump to something like 42%.

It correctly tells you that the second game is twice as risky, even if the expected payoff is the same.

That's the number the manager meets.

It's important to remember, though, that this volatility isn't constant.

It changes over time.

Oh, absolutely.

Figure 7 .9, which shows market volatility over time, is really telling.

You have these calm periods, like the mid 90s, where the market standard deviation was under 8%.

And then you have these moments of sheer panic.

The 2008 financial crisis, the COVID crash,

the standard deviation spiked to well over 30%.

And the single day drop on Black Monday in 1987 was so extreme, it was equivalent to an annualized standard deviation of 89%.

Just unbelievable temporary chaos.

OK, so the final and maybe most important point in this section is what happens when we move from looking at the whole market to looking at individual stocks.

This is the real puzzle.

The market as a whole recently has a standard deviation of around, say, 12%.

But if you look at individual stocks, like in table 7 .3 and 7 .4, their risk is vastly higher.

Like Tesla was at 48%, US Steel was at 76%.

Even a big stable bank like Volkswagen was over 21%.

So here's the question that sets up everything that comes next.

If the market is just a collection of all these incredibly volatile individual stocks, where does all that risk go?

How can the components be so much riskier than the whole?

It feels like some kind of financial magic.

It's not magic.

It's the single most powerful concept in modern finance.

Yeah.

Diversification.

OK, so this idea that you can combine risky things and end up with something less risky, it's maybe the most important lesson for any investor to learn.

It changes everything about risk management.

It really does.

Let's start with the perfect clean textbook example to build the intuition.

Example 7 .1, Izy's Ice Cream and Ursula's Umbrellas.

A classic.

So Izy's Ice Cream makes a lot of money when it's sunny, say a 14 % return, but not much when it rains, maybe a 6 % return?

And Ursula's Umbrellas is the complete opposite.

When it rains, she's making a 16 % return.

When it's sunny, only an 8 % return.

So if you own just one of these companies, your return is totally at the mercy of the weather.

That's a huge specific risk.

Exactly.

But now, what if you put half your money in Izy's and half in Ursula's?

Let's see what happens.

If it's sunny, your portfolio return is half of 14 % plus half of 8%.

That's 7 plus 4, which is 11%.

And if it rains?

If it rains, your return is half of 6 % plus half of 16%.

That's 3 plus 8, which is also 11%.

So the risk is just gone.

You get a guaranteed 11 % return no matter what the weather does.

The specific risk associated with the weather has been completely eliminated.

That is the superpower of diversification in its purest form.

Now, that perfect cancellation happened because the two businesses were perfectly negatively correlated.

They moved in exact opposite directions.

Which almost never happens in the real world.

But the good news is, stocks also almost never move in perfect lockstep either.

So to see how it works in reality, we have to look at the portfolio variance formula.

For a two -stock portfolio, figure 7 .10ZO visualizes it as four boxes.

The two boxes on the diagonal are the variance terms.

They just represent the individual risk of stock one and stock two, weighted by how much you own of each.

But the real action is in the other two boxes.

The two off -diagonal boxes, those are the covariance terms.

And this is the entire secret to diversification.

Covariance measures how two stocks move together.

So if they tend to move in opposite directions, the covariance is negative, and it actually subtracts from the portfolio's total risk.

Precisely.

And if they move together, it's positive, and it adds to the risk.

We formalize this using the correlation coefficient, which is written as rho or O.

Right.

And correlation runs on a scale from plus one to nanks one?

Exactly.

If the correlation is plus one, they move in perfect lockstep, the covariance is high, and you get zero benefit from diversification.

The portfolio risk is just the weighted average of the individual risks.

And if it's angadish one, like our ice cream and umbrellas, they move in a perfect opposition, and you can potentially eliminate all the risk.

Let's put some real numbers on this.

Let's take Southwest Airlines with a standard deviation of about 30 .5 percent, and Amazon at 28 .3 percent.

And let's say we have a 60 -40 portfolio.

Okay.

If they were perfectly correlated, if ice was plus one, the portfolio standard deviation would just be the weighted average, which is 29 .6 percent, no benefit.

Right.

If, in a perfect world, they were negatively correlated with Osweat Magus 1, the portfolio risk would plummet down to just 7 .0 percent, almost all the risk would be gone.

