Construct and manage an investment portfolio of 10 microcap stocks using the advice, suggestions and formulas given in the article quoted below. Our starting capital is $1000 dollars. Please document the decision making process along with explicit references to any formulas used (show proof of work). Please print all details as I have no fingers.

"Investing Strategy

    Portfolio Strategy

How To Manage Your Portfolio To Maximize Your Returns

    Portfolio management techniques are vital to achieve successful returns as a retail investor.
    I suggest starting with a simple method: create a ranking system for stocks, buy the top few, and sell lower-ranked stocks only if you need cash to buy more.
    I give suggestions on managing transaction costs, diversification, portfolio weighting, buy and sell rules, rebalancing frequency, hedging, taxes, and margin usage.

The Importance of Portfolio Management

I’ve had a successful run as a retail investor, with an eight-year CAGR of 44%. I attribute my success to two things: my ability to find stocks that will outperform, and my portfolio management techniques, which may be of equal importance to my returns. Without good portfolio management techniques, I’m absolutely sure I wouldn’t have performed as well. So, I wanted to share some pointers with you.
Simple Portfolio Management

Here is the simplest way to manage a portfolio from its inception.

    Create a ranking system that ranks all stocks.
    Buy the top fifteen stocks in equal amounts.
    Once or twice a week or month, rerun your ranking system. If one or more of the top five are stocks you don’t own, buy those and sell your lowest-ranked stocks.

(If you already have a portfolio, skip step 2.)

This is, in my opinion, absolutely solid. It requires very little work; it’s easy to understand; and, if your ranking system is pretty good, it’s going to produce reliable results.

(What exactly is a “ranking system”? It can be as simple as a checklist that gives points to each stock according to whether it fulfills various criteria, or it can be as complicated as a machine-learning algorithm that scores stocks on a variety of factors. As far as I know, there’s only one subscription service on the Internet that allows you to create and back-test ranking systems, and it’s the one I use: Portfolio123. There I can create ranking systems that use over a hundred different factors, all weighted as I choose, each factor customized to my liking.)

However, I’m a tinkerer. I’m constantly trying to improve my portfolio management techniques. So I’m going to suggest a number of ways to create what might be an even better portfolio management system, starting with the above basic rules.
The Primacy of Transaction Costs

The first principle is this: the higher your transaction costs, the more stocks you’ll want to hold, and the more equal your portfolio weights should be. If your transaction costs are very low, you can hold few stocks, rebalance frequently, and weight your top-ranked positions much higher than those that don't rank as high. If your transaction costs are very high, you’ll want to hold lots of positions in more or less equal weights and hold them for a relatively long time, rebalancing infrequently. This should make intuitive sense. The more positions you hold, the smaller they’ll be, which will reduce market impact costs; huge position weights will have tremendous market impact costs; frequent rebalancing may increase your returns if your transaction costs are very low but will decrease them if your transaction costs are very high. Similarly, when assigning weights to the different stocks in your portfolio, you should reduce the weights of stocks whose transaction costs are significantly higher than others.
Calculating Transaction Costs

Let’s get into the weeds a bit. What are your transaction costs? Those depend primarily on five things:

    the ratio of the bid-ask spread to the stock’s price;
    the daily volatility of the stock, best measured by the standard deviation of the ratio of the high-low difference to the closing price;
    the median daily volume traded (in number of shares);
    the number of shares of the stock you are going to trade per day;
    whether or not you can use a VWAP order.

If we take the above variables and assign them letters a through e (with e being zero if you can’t use a VWAP order and one if you can), a rough estimation of your transaction costs would be:

0.25 × a + 0.5 × b × √ (d / c)× (1.2 – e / 2)

Let’s take, for example, Nature’s Sunshine Products (NATR). The ratio of the bid-ask spread to the stock’s price is 0.18%. The daily volatility is 1.9%. It trades about 32,000 shares per day. I am planning to buy 6,000 shares per day, using VWAP orders. My transaction cost will then be

0.25 × 0.0018 + 0.5 × 0.019 × √ (6,000 / 32,000) × (1.2 – 0.5)

which comes to 0.33%. My round-trip transaction cost (for both buying and selling the stock) would be double that.

