The mathematics of diversification

Readers that have followed my blogs for a while will know that I’m a big fan of diversification. It’s one of the central tenets of good investment management, but it’s commonly ignored in local authorities today.  So let me take you on a numerical tour of the subject, with apologies to those whose A-level maths is a bit rusty.

It’s quite common at the moment for local authorities to spread investments across just five big UK banks, keeping maturities quite short and earning around 0.50% interest.  As everyone knows, to increase returns you have two basic options – extend maturity dates or widen credit criteria.  Both avenues normally come with more risk, but it’s not always appreciated how much this extra risk can be minimised if you also increase diversification.  Luckily there are more potential counterparties with lower credit scores, so it’s quite possible to do just that.

Firstly, let’s consider lending fixed 3 year deposits to our favourite five banks at 2.5% per annum.  The rating agencies tell us that the risk of an A-rated bank defaulting in that time period is around 0.40%, or 1 in 250.  With five banks then, we have a 1 in 50 chance of suffering a default. I don’t know how many authorities express their credit risk appetites in those terms, but I suspect most would want a higher probability of maintaining capital value than that.

Now let’s loosen the credit criteria by one grade and lend for the same term to 25 smaller banks and building societies, maybe with a BBB credit rating. With some careful selection, we can earn around 3.5%, but the default rate for each counterparty is now 1%.  Our chance of being exposed to a default is now around 25%, or 1 in 4.  But the key is that one default won’t break the bank – losing 4% of our principal is more than covered by the 3 years of 3.5% interest earned on the other 96% of our investment.  In fact, even 2 defaults still yields an adequate return of (92% x 3.5% x 3 – 8%) = 1.66% over 3 years or 0.55% a year.  We would need 3 defaults before we started losing money.  Remembering our A-level probability theory we know that the chance of exactly three defaults is (0.99^22 x 0.01^3 x 25 x 24 x 23 / 6) = 0.18%.  Including the ever remoter chances of 4 or more defaults gives us a failure rate of 0.20% or 1 in 500. That’s starting to sound like the type of risk we could run with.

Moving down another level, we could lend our money to 100 small businesses, maybe via a peer-to-peer lending website.  Returns are at least 6% a year, with expected lifetime defaults of around 4%.  If we define our success criteria as earning more than the base rate of 0.50% (which is close to the 3 year gilt yield or risk free rate) then we can see that we now need to suffer 17 defaults out of 100 to be a net loser from this strategy.  With bigger numbers like these, we can use a binomial distribution to show that the chance of 17 or more defaults is one in 2 million.

We can go on. You could invest in high yield corporate bonds with gross returns of 7% and an 8% default rate. Far too risky for most authorities to make single bond purchases, I’m sure.  But a highly diversified fund of 250 bonds would need a fifth of the fund to default before it returned you less than base rate. The chance of that is 1 in 200 million while the average default rate remains 8%.

Of course, company default rates over the next three years might be rather higher than the recent average, increasing the risk of getting a lower total return.  But don’t forget that more bankrupt companies and redundant employees will increase banks’ losses too and raise the chances of them defaulting as well.  Economic downturns affect everyone – even the government.

Now, some would say that the overuse of mathematical models was one of the causes of the financial crisis.  But this was exacerbated by the banks’ massive leverage – some banks lent £50 for every £1 of their own money, borrowing the rest from depositors and bondholders.  This meant that a 1% loss on their assets translated to a 50% drop in their reserves.  Local authorities have a much more prudent balance sheet structure, so aren’t nearly as susceptible to being wiped out by small losses.

This simple model can be extended to include a range of default probabilities, the effects of compound interest, and sums recovered from companies in default, but the underlying picture remains the same.  With wider diversification, you are more likely to get an average level of failures, which is easier to budget for and build into the required rate of return, and you’re more likely to earn yourself a higher overall yield.

David Green is Client Director at Arlingclose Limited.  This is the writer’s personal opinion and does not constitute investment advice.

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