So what is it in reality?

In reality, their correlation was about plus 0 .38.

They tend to move in the same direction, but not that strongly.

When you plug that real number into the formula, the portfolio standard deviation comes out to 24 .9 percent.

Okay.

So it's not 7 percent, but it is a lot lower than the 29 .6 percent we started with.

The diversification still worked, even with a positive correlation.

As long as the correlation is less than a perfect plus one, you will get some diversification benefit.

So the fact that even in that real world example, we were still left with a pretty high risk of almost 25 percent,

that's the key, right?

Diversification helps, but it doesn't solve everything.

It doesn't.

And that's because all risk can be broken down into two types.

The first is what we call specific risk.

This is also called idiosyncratic risk or diversifiable risk.

Right.

This is the stuff that affects only one company or maybe one industry.

It's a pilot strike at Southwest.

It's a key drug failing its clinical trial at a pharma company.

It's an asbestos lawsuit.

These are the risks you can get rid of just by owning lots of different unrelated companies.

Correct.

The other type of risk is systematic risk.

This is also called undiversifiable or market risk.

And this is the stuff that hits everybody.

Almost everybody.

A big recession, a spike in interest rates, a global pandemic.

Since almost every company is exposed to these big macro forces, you can't diversify this risk away no matter how many stocks you own.

Let's bring this back to the ice cream and umbrellas.

We got rid of those specific weather risk.

How do we introduce systematic risk?

We can add another layer of uncertainty.

The economy.

Let's say in a boom year, our diversified portfolio makes it stable 11 percent.

But what if it's a recession year?

In a recession, maybe people buy less of both ice cream and umbrellas, and the portfolio only makes a stable 7 percent.

Ah, I see.

Even though we've diversified away the weather risk, the portfolio is still completely exposed to the economic risk.

Exactly.

The specific risk is gone, but the systematic risk remains.

And the math behind this is actually really elegant.

If you build a giant portfolio with n stocks in it, the total risk is made up of n individual variance terms.

And a whopping n squared minus n covariance terms.

So as you add more and more stocks, as n gets really big, those individual variance terms, the specific risk, they just get swamped.

They shrink towards zero.

The unique quirks of any single company become irrelevant.

But the covariance terms, which represent how all the stocks move together, they start to dominate.

The portfolio's overall risk doesn't go to zero.

It gets closer and closer to the average covariance of all the stocks in the market.

And that average covariance is the systematic risk.

That's it.

You can see this perfectly in Figure 7 .2.

When you start with one stock, the risk is high.

When you add a second, a third, a tenth, the risk drops really fast.

But then the curve starts to flatten out.

After about 20 or 30 stocks, adding more doesn't really reduce the risk much at all.

You hit a floor.

That floor is the systematic risk.

And this leads to the most important lesson in this entire deep dive, the one that governs all of finance.

You can only diversify away specific risk, not systematic risk.

And because you can get rid of specific risk for free, the market does not reward you for bearing it.

You only get a risk premium for bearing the systematic risk you can't escape.

So we've established that only systematic risk matters.

But now we have this massive practical problem.

How do you actually measure the systematic risk of one single stock, like say Coca -Cola?

Right.

You can't just calculate its covariance with every other stock in the world.

That seems impossible.

It's a computational nightmare.

If you take the 6 ,000 or so stocks on the major US exchanges, building the full variance -covariance matrix would mean calculating nearly 18 million unique covariance terms.

It's just not practical.

So we need a shortcut, a theoretical breakthrough.

And we got one from Nobel laureate Harry Markovitz.

He came up with the idea of the efficient frontier.

Okay, let's build this up.

We saw that if you combine just two stocks, like Southwest and Amazon, in different proportions, you trace out a curve of possible risk and return combinations.

Right.

Now, imagine doing that with all 6 ,000 stocks.

You get this huge messy cloud of possible portfolios, as you see in figure 7 .14.

But here's the key.

A rational investor only cares about the portfolios on the upper left edge of that cloud.

The ones that give you the highest possible return for a given level of risk or the lowest possible risk for a given level of return.

Exactly.

That outer edge is the efficient frontier.

It immediately narrows our focus.

We don't have to worry about any of the portfolios inside the cloud, only the efficient ones on that red curve.

