Hello,

Recently in an interview, I got asked the following question. We have two strategies S1 and S2 with the following daily return averages and standard deviations.

S1: 2% (5%)

S2: 1% (1%)

Which strategy do you prefer? How would you combine them?

The interviewer didn't seem satisfied with my answers and I'm curious to know how you would approach that problem.

Can someone mathematically prove that S2 will make more money?

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S2 has higher risk adjusted returns

Why is that a good metric? What if what I care about is making the most money?

S1 = 2/5 = 0.4

S2= 1/1 = 1

If you could only choose one strategy, you would choose S2. If you cared about making most money, you would borrow to leverage your investment in S2 rather than investing in S1.

Ideally, you would invest in both, and assuming the correlation between the returns of S1 and S2 is not high, you could reduce your risk below that of S2. So the ideal solution would be to invest most of your money into S2, and a little into S1 to achieve best risk / return trade off (in financial theory terms - market portfolio, which would depend on risk free rate) and then borrow / lend to adjust for ones risk preferences.

He wanted you to create a portfolio of two strats which maximizes something like Sharpe ratio. So, to find weights w1 and w2 such that mean divided by var of linear combination is max. Easy calculus.

I understand but why is the Sharpe ratio a good metric to optimize? If I want to make the most money why should I optimize the Sharpe ratio?

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Well, you shouldn't. In the risk-neutral world with unlimited capacity Sharpe gives you the optimal leverage (Kelly).

Your risk appetite is not the same as that of most HF LPs (investors). The latter focus A LOT on Sharpe, drawdowns etc.

If you double your money over 2 yrs but you have suffered a 50% drawdown in between, you will find a lot of your investors have jumped ship along the way. This is something many junior people in the industry don't realize and only learn with experience.

Scenario 2 structure.  I assume the question means that a 2stdev move is -1 to 3 with a mean return of 1 over an ANNUAL period?  Assuming the underlying is 100.

1.  Lever up.

2. Sell out of money puts (99p) and out of money calls (103c) for size.  Collect premium.

3. Buy portfolio insurance (so 80s of me!), synthetically swap the portfolio with a bank (hello softbank!), or the like with the premiums from the options.

4. Profit.

You can do the same with scenario 1,  with different strikes, but the vol will be costlier and you would have probably have to sell more calls than puts.

Namaste.

D.O.U.G.

• 1

I did not follow very closely because of my lack of finance background. But from a high level point of view, it seems that you're claiming that S2 is better than S1 because you found one strategy in which S1 would cost you less. That doesn't exclude the possibility that there could be other types of strategies where S1 is actually better than S2.

You changed a question from evaluating strategies to a vol question, which added more complexity and was not straight to the point of what the interviewer asked originally though.

With just your information, then most cases, people will go with the strategy with higher Sharpe. Why is Sharpe a good choice? With no assumption, Sharpe is what you get paid per unit of risk, with risk here being the volatility of returns. Strategy 2 offers a better payoff per unit of risk than strategy 1, so it is straightforward to pick 2.

To combine the 2 strategies, we need to specify the correlation between the two. This goes into portfolio optimization theory. Many ways to do it: be it mean-variance or risk parity. The goal should be that the portfolio has better performance statistics (Sharpe, etc) compared to each individual strategy, thanks to diversification benefits.

In a professional investing context, it is quite easy to access cheap leverage, as long as the strategy has enough capacity. Therefore, the returns that you quoted can always be leveraged to reach a certain level of risk/volatility(at 5% vol, strategy 1 offers 2% return whereas strategy 2 offers 5% return).

Now, if we go a level deeper, we can ask more questions, such as:

- what is the distribution of returns: is it negative skew? we can sell vol with high Sharpe until we blow up.

- what markets are these two strategies trading? (related to liquidity of these strategies, as well as cost)

- what are the max capacity that these two can trade? (related to whether we can leverage or not, and by how much)

- etc

These questions can change how one answers the original question.

That makes sense. Why is volatility a good measure of risk? Here's an example of I mean:

Let's say:

- S'1 has average 4% and std 2%

- S'2 has average 1% and std 2%

According to your reasoning, these two strategies are equally risky. But intuitively, these two 2% should not be counted as the same amount of risk. Don't you agree? Why should standard deviations that happen around very different average returns be seen as the same?

Volatility is a decent measure of risk, but not comprehensive. It is good because it is easy to compute and understand. For a very quick evaluation, Sharpe (with volatility in the denominator) is a good choice. Obviously, you need more measures to cover more dimensions such as what happened at the tail, drawdown behavior, correlation etc.

Regarding your question, why are two strategies not equally risky/volatile?

Let's draw this on paper. The mean return is a drift/slope of a straight line. Obviously s1 with mean return at 4% will have a higher slope than s2 with 1%. Overtime, we should expect to s1 to compound/reach a higher end value than s2.

However, it is not a deterministic/straight path. This is where volatility comes into the picture. These two strategies vary/zig-zag/hover at the same amount around the trend line. In other words, the volatility has nothing to do with the mean return. In that sense, 2 strategies are equally risky/volatile.

To judge which one is better, that's when Sharpe ratio coming into the picture. By combining the two measures (mean return and volatility), we have an easy way to judge the quality of a return stream.

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