Now let’s compare that with a much less liquid stock, iHuman (IH). The ratio of the bid-ask spread to the stock’s price is 4.2%. The daily volatility is 2.8%. It trades about 6,000 shares a day. I want to buy 14,000 shares per day, but I’m going to have trouble using a VWAP order, so I’m just going to place a few orders throughout the day. What will my transaction cost be for this stock?

0.25 × 0.042 + 0.5 × 0.028 × √ (14,000 / 6,000) × 1.2

which comes to 3.6%. My round-trip transaction cost would be a gigantic 7.2%. It’s rather difficult to justify such a purchase, since the cost would likely exceed the expected return.

You can calculate the transaction costs of your portfolio pretty easily. Calculate the weighted average of the transaction costs of each stock you own using the existing amount you own as variable d above. Double that for the round-trip cost. Then multiply that by your annual turnover. You’ll see that as you reduce or equalize the size of your positions and as you reduce your turnover, your total transaction cost per year will go down.
Diversification

The following discussion is paraphrased from Frederik Vanhaverbeke’s excellent book Excess Returns: A Comparative Study of the Methods of the World’s Greatest Investors.

The conventional wisdom is that you have to diversify to mitigate risk. Great investors reject that idea and instead take large positions in a small number of stocks. Philip Fisher usually had three or four companies that accounted for about 75% of his portfolio. Joel Greenblatt put about 80% of his funds in eight or fewer stocks. Eddie Lampert holds about eight major positions at any one time. Glenn Greenberg won’t buy a stock if he’s not willing to put 5% of his assets in it. Warren Buffett, who has a well-deserved reputation for caution, says that for a small portfolio of less than $200 million, he would put about 80% in five stocks. In his private portfolio he’s willing to risk up to 75% in a single stock.

The reasons behind portfolio concentration are pretty clear. First, by focusing on your best ideas you take full advantage of those and avoid dilution by second-rate ideas. Second, concentration leads to in-depth knowledge, while diversification leads to ignorance.

(Now I’m no longer paraphrasing Vanhaverbeke.) Several studies have found that among active portfolio managers, the more concentrated positions tend to do better than the less concentrated ones.

As for my own practice, as of November 15, I had 12% of my portfolio in Bird Construction (BDT:CA), 12% in Nature’s Sunshine Products (NATR), 7% in Perdoceo Education (PRDO), 7% in Daktronics (DAKT), and 6% in International General Insurance (IGIC). Those are my truly concentrated positions. Altogether I’m holding twenty-four stocks that each make up more than 1% of my portfolio, and fourteen more that are under 1% each. This kind of imbalance isn’t uncommon: Peter Lynch sometimes had more than 1,000 stocks in his portfolio, but still had some very large positions among them.

I do advocate diversification among industries or subsectors. I’m currently overly concentrated in “industrial services” (transportation, infrastructure, delivery, logistics, distribution, facilities, construction), so I’m taking some steps to limit my investment there.
Portfolio Weighting

If you use ranking systems, it makes sense to scale your positions by rank, with the highest-ranked positions having the greatest weight. There are any number of ways to specify this in a formula: if a stock’s rank position is r (i.e. for your top ranked stock, r = 1 and for your second ranked stock, r = 2), some formulas might include 25 – r, 1 / r, e –0.2 r, ln (25 – r), √ (25 – r), and so on. I advocate taking transaction costs into account when setting a stock’s weight. One way to do this would be to use one of the formulas above and multiply it by (0.06 – x) / 0.06, where x is twice the stock’s transaction cost and 6% is the average expected return of a stock. (You can adjust the expected return to match your own historical returns.)
Buy and Sell Rules

Besides the very basic buy and sell rules outlined above—buy a stock if it ranks in the top X and sell it if you need money to buy another and it ranks lower than other stocks—a few others are worth considering.