So we could have a very aggressive portfolio, let's call it the lion portfolio, way up on the right.

High risk, high return.

Or a super cautious one, the chicken portfolio at the far left tip, which is the absolute minimum risk combination.

But this creates a new problem, a huge dilemma.

Which point on the frontier do you choose?

It seems like it depends entirely on your personal risk tolerance.

The lion picks one, the chicken picks another.

And if everyone has a different optimal portfolio, then the required return for any single stock would be different for every single person.

It would be chaos for a manager trying to pick one discount rate.

This is where we bring in an old friend.

The idea of borrowing and lending, just like we did with the Fisher separation theorem,

introducing a risk -free asset solves the disagreement.

So we add the ability to borrow or lend cash at the risk -free rate, for far over saying.

And remember the geometry here.

Combining two risky assets gives you a curve.

But combining one risky asset with the risk -free asset gives you a straight line.

Let's visualize that.

Let's say the risk -free rate is 2%.

If we take our lion portfolio and put half our money in it and half our money in risk -free T -bills, we create a new portfolio that's halfway along a straight line connecting them.

Lower risk, lower return.

That's the lending portion of the line.

But you can also go the other way.

You can borrow money at the risk -free rate to invest more than 100 % of your own money in the lion portfolio.

That lets you move up the line to a point of even higher risk and higher expected return.

Okay, so every portfolio on the efficient frontier has its own straight line connecting it back to the risk -free rate.

And the slope of that line is incredibly important.

It's called the Sharpe ratio.

It's the risk premium of the portfolio divided by its standard deviation.

It basically measures your bang for your buck, how much extra return you get for each unit of risk you take on.

And a rational investor wants to maximize that.

They want the most bang for their buck.

They're looking for the line with the steepest possible slope.

Which means, if you look at figure 7 .15, that the optimal line is the one that just kisses the efficient frontier, the one that is tangent to it.

And that single point where the line touches the curve is called the tangency portfolio, or T.

And this is it.

This is the core insight of modern portfolio theory, and it's called the two -fund separation theorem.

What does it say?

It says two revolutionary things.

First, all investors, no matter if they're a lion or a chicken, should hold the exact same basket of risky stocks.

The tangency portfolio, T .Y.

Because it offers the highest possible Sharpe ratio, the best risk -adjusted return in the entire market.

Wait, so my personal feelings about risk don't affect which stocks I should own?

Not at all.

That's the second part of the theorem.

You satisfy your personal risk preference not by picking different stocks, but by deciding how much to mix that single tangency portfolio with the risk -free asset.

So the chickens put most of their money in T bills and a little bit in portfolio T.

And the lions take out a loan to invest like 150 % of their money in portfolio T.

Precisely.

The recipe for the stock portfolio is the same for everyone.

The only decision is how much of it to own.

It completely separates the investment decision from the financing decision.

That's incredible.

And it leads to an even bigger conclusion, doesn't it?

If every single investor is holding the exact same portfolio of stocks,

T.

And since every stock in the market has to be owned by someone.

Then the tangency portfolio, T, must be the market portfolio.

The portfolio of all available stocks weighted by their market value.

That's the final leap.

And with that, the whole picture snaps into focus.

Because if every rational investor is holding the entire market, then by definition, they have diversified away all possible specific risk.

The only risk that remains, the only risk they are compensated for bearing, is the systematic risk of the market itself.

Systematic risk is market risk.

So that straight line, the best possible one, running from the risk -free rate through the market portfolio.

That's the capital market line, or CML.

It's the ultimate efficient frontier for every investor.

It shows the best possible return you can get for any level of market risk you're willing to take on.

And the equation for it, equation 7 .4, it just perfectly captures this logic.

Your expected return is the risk -free rate.

Plus a reward for the risk you're taking.

And that reward is the market price of risk.

The Sharpe ratio of the market multiplied by how much market risk you've chosen to bear.

It's beautiful.

It gives us a complete framework.

And it proves that the right measure of risk is standard deviation.

But only for a fully diversified portfolio like the market itself.

This sets us up perfectly to finally figure out how to find the discount rate for a single individual project.

Okay, so we've just spent all this time showing that diversification is this incredible financial superpower for investors.

Right, so the logical next question is, shouldn't companies do it too?

Shouldn't a company's managers actively try to diversify the business by, say, buying up firms in totally unrelated industries?

