What period to compute the most accurate expected returns to do a covariance matrix for the optimal portfolio according to Markowitz method?

Question

I am able and willing to invest some money into risky assets that may have high expected returns on at least one year: I want to invest my personal money into crypto currencies. I use a solver made by friends during Finance classes in order to know what would be the balance. Everything is okay but I'm not sure about the period to use to create the expected return. I use a three month period for all cryptocurrencies as far as one cryptocurrency is priced in my currency (the Euro) only for this time.

• What would be the best period to compute the most accurate expected returns in order to do a covariance matrix used by my solver to find the optimal portfolio according to Markowitz method ?
• A sub question would be : Should I close my eyes on the fact that these cryptocurencies aren't perfectly priced in my currency and use another one (such as the dollar) or should I use different time period for different assets (which I think is a bad idea because it therefore becomes impossible to compare).

What currency should I use as a reference ?

PS: (I'm okay to receive other tips over how to invest one's money when being young (I am over 20 and under 25 and I have no loans) ;) ).

Answer 1

Any time you are optimizing a portfolio, the right horizon to use for computing the statistics you will use for optimization (expected return, covariance, etc.) will be the same as your rebalance/trading frequency. If you expect your trading strategy to trade once a day, you should use daily data for optimization. Ditto for monthly or quarterly. If at all possible you should use statistics across the board that are computed at the same frequency as your trading.

Regarding currency pricing, I see no reason you can't take the reported prices and convert them to whatever currency you want using that day's foriegn exchange rate. Foreign exchange rates are available for free at the Fed and elsewhere. Converting prices from one currency to another is not rocket science.

Since you are contemplating putting actual money behind this, note that using data to compute statistics is less reliable for lower statistical moments. The mean (expected return) is the first moment, so using historical returns is extremely unreliable at predicting future returns. The variances and covariances are second moments, they are better. Skewness and kurtosis, yet better. The fact that the expected return can't reliably be estimated from past returns is the major downfall of the Markowitz method (resulting portfolios are often very crazy and will depend critically on the data period you use to set them up). There are approaches to fixing this, such as Black-Litterman's (1992) method, but they get complicated fast.

Answer 2

@farnsy has provided a good answer. I'm only addressing my comment about the data quality.

The portfolio optimization technique you employed is very sensitive to the inputs. In particular, it relies entirely on the mean and (co)variance assumptions (i.e. the first two moments) and the results could change drastically with very small amount of change in the inputs. To see that, you can make up some inputs for the solver you have, and try adjusting the inputs a little bit and see the results.

Therefore if you decide to take this approach, data quality is very crucial.

EDIT: What I meant by "data quality"

1. Are the data source you are using (i.e. cryptocompare.com) reliable? Is its data consistent with what was observed in the market?

I have no experience with this website but this should be easy to spot check.

2. Does the market data reflect the fair value of the assets?

The answer is usually "yes" for liquid assets. Illiquid assets can often be priced at a level with no volume, and the bid-ask spread could be huge.

Should I close my eyes on the fact that these cryptocurencies aren't perfectly priced in my currency and use another one (such as the dollar)

You seem to have concern about data quality in at least the price quoted in your currency and are thinking about using data quoted in USD, but would it be any better? The law of one price tells us that there shouldn't be any discrepancy between prices in different currencies (otherwise there would be arbitrage).

In addition, (when compared to traditional assets) cryptocurrency price data has a shorter data history, and with lower liquidity in the market. The short history means you have less data to infer the characteristics of the price behavior. Low liquidity means the volatility may well be underestimated.

So we have an input-sensitive technique combined with not-so-perfect data. I wouldn't allocate my money solely based on the result of this exercise.

EDIT: I have quite some reservation about doing portfolio optimization for cryptocurrency. Personally I'm not a fan of the technique as is. The optimization has an underlying assumption that returns follow a certain distribution, and correlation is fixed. I don't know if you can make such assumption for cryptocurrencies. From what I read about BTC for example, it seems to have a high risk exposure concerning Chinese monetary policy. For that kind of assets perhaps a fundamental analysis approach is a better one.

Also if you would like to learn more about portfolio optimization, try quant.SE

Answer 3

If I were you and wanted to make as much as possible in Crypto for the next year, I'd commit to weekly rebalancing. With this commitment, I'd look at weekly returns of the previous week and plug them into my code. I'd use USD data sets because of its liquidity and volume.

I'd also read up on swing trading enough to learn how to put stop loss orders to contain losses.

As a 25-year-old, I'd put some money in SPY because last time I googled how many top traders can do better than S&P 500 in a 10 year period, I wasn't able to find too many.