    Sell rules. You might want a minimum holding period, which would mean selling a stock only if it has been held more than X days. You might want to sell companies going through merger proceedings. I would strongly advise against using stop losses. If a stock’s price goes down but its ranking remains the same or goes up, you probably want to buy more of it. Remember that a stock’s prospects have nothing to do with the price you paid for it, since everyone who owns shares paid a different price for them. Using stop losses virtually guarantees that you buy high and sell low. Studies have shown that most active managers sell positions that have either gone way up in price or gone way down in price and hold positions whose price remains the same. Don’t do things like that. Instead base your selling on a) rank and b) what you need in order to buy something else.
    Buy rules. Instead of buying stocks simply based on their rank in your ranking system, you might want to buy them based on their rank in terms of expected excess returns. (In other words, instead of buying the five top-ranked stocks, buy the five stocks that will give you the highest expected excess returns. The difference between the two comes down to transaction costs per stock.) This gets complicated: first you have to determine how much more money a stock ranked #1 will make versus a stock ranked #5, and then you have to subtract double the transaction costs for each stock to come up with a total expected excess return. Here’s a formula you’re welcome to use. If r is the rank position of the stock and c is the transaction cost in percentage points, your expected return may be approximately 0.1 – 0.025 × ln (r) – 2 × c.

Rebalancing Frequency

I tend to rebalance or reconstitute my own portfolio every day and my kids’ portfolios twice a week. Their portfolios have outperformed mine pretty consistently. On the other hand, theirs have less market impact, being much smaller than mine. Twice a week is probably good enough. More frequent rebalancing gives you the opportunity to take advantage of strong and sudden price moves, but it also increases churn.
Holding Period and Rebalancing Size

I tend to hold positions on average eighty or ninety days, but you might want a longer or shorter holding period. In addition, I’m often adjusting position weights, so that even when I’ve held a position for five hundred days, there are portions that have been held much less and others that I might have already sold. This creates additional transaction costs, and lowers my returns. It’s therefore important to not diminish or increase positions simply because there’s a small difference in rank, but to place strict limits on rebalancing and do it only when the discrepancy is rather extreme (say more than a third of the position).
Hedging

I like to hedge my long positions with put options.

Shorting has a very lopsided return profile: you can never make more than 100% on a short position, but you can lose more than 500%. I wouldn’t recommend taking short positions, and I don’t short stocks myself. But perhaps if done very carefully, it can smooth and even increase returns. That’s still something I’m investigating. So far, I haven’t found a good method of making it work.

Put options have the opposite return profile: you can easily make far more than 100% and you cannot lose more than 100%. I favor three-to-nine-month out-of-the-money put options with a price that’s usually well under the Black-Scholes or binomial price (or, put another way, whose implied volatility is significantly less than the stock’s historical volatility; I’m willing to pay a bit more for stocks with relatively low historical volatility). Cheap puts like these are hard to find; during high volatility periods, you might only buy puts on 5% of the stocks you’re interested in. At any rate, this is by far the best hedge I’ve come across, and has increased my own returns significantly. I’ve only been buying puts for about 21 months, but my annualized return so far is about 840%. As with shorting, you want to choose your stocks carefully: buy puts on the stocks with the greatest odds of underperforming, and use a good ranking system designed for this. As for selling them, I usually wait until the week before expiration, unless my ranking system tells me that they’re no longer likely to fall in price, in which case I sell them for whatever I can get.

To keep your sanity with out-of-the-money options, it’s important to realize that

a) they’ll often expire worthless;

b) their value can plunge about ten times faster than stocks; and

c) when the market is in high-risk mode, their value may all plunge at the same time.

It’s such a rocky ride that you may not want to put more than 25% of your money in them.

Many people like to hedge only under certain market conditions. I have not found any system of reliably predicting market conditions, and have always found market timing and/or tactical asset allocation unrewarding. Almost all such systems are overfitted. Yes, of course, there are certain times when it’s better to be out of the market or refrain from shorting anything. But predicting those is a very tall order, and if you happen to be even slightly wrong you can squander your best opportunities. It’s far easier to use a relatively constant hedge: 75% to 80% long and 20% to 25% short/puts.
Easing into Positions

If you’re facing massive market impact, it’s far better to ease into and out of positions by buying or selling a small amount daily than trying to get into positions all at once. On the other hand, there may be a small cost of delay, especially right after an event or earnings announcement. These are the considerations to weigh when you’re thinking of easing into or out of positions.
Taxes

It’s always best to invest in retirement accounts, where taxes aren’t an issue.