It seems logical, but the answer is a firm no.

For the most part, corporate diversification is completely redundant for shareholders.

And managers who do it just for the sake of diversification are probably actually destroying value, not creating it.

The reason is all about cost and efficiency.

An individual investor can diversify on their own, instantly, and for almost no cost.

Sure, you can buy an S &P 500 index fund.

Or you can just buy shares of an airline and shares of a pharmaceutical company.

The transaction costs are tiny.

But for the airline to go out and acquire the pharmaceutical company, that process is incredibly expensive.

You're paying huge fees to investment banks.

You almost always have to pay a big premium over the stock's current price to get the deal done.

And then you have the massive headache and cost of trying to merge two completely different businesses and cultures.

Exactly.

And since the investor can get the exact same diversification benefit for free at home, they are not going to pay a premium for a company that does it for them in a very expensive way.

This brings us to a really crucial principle in corporate finance,

value additivity.

Yes.

The idea is that the market value of a combined diversified firm is worth no more than the sum of its parts if they were separate.

The present value of firm AB is just the PV of A plus the PV of B.

The whole is not worth more than the sum of its parts.

And this result is incredibly freeing for a financial manager.

It's fundamental to good capital budgeting.

It means that when you're looking at a new investment, you can and you should analyze it completely on its own.

So if my company is, say, a car manufacturer and I'm thinking about getting into the restaurant business.

You just need to calculate the net present value of that restaurant project, discounting its cash flows at a rate that is appropriate for the risk of the restaurant industry.

And if the MTV is positive, the value of my company will go up by exactly that amount.

That's it.

It doesn't matter one bit that the restaurant business might be negatively correlated with the car business, which would make your overall company's earnings more stable.

The market doesn't care about that corporate -level diversification.

Because investors can do it themselves.

All the manager needs to do is find and execute positive NPV projects.

The market will handle the rest.

This means the NPV rule we learned about still works perfectly, even in a world full of uncertainty, all thanks to value additivity.

We have covered a massive amount of ground today.

We went from a pretty vague idea of risk all the way to a very rigorous mathematical framework for valuing assets.

Let's quickly recap the biggest principles.

Sounds good.

I think there are five core takeaways.

OK, what's number one?

First, the historical link between risk and return is undeniable.

Stocks have delivered a huge risk premium over safe assets, and we must use the arithmetic average of those past returns to estimate the opportunity cost of capital.

Second, risk measurement.

We now have a ruler for risk.

It's called the standard deviation, or sigma, and it quantifies the spread of possible outcomes.

Third, and this is a big one, diversification's limit.

Diversification is a free lunch, but it only gets rid of specific risk, the unique risks of a single company.

You are always left with unavoidable systematic risk.

Fourth,

the optimal portfolio.

The two -fund separation theorem tells us that every single investor should hold the same basket of stocks,

the market portfolio.

This proves that systematic risk is market risk, and the capital market line, the CML, shows us the reward for bearing it.

And finally, fifth, the lesson for corporate finance.

Because investors can diversify on their own, corporate diversification is redundant.

Managers need to follow the principle of value additivity and just focus on finding projects with a positive standalone MPV.

Okay, so that's the framework, but let's end with a final thought here.

We've shown that the most sophisticated financial theory from Markovitz says that all rational investors should agree on what the single best portfolio is, the market portfolio, Auntie.

Right, the math is pretty clear on that.

And yet, we started this whole deep dive by admitting that trying to estimate the future expected return of that very same portfolio,

three dollars ours, is much more art than science.

It's based on these fuzzy forward -looking assumptions about the risk premium.

So we have this tension.

The math about the composition of the portfolio is absolute, but the assumption about the return of that portfolio is completely subjective and debated.

So if everyone agrees on the recipe for the perfect portfolio, but they all fiercely disagree on how much that portfolio is going to return in the future, what does that tell us about all the buying and selling we see in the market every single day?

It suggests that, you know, a lot of the game isn't about having a better formula.

The models are all pretty similar.

The game is about who has the right artistic assumption about the future.

The models are only as good as the inputs you feed them.

So the final question for you is this.

What assumption about the future risk premium are you willing to stake your investment decisions on?

It's something to think about until our next deep dive.

Thanks for joining us.

We'll see you soon.