- Stuart Fowler

# Is Clare being realistic about equity risk?

Ask No Monkey Business clients what they most value about our approach and they mostly answer ‘more certainty’. On the surface this is odd as one thing singling us out, which stems directly from how we model real equity returns, is that we tell people just how uncertain are their wealth outcomes when partly relying on the systematic risk inherent in equities.

Most investors in financial markets act as though they believed equity returns were ‘mean reverting’ but most agents (financial advisers or portfolio managers) then use that to play down the degree of risk, possibly deliberately. At the opposite end of the spectrum are agents (and academics) who believe that the equity return process is entirely random, and hence much riskier and far less useful to investors. Between the two extremes is a definition of equity risk, and its relation to time, that we believe is realistic.

In this item we expand on a short article of mine published in the FT on 9th November which used as a peg for the random versus mean reverting debate the example of the recently-announced decision of Clare College, Cambridge to make a leveraged 40-year bet in equities in their endowment fund. Subsequent emails suggest we need to explain this a little more fully. If you do invest in equities and you do not want it to be a blind bet (whether on agents or markets), this item will help give you some of the ‘certainty’ you are looking for.

**Clare College’s bet**
Clare College, Cambridge, like many Oxbridge colleges, enjoys an endowment fund that is invested in real or financial assets and which it can draw on to contribute to its operating budget or to meet exceptional expenditures, such as adding to or repairing the fabric of its ancient buildings. College endowments are typically managed by trustees and advised by experts (often including former alumni).

The heavy weight of history bears down on successive generations of stewards of this capital. Nobody wants to be the incumbents that blew the capital so carefully built up by their predecessors over several hundred years. So when Clare announced that it had borrowed £15 million for 40 years at a fixed real rate of interest (ie at a rate that will not be affected by either changing nominal interest rates or changing inflation) to invest in UK equities, this was bound to provoke very different reactions according to people’s view of the risk in equities. Was it a rational choice that maximises the utility (or benefit) of a permanent endowment fund or yet another example of a false free lunch? Was it wild speculation or wise investment?

Predictably this clash of views duly emerged in two letters in the FT in the last two weeks, one from consultant John Ralfe, a reliable mocker of free lunchers, and one from Norman Cumming who was one of several highly experienced investment professionals who advised the College on this occasion.

**Mean reversion**
Clare College is betting consistent with a view of the capitalist system that makes the equity return-generating process ‘mean reverting’. Deviations from long-term trends in achieved returns may be large and prolonged but they are also bounded and eventually reverse. Most capital owners have acted explicitly or implicitly as if this was the way the world works and much wealth has been created by doing so. This is also the view expounded by Mr Cumming in his reply, a view he attributed to both theory and statistical evidence.

Clare is specifically betting on a real return process, funding the bet at a fixed real rate of interest. Since inflation is the biggest known risk that a permanent endowment faces, this is consistent with its utility. But deflated equity returns also provide the strongest support, both empirically and theoretically, for a mean reversion view.

**Damned lies and statistics**
Mr Ralfe argues that Clare has fallen for an illusion that equity risk declines with time. My FT article suggested there may be people who mistake the declining annualised standard deviation of equity returns (when applied to different holding periods) for a shrinking band of future probable wealth outcomes. In fact, I can tell this from my emails. So what exactly was I referring to?

The normal means of expressing equity risk is a statistical measure of the standard deviation of returns relative to an observation of their central tendency, such as a simple average. Standard deviations are usually measured from short-period observations (such as monthly) but may then for convenience be applied to longer period returns (such as annual) with an adjustment. The adjustment is based on the relationship between a normally distributed random series of returns and the length of the time series. Mathematically, the standard deviation changes as a function of the square root of time, so a 1-month standard deviation scales up to a 12-month standard deviation with a multiple of 3.46, not 12. Similarly, multi-year standard deviations when annualised will show declining values the greater the number of years.

One of the most common uses of a single-period measure of risk is in conjunction with growth rates, or mean returns, which are themselves expressed as comparable rates, such as annualised. By combining the mean rate with its standard deviation, it is possible to show ranges of growth rates that decline with time. The problem with this is that they are still growth rates, compounding over time. If those rates were applied to a particular level of starting wealth, they would show expanding ranges of probable wealth levels the longer they compound. Few people looking at exhibits of outcomes would conclude that equity risk declines the longer the holding period, because most of them rightly relate more readily to uncertainty of levels of wealth than to a statistical measure of risk.