If you have both a cash account and a retirement account, I advise trying to make as many of your yearly losses occur in the cash account as possible and as many of your gains in the retirement account as possible. The following practices may help you avoid paying capital gains taxes.

    If you can move stock into a retirement account, a business account, a foundation, or as a charitable donation, move the stock that has appreciated the most (in dollar terms, not percentage).
    If you’re going to sell a portion of a position that’s held in both your cash account and your retirement account, first look to see if the stock has appreciated in the cash account. If it has, sell the portion in your retirement account; if it has lost money, sell the portion in the cash account.

Last year I made 22% on my investments, but because I followed the two rules above, I paid no capital gains tax at all.
Margin

It’s not a bad idea to use margin in your non-retirement accounts, as long as you do so carefully. Here are a few recommendations.

    Negotiate a good margin interest rate with your broker. Interactive Brokers charges the lowest margin rate, and you can always threaten your broker to leave and use IB instead.
    Keep track of which stocks are marginable and which are not. Buy the marginable ones in your margin accounts and the non-marginable ones in your retirement accounts.
    When allocating how much to spend on a stock, take margin into account. It’s better to buy larger amounts of marginable stocks.
    Watch market volatility. I use CBOE’s RVIX index, which is a lot like VIX except it gauges the Russell 2000. When the VIX or RVIX goes up, decrease your use of margin.
    Always keep a cushion. When calculating the amount of margin to use, add 5% to your margin requirements for each stock. The idea is to avoid margin calls.

Here’s a quick primer on how I manage margin.

Let V be the market value of all the stocks in your margin account, let R be the weighted average margin requirement of those stocks, and let D be your total debt. Your surplus will then be V – D – R × V, or (1 – R) × V – D.

Next, let’s calculate your worst-case expected future drawdown (this is based on your current assets, not on your high-water mark). I use 0.65% of the VIX minus 1%. So if the VIX is at 20, your expected drawdown is 14%, and if the VIX is at 40, it’s 29%. We’ll call that number dd.

Now recalculate the value of your portfolio if the drawdown happens: V × (1 – dd). Then calculate the margin requirement of your stocks at that lower value if your margin requirements went up by 5% (since some margin accounts will increase margin requirements in the case of high risk): (R + 0.05) × V × (1 – dd). Your new surplus, after the disaster, would be V × (1 – dd) – D – (R + 0.05) × V × (1 – dd), which simplifies to (0.95 – R) × V × (1 – dd) – D. When I’m buying and selling stocks, I try to keep that new surplus above zero at all times.

For example, let’s say the market value of all the stocks in your account is $750,000, your weighted average margin requirement is 36%, and you have $350,000 in margin debt. Your surplus is therefore 0.64 × $750,000 – $350,000 = $130,000.

Now let’s say the VIX is at 20, giving you an expected drawdown of 14%. Your new surplus will be 0.59 × $750,000 × 0.86 – $350,000 = $30,550. You can therefore spend $30,000 on buying new stocks with a margin requirement of 100%, and quite a bit more on new stocks with a lower margin requirement. As long as you do this kind of math, you should never get a margin call. I never have, even during the COVID crash.
Conclusion

All of this can be overwhelmingly complicated. That’s why I began this article with a very simple portfolio management method. I still think that it is an excellent foundation, and everything else is icing on the cake. But many people never even think about portfolio management. They keep a huge amount in cash in case they see a stock they want. They buy on whim, and base the weight of their new buy on things like how much cash they happen to have. They sell when stocks go way up or way down, disregarding their prospects. They get margin calls; they get into and out of hedges on a moment’s notice; they overdiversify; and so on. All of this will diminish their investing success.

Systematic and thoughtful portfolio management should be a key part of your investing process. Don’t let your stock-picking talents go to waste by not managing your portfolio well."