100921 LargePortfolios

Large portfolios reduce risk? Really? How much?

Are you looking for the 1 in 10 investments to return x10 to pay back all the failures? And expecting another 4 to return the money invested? How many investments do you need to make to have an expectation positive return? How big should your portfolio be? Business Angel investment is said to be a game for High Net Worth (HNW) individuals – just how high must the net worth be? We look at the maths behind investment probabilities to examine the realities of investments in SMEs. And the conclusion is that the HNW should be worth…

Those advising business angels tell them that they need a portfolio in order to spread risk because, out of ten investments, they can expect five Failures, four to repay the money invested with no uplift (Repay) and one to be a Star, returning ten times the investment.

So, assuming all other things are equal, what impact does portfolio size have? What is the likelihood a portfolio will make a return of > 2x the original investment?

It’s just a bit of maths and statistics. Let’s work through the case for a portfolio of two investments before we compare portfolio sizes.

With two investments having three outcomes (Failure, Returns money, Star performer) we have nine possible outcomes for our portfolio. For each of these we can work out the return and probability of that return:

Investment 1 Investment 2 Return Probability
Star Star 10 + 10 = 20 10% x 10% = 1%
Star Repay 10 + 1 = 11 10% x 40% = 4%
Star Failure 10 + 0 = 10 10% x 50% = 5%
Return Star 1 + 10 = 11 40% x 10% = 4%
Repay Repay 1 + 1 = 2 40% x 40% = 16%
Repay Failure 1 40% x 50% = 20%
Failure Star 10 50% x 10% = 5%
Failure Repay 1 50% x 40% = 20%
Failure Failure 0 50% x 50% = 25%

Removing the repeats (i.e. it doesn’t matter if you have Start and Failure or Failure and Start) we have six possible outcomes:

Investment 1 Investment 2 Return Probability
Star Star 20 1%
Star Repay 11 4% + 4% = 8%
Star Failure 10 5% + 5% = 10%
Repay Repay 2 16%
Repay Failure 1 20% + 20% = 40%
Failure Failure 0 25%

There are several observations:Since we spend 2 units of money on our portfolio, the chance of failing to get that return is 25% + 40% or 65%. The chance of equalling it is 16% and the change of exceeding it is 10% + 8% + 1% = 19%. In other words, the probability that the return will exceed x2 is only 19% although that is better than 10% for a single investment.

Ok, enough of the figure work, getting a computer to work it out, we find the following table for portfolios of up to ten investments:

Investments < investment >= investment >= 2x investment >= 3x investment
1 50% 50% 10% 10%
2 65% 35% 19% 19%
3 73% 27% 20% 20%
4 71% 29% 29% 8%
5 59% 41% 38% 7%
6 54% 47% 25% 10%
7 48% 52% 17% 12%
8 43% 57% 18% 5%
9 39% 61% 23% 5%
10 35% 65% 26% 7%
  1. Not until you reach seven investments does the likelihood of getting the investment back exceed even odds.
  2. The likelihood of losing all the investment for a portfolio of ten investments has reduced from 50% to 35%.
  3. These figures suggest a sweet spot of five investments with a 38% chance of getting twice the investment back.

Given everything else is equal (which it isn’t), the optimum portfolio size is five investments (below ten). This gives the investor maximum likelihood of greater than 2x return and is manageable.

Investing £100K per time (so transaction costs do not skew the results too much), that is an Angel pot of £500K. In general it is recommended that the Angel pot is approximately 10% of the net wealth. So to play this game for profit, an Angel’s net wealth needs to be over £5 million. And that Angel has a 38% chance of at least doubling his money over ten years.

This Post Has 2 Comments

  1. The timing of this piece amused me greatly, having just done a review of my recreational expenses and an analysis of my portfolio performance. As a private pilot and amateur sailor, the hobby related spreadsheets were surpisingly complex, and relied heavily upon certain assumptions, historical data, hopes and fears (probabilities) for the next year.

    Yesterday also happened to be our 35th wedding anniversary. My efforts have done no more than prove that if it flies, floats or f***s, renting is 65% more probable to be cheaper. Many have come to this conclusion before me but, despite the numbers, I persist in (co-)owning two of the three options available to me.

    I enjoyed your maths illustration (really – I did), but the moral I took from it is that we all rationalise towards our beliefs. Whether it is the Black-Scholes pricing model, optimum portfolio size, investment level or whatever, we can always adjust the numbers to give us the outcome that we think we want. Your sweet spot could be somebody else’s sweat spot. At my age, a 60% chance of taking out less than I put in doesn’t really appeal, even for a 40% chance (when?) of doubling.

    You did say that ‘ceteris paribus’ seldom applies in real life. From what I’ve seen, these perturbations often dominate the outcome well beyond the probability field you outlined. They are embraced by the ‘feel’ (or ‘gut’) factor.

    It would be interesting to see the data that yields the “failure”, “repay” & “star” proportions. It’s often quoted in the guise of a business axiom, but can you point me at a source?


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