Learning outcomes.

After watching this video,

you will be able to,

one, understand why there is a need to measure performance.

Two, define Sharpe ratio.

Three, list down the issues with Sharpe measure.

The next part of this lecture is all about performance measurement.

Now performance measurement, as opposed to the previous section,

is relevant to everybody,

not just to students of finance.

And I will connect efficient markets to performance measurement very quickly.

But before we do that,

let's understand what is performance measurement.

Essentially what we are saying is,

there is a bunch of us who invest in

notably equity markets or stock markets through vehicles called mutual funds.

Now these mutual funds come in two flavors: active and passive.

Active mutual funds are those mutual funds which believe that they can,

again, process or acquire information smarter.

They're smarter than everybody else,

so they can earn more money than what the average market return is.

Now, passive mutual funds are those that seek to mimic a certain index.

So, for example, there are S&P index funds which have

the sole purpose of mimicking the return on the S&P 500.

Essentially, if the S&P 500 goes up five percent,

this fund aims to go up five percent too.

Now, where do the active and passive mutual funds come from?

Where do these philosophies come from?

They come directly from what we talked about just now,

which is the market deficiency idea.

If you believe, let us say,

that markets are largely efficient or,

close to 100 percent efficient,

then the idea is obviously getting information, processing it,

hiring fund managers and so on,

is not going to buy you much by way of extra return.

In which case you would say,

I would go for a passive mutual fund,

which is to say, I will be a passive indexer.

I will just put my money into an S&P 500 fund,

and let the money ride according to the fortunes of the S&P 500.

However, if I believe that

a particular manager is really exceptional at beating the market,

I would actually go and invest my money with an active mutual fund manager.

These ideas are closely connected to the efficiency of the market.

Now, in terms of active management,

there's about $10 trillion under active management in the U.S. alone.

In India for example,

that number is about $200 billion.

Now, obviously these active fund manager,

if you believe that he can outperform the market,

and deliver you an extra market return,

he or she is obviously going to charge you a fee.

Now, these fees are typically obviously higher for

active mutual fund managers compared to passive mutual fund managers.

Now, let's assume an average fee of 0.4 percent.

If you think about the $10 trillion number I just stated,

you would see that investors spend upwards of about $40 billion.

And this is $40 billion,

with a B, on fees to these fund managers.

Now, if you are in aggregate paying these guys a lot of money,

now we need to actually go back and investigate and figure out,

are these guys making money for me?

Because unless they're making money for me,

I will not be willing to part with the fee for the management of the fund.

Now, there's a battery of measures which is used to test

whether a particular fund/fund manager has done better or worse than the market.

These come from the theory that we have studied called Mean-Variance theory.

The first measure I'm going to talk about is something called the Sharpe measure.

The Sharpe measure is very simply an average return on the fund,

so R_P bar is simply the return on a particular fund.

R_F bar is the average risk-free rate over the same period.

So take the average return of the fund,

subtract the average risk-free rate.

That's how much you earn over the risk-free rate.

Now divide that by the standard deviation of the portfolio's return, or the funds return.

And what do you get you get?

You get something which is very clearly a reward-to-risk ratio.

This particular reward-to-risk ratio is called the Sharpe ratio;

a Sharpe measure to be more precise.

Now, suppose we do this with a bunch of funds.

Obviously, if there is a fund which has

a Sharpe ratio of five and another fund which has a sharp ratio of two,

obviously the one with the Sharpe ratio of five is set to perform better because,

for the same risk it's giving you a higher reward.

So essentially it's a standardized reward-to-risk ratio.

Now, in isolation this number five,

or three, or two, what does it mean?

Well, there are statistics that can test

whether you are performing better than a particular benchmark.

So I could just as well take the Sharpe ratio off a particular index.

Let's say there are simply 500 index.

In other words, take the average returns on the S&P,

subtract the average risk-free rate,

divided by the standard deviation of the S&P.

And then, compare that to the Sharpe ratio of a fund.

Then I'm in a position to figure it out,

using the Sharpe measure,

whether my particular fund,

or my particular portfolio,

has actually beaten the index or not.

Now, the Sharpe measure,

as you might notice, has two immediate problems.

Number one, the numerator contains the average fund performance;

what I call R_P bar. What is that?

It's simply an arithmetic average of about 60 months,

or 48 months, or 12 months of the fund returns.

So arithmetic averages can be notoriously misleading.

Think of a situation where you have $100 to begin with.

You have a 50 percent gain in year one that takes you to 150,

and a 50 percent loss in year two.

That means your 150 has gone down to 75 at the end of year two.

Which means over the two years what has happened is,

your 100 has become 75,

which is a net loss of 25 percent.

But if you take the arithmetic average of the two returns over the two years,

50 percent, negative 50 percent,

average the two, you get zero.

So basically arithmetic average is not

a true or accurate measure of the overall return over this period.

Now, after all, investors really care about overall returns.

They don't really care about the arithmetic average.

That's something a statistician would be interested in, not an investor.

So that's an issue with the numerator.

With the denominator, we have a slightly tangled issue,

which is that in the denominator we have

the standard deviation. What is standard deviation?

It's simply a measure of how spread out

the distribution of returns is around the mean or average value.

Now, obviously, you can see this most clearly in a normal distribution where,

if you have a tight normal distribution around the expected value or average,

versus a more spread out distribution,

typically we say the more spread out

distribution has a higher variance or standard deviation.

Notice that the variance or standard deviation,

which we're calling volatility here,

is really a measure of uncertainty.

What we really want in the denominator is a measure of risk.

Now, by using standard deviation in the denominator,

what we're doing is,

we are essentially saying volatility equals uncertainty equals risk.

Now, uncertainty statistically is indeed standard deviation,

but it is not risk in the way usually investors understand it.

What is risk to you?

If you think for a moment,

what you will find is,

risk is really the loss of losing principle.

In fact, if I perform better than average,

it would still be counted in the standard deviation calculation,

but that upside volatility is really not

something that I traditionally think of as risk in my portfolio.

What I really think of risk,

or what I fear most in a portfolio context, is the downside.

That is the possibility that I might perform below average.

So we need to fix that too.

And several measures have been suggested to fix this,

and we will discuss this soon.