It may not be accidental that we see few predictions of future wealth and lots of predictions of possible growth rates in wealth. Mr Ralfe has many times argued that the financial services industry is complicit in using the risk measure to play down wealth uncertainty when investing in equities. I said the same in my book. However, in Clare’s case it seemed ridiculous to argue there was any hoodwinking or room for illusion, given the quality of the advisers.

**Mean reversion, risk and time**
These observations hold for a random time series but a characteristic of mean reversion in real equity returns is that the band of probable real wealth outcomes expands less rapidly with time than if the return process is purely random. This argument does not go away with a change in the way we measure risk.

I did not include in my FT article any illustrations but, at the risk of introducing more maths, I am adding here some graphical outputs of two simple models of the equity return generating process, based on running 250 simulations of each.

In the first, we show what can happen to the starting value of a level of wealth if growing randomly but with an upward drift of 6% pa, which is about the level of the upward trend in real equity returns (including income reinvested) observable in the UK over the last century.

In the second we assume changes from year to year are not independent (the assumed reversion coefficient based on historical observation is between .8 and .9).

Both sets of simulations show an ‘expanding funnel of doubt’ but the rates at which they expand are very different. This is fundamental to the Clare bet but is also fundamental to the rationale of any equity investor.

The model parameters assuming mean reversion are very similar to those in the return-generating model No Monkey Business uses, which are derived from observations of the actual achieved real return over as long a history as possible, plus or minus an adjustment for where the starting level of the market is high or low relative to that trend. If we plot the standard deviation of the resulting ratio over different holding periods (actually the log of that ratio), which is not affected by the deviation from trend in the starting conditions, we see that the standard deviation rises very rapidly and then levels off as mean reversion takes effect.

The obverse of the actual standard deviation is also illustrated below, as the decline in annualised standard deviation of a random series compared with the mean reverting model.

**Option pricing as evidence of equity risk**
Mr Ralfe belongs to the ‘fair value’ accounting school that rightly has argued that the valuation of assets and liabilities should, as far as possible, be based on prices established in free and open markets between willing buyers and sellers.

Using information-efficient markets as the reference point clearly brings transparency and consistency to the valuation process. However, it is not universally accepted if there is no direct market reference price and if the process therefore has to adopt, by convention or diktat, some proxy. For instance, the application of fair value accounting to occupational pension schemes requires future pension promises, as deferred pay, to be discounted to a present value using a corporate bond rate, on the basis that the liability is similar to a bond, that the covenant is similar to a corporate borrower and that there is a readily available yield in the corporate bond market for such a bond. Not everyone agrees that this logic holds.

The fair value approach is relevant to the relationship between equity risk and time because the cost of options provides market-based evidence of the scale of the risk. In his letter to the FT Mr Ralfe reiterated his frequent argument that ‘theoretical’ option prices support his view that the degree of equity risk is being understated.

Well, it would, if you extrapolate from actual option prices at one horizon the theoretical prices at another while still assuming a random process. This is simply because option pricing generally does assume randomness. This is because over the relatively short periods of the vast majority of contracts, including equity options, this is the most sensible assumption. As we saw above, even a mean reverting equity process looks barely distinguishable from a random process over a few years. So extrapolating from actual option prices ignores the reality that there are conflicting views of the return process at different horizons. It cannot prove anything.

**Investor utility**
It is possible for an investor to hold that short-term nominal returns are essentially random and that long-term real returns are mean reverting but it implies a need to select between them when defining utility.

Clare is effectively saying ‘we and future generations of trustees can and will bear the interim volatility in order to accept this degree of probable future incremental wealth’. In doing so, they are saying something about the way they think the world works but also about their natural advantage in it.

I know other college endowments who have left the same bet on the table solely because they doubted successor committees’ willingness to stick to the strategy through thick and thin. That would be enough to make path volatility, rather than long-term payoffs, dominate the strategy, even if their implied utility, given their natural advantage as an independent permanent institution, is dominated by real wealth outcomes.

The impact of fair value accounting on investment practice has been highly significant precisely because it has changed the utility definition for two of our largest ‘democratised’ capitalists: life insurance companies and pension funds. Between them, they held much of the long-horizon wealth of the nation. Accounting changes redefined their utility in terms of short-term nominal volatility at the expense of real wealth outcomes. I leave it to others to judge in due course whether accounting clarity was worth the social cost of effectively preventing democratised pools of capital from acting like capitalists. I am sure, though, that we do not want to drive other investors whose utility is not dominated by path risk to stop taking equity risks. I place both endowments and individual investors in that category.

**Why leverage the equity bet**
Mr. Cumming points out that that the rationale for the bet partly rested on the low level of the fixed real borrowing cost (fractionally over 1%). Ironically, this low rate itself probably reflects the imposition of fair value accounting on UK pension funds, causing them to hedge inflation-linked liabilities by buying index linked gilts. Rather than issue mich more stock, the Government has profited from a false market by keeping its marginal borrowing rate low (a form of the fallacy that it is independent of the people it represents).

Because Clare knows its real opportunity cost, the bet is directly comparable with an optimal asset-allocation choice for individuals who are planning in real terms and whose time horizon is long enough for all or much of the range of probable real equity outcomes to exceed the risk free alternative of index linked gilts.

In terms of the chart illustrating the expanding future wealth outcomes with mean reversion, the slope of that funnel-like distribution is comparable with a straight line whose slope is equal to the real yield on index linked gilts. In our own real return-generating model, the cross-over, at which all the equity outcomes are above the real gilt yield (at 99% confidence), currently arises for the UK All Share Index (including the cost of tracking) at 22 years. This is shorter than ‘usual’ because the market (at the end of October 2008) is about 20% below its long-term trend.

At its own real cost of borrowing, Clare should not expect to need to wait 40 years for its bet to pay off, even in the event of a dire economy and stock market.

At typical levels of risk aversion, portfolios for individuals with a 22-year horizon will be fully invested in equities. Yet their portfolios would be suboptimal (assuming the same utility as Clare) if they could also borrow at that rate and chose not to.

In practice, individuals cannot fix their real cost of borrowing, cannot borrow as cheaply as Clare and may not be as confident as Clare that they can cover the interest payments out of the portfolio without a serious impact on their household budget. These disadvantages suggest individual investors will accept lower long-term real wealth outcomes than appear to be achievable given their risk appetite. They may not have an endowment’s problem of worrying that their utility will be defined differently by a different set of decision makers before the plan is over. But because individuals’ attitudes may shift (such as with age) and circumstances alter (with events like divorce), they may not be able to hold their course either.

Consider, however, that any family taking out a long-term mortgage loan (whether for their primary residence or for buy-to-let) has implicitly acted consistent with the same utility as Clare. If, instead of renting property, someone is willing to rent money to buy property, they should be equally open to renting money to buy equities. In practice, those who have taken out a mortgage probably thought they were leveraging a property investment but as a financial planner, thinking in terms of how to maximise their utility or welfare, we could re-express that as leveraging its financial assets, earnings power or overall balance sheet.

**Conclusion**
Equities are dangerous but, managed properly, they can help individuals, trusts and endowments to achieve outcomes they value highly enough to justify the risk. The key to proper management is matching the risk to the time horizon. Equity returns are close to random over a 5-year period that many agents describe as ‘long term’! You would probably not bet the housekeeping money on a 50:50 bet but that is what many mismatched portfolios effectively do.

If you do not believe that mean reversion explains the way equity returns are generated, and that the historical evidence is either statistically not proven or not valid as a basis of future predictions, you are not alone. But the implication is that you should not be invested in equities for any horizon. More realistically, unless you are wealthy enough to achieve your goals using only index linked gilts, you will end up having to gamble on something else that can produce higher real returns (perhaps more property, or hedge funds as the ‘new’ alternative).

Finally, if enough natural risk takers are driven out of the game by rules that shift their utility away from wealth outcomes to path volatility, and from real assets to nominal assets, even mean reversion assumptions could go out of the window. Without capitalists there is no capitalism and it is capitalism that is the root of mean reversion.