The magazine of the Institute & Faculty of Actuaries
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# Discounting actuarial theory

In this article we take a fresh look at the core actuarial skill of discounting cashflows to calculate present values. Some of the early conclusions may well shock those who had the importance of consistency between the asset and liability discount rate drilled into them in the actuarial exams.

Common ground
Before we launch into the complexities of compound interest, we need to establish a bit of common ground. Most readers will probably agree that the price of a portfolio is the sum of its parts, so that, for example:
Value of A + Value of B = Value of (A+B)
With financial assets, the proposition is verified by the argument that if it were false, then infinite wealth could be created simply by buying and selling combinations of As and Bs.
To see the relevance of this simple (but abstract) linear algebra in day-to-day actuarial work, we can think of A and B in terms of a pension fund, say. For example, consider a scheme closed to new entrants. Ignoring complexities such as expenses and costs of recovering terminal surplus, we could express the total aggregate (past and future service) value of the liabilities to the existing members as L and the assets as A. We can then say:
Value of L - Value of A = Value of (L-A) = Cost of future contributions
If we start at the left-hand side then it is quite clear that the value of £100 of equity assets (A) is equal to the value of £100 of bond assets. It should also be clear that the liability value (L) is independent of the choice of assets (hint: think of an unfunded pension scheme, or think of a pension liability as similar to any other corporate liability). Thus if the left-hand side is independent of the way the assets are invested, so is the right-hand side. So why do common actuarial valuation techniques appear to show that the cost of future contributions reduces as more is invested in equities? Confusion seems to arise when actuaries value (L-A) as a single entity, and this is where the choice of discount rates becomes rather crucial.

The big mistake
The trouble starts with the use of a single deterministic scenario in a valuation, whereupon we all seem to be programmed into the idea that if we use a single scenario then the ‘average’ outcome is the one to use. So if we think that bonds will return 4.5% per annum, while equities will on average return 9% per annum, and we plug these higher equity returns into our valuation of (L-A), we might conclude that the cost of future contributions is reduced by investing in equities instead of bonds. However, this is only evident if we use the same discount rate (say 4.5% per annum) to convert the liability and asset cashflows into a present value in both cases, and herein lies the big mistake in conventional actuarial practice.
If we return to the left-hand side and discount the proceeds from £100 of equity investment (growing at 9% per annum if income is reinvested) at 4.5% per annum in isolation, then we would find that £100 of equities has infinite value, if we hold on to it for long enough (1.09/1.045)N for large N and yet its market value is £100! But of course if we lived in a simple (single scenario) deterministic world, then it would be impossible for two riskless assets to exist with different (single outcome) expected returns nobody would buy a bond returning 4.5% per annum if equities were equally certain to return 9% per annum.

Standard approaches
There are two different standard approaches in the valuation literature to ensure that we do not arrive at contradictions such as this.
– We discount the proceeds of equity investment at 9% per annum as an implicit recognition that the asset is more risky (so that we return to £100 of equities being equal to £100 x (1.09/1.09)N = £100 for all holding periods). To make the books balance in a combined projection of asset and liability cashflows, we might well need to discount future contributions at a low or even negative rate of interest.
– We assume that equities return the same as bonds in our single-scenario world. The risk adjustment is then effectively applied to the equity cashflows themselves, and we can justify valuing all cashflows at bond yields. This approach is not as strange as it sounds. However, it does take a while to realise that there is nothing particularly special about the arithmetic or geometric average outcome, or to realise that the median (50th percentile) investment return is no more special than any other percentile. The choice of average here is just an arbitrary mathematical construct. Indeed, if in practice there is a range of possible returns for equities, then one possible outcome could be that equities do indeed return the same as bonds. Picking this outcome as our risk-adjusted average scenario is therefore just a different way of choosing an average, with the added bonus that using this average actually gives consistent answers.
In practical terms, if we are calculating the present value of combined asset and liability cashflows to value (L-A), the second approach is probably the most feasible. It does not mean that we ‘expect’ equities to return the same as bonds, it just means we get an internally consistent answer most easily if we base our calculations on this hypothesis.

How to proceed
For those interested in how this extends to a more realistic model, where we recognise the volatility of equities in an asset and liability model with multiple scenarios, we find that these conclusions do have generalisations. In particular, there is a wrong way to proceed. For example, we could look at the expected outcome from equity investment (or indeed pick on any arbitrarily chