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The Actuary The magazine of the Institute & Faculty of Actuaries
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Answering a costly question

THERE IS A CURRENT FASHION in certain quarters of the actuarial profession which seeks to challenge the basic elements of actuarial thinking. A prime example is Exley and Smith’s article, ‘Discounting actuarial theory’ (The Actuary, October 2001), where the authors say, not once, but twice, that the cost of funding a liability cannot be reduced by investing in equities, rather than in bonds.

The economics used by Exley and Smith are impeccable. Where they go wrong is in asking themselves one question and then treating the answer as though it applied to a multitude of other, quite different, questions. To mis-use a phrase: they are truthful with their economics, but are they, perhaps, a little too economical with their truths?

Savings fund example
Let’s take a simple example. My client tells me that he wishes to save up for an event 12 time periods from now. The cost of the event when it arises is expected to be £1,200. He wants to know how much to set aside in a ‘savings box’ at the start of each time period. He prefers to save equal amounts on each occasion, as far as is possible. The savings vehicle, he tells me, is nothing more adventurous than a jar of money above the fireplace.

Given that question, my answer is that he should save £100 each time period. I attach a caveat: that he should keep an eye on the estimated cost of the event, and ask me to review the savings plan if the estimate varies from the figure of £1,200 that I used for my calculations.

But before the savings plan swings into action, my client comes up with a variation on the original question. He has just heard about deposit accounts at savings institutions, and would like to put the money in one of those, instead. He tells me he can expect a return of 0.5% per time period, and he wants to know if that alters my recommended level of saving.

Using a well-known formula based on a geometric progression, I calculate that £1 per time period, invested at 0.5% per period, will grow to £12.40 at the end of 12 time periods and I tell him that he can reduce his periodic savings to £96.80 per period. I repeat my previous caveat and add a new one: that if there is any reason to suppose that his estimated return of 0.5% per period will not be maintained, he should invite me to review my calculations.

So, by switching from the jar above the fireplace to an investment vehicle guaranteed to return slightly more than zero per period, my client can reduce the periodic amount of his savings plan. That much is unarguable. But has he reduced his cost? Under either scenario, he will still have to shell out £1,200 at the end of 12 time periods and, if he saves via a deposit account, he may have too little or too much at the critical moment. He will have the exact amount only if the actual rate of return turns out to be equal to the assumed rate.

In economic terms, if my client exercises the option to invest in a jar above the fireplace, the extra amount of cash he needs to set aside each period is not part of the cost of the event. The extra amount is the cost incurred in electing not to use an efficient savings vehicle. [NB: I’m not saying the deposit account is the most efficient vehicle, but it’s more efficient than the jar above the fireplace.”

But splitting hairs between the cost of the ‘event’ and the cost of the ‘choice of savings vehicle’ is too subtle for most people. So far as my client is concerned, the cost of the savings plan is £100 per period if he uses the jar and £96.80 per period if he uses the bank.

There are many other ‘costs’, depending on alternative investment schemes. A particularly interesting cost is the amount per period if my client finds a third party willing to guarantee to provide the event in exchange for a fixed periodic charge in the meantime. Provided that the third party is safe and secure to do business with, that any shortfall in the savings plan will be met by the third party, not by my client, and that the third party is pricing his service at the going rate, we can call this third cost the ‘true economic cost’ of the event. The difference between this ‘true’ cost and the cost under any other savings plan reflects the cost of electing to adopt an alternative savings strategy.

Equities
Now, what about those equities? Given that my client’s plan is to save for a single event at the end of 12 time periods, equities introduce an enormous risk. There might be too much in the fund, or there might be too little, and, if this will have to be made good, the cost estimate using an equity savings vehicle will have to include the cost of any final top-up (or refund) in the event that the actual rate of return does not match the assumed rate. For my client, that could be a heavy risk.

But let’s change the liability to something more familiar to actuaries. Let’s suppose that my client sets up a joint fund with everyone in his town, all of whom agree to make similar savings. Under their scheme, a ‘time period’ is a month. The event is each individual’s birthday (which can be assumed for these purposes to be distributed randomly throughout the year) and the £1,200 is spent on a birthday party.

Let us add some additional features. The birthday parties will take place not just once per person, but every year, for all the town’s people (including future generations), so the fund will keep going indefinitely.

There are no separately identifiable ‘accounts’ within the fund all money is pooled jointly across individuals and across time periods. (In other words, any surpluses are reinvested in the scheme and any deficits are met by drawing on the assets within the fund.)

In the event that the scheme is ever discontinued, the assets held at termination will be divided among the scheme members according to some agreed rule. So long as the scheme continues, there will be an annual re-assessment of the savings plan, having regard to the latest estimated cost of a birthday party and the expected return on investments.

With a scheme like that, the use of equities is now not looking so inappropriate. True, there is no matching between existing assets and liabilities, but there is no need for it. Given the indefinite time horizon of the fund, the only risk is that the equity market will fall so low that funds will run out before the next savings reassessment. And that has been allowed for in the rules: everyone agrees that, if the scheme stops, they will share out what is left recognising that there may not be anything left. Faced with such a scheme, the economist will realise that the periodic saving amount is not simply the cost of a birthday party, but rather the aggregated cost of:

>> a birthday party;
>> the decision to gamble on whether the scheme will be able to fund a birthday party in every year; and
>> the freedom of the scheme members (collectively) to choose an investment vehicle.

So long as we accept that wider definition of the thing we are costing (party + gamble + investment freedom), how does the cost vary according to the choice of investment vehicle? Here is a selection of three answers out of the infinite variety of possible variations:

1 With all moneys invested in a portfolio of fixedinterest securities selected to mature on the various dates of scheme members’ birthdays, the cost of the gamble can be reduced to zero.

2 With the moneys put in a jar above the town’s collective fireplace, the insolvency gamble is held at zero, but the cost arising from the freedom to select a zero-return investment vehicle makes the overall cost higher than in scenario 1.

3 But switch to equities and we can expect to make a saving on the investment selection element of the cost, at the expense of an increase in the gamble. In the long run, the cost will be lower when equities are used as the investment vehicle than when fixed interest (or a jar) is used. If the scheme continues indefinitely, the funds may never actually run out, but in those scenarios where the funds do run out, the members will experience the consequences of the right to gamble which was part of the scheme.

And the moral of the story is: It is not possible to reduce the economic cost of a pure liability by switching out of a matching asset into a non-matching asset. But it is possible to reduce the cost of something comprised of liabilities + gamble + investment freedom.

Companies with defined benefit pension schemes are generally very well aware that their pension trust includes investment freedom. They also know that, although there are regulations to prevent them gambling with the solvency of the scheme, they are allowed to take their chances with the cost. So why is there a desire among certain actuarial economists for quoting costs based on a pure liability, rather than the cost based on the client’s scheme as it really is, allowing for the actual selection of investments, rather than a hypothetical matching asset?

Shouldn’t we be answering the question our clients are really asking, not some arcane economic question that looks similar to the client’s question, but is actually rather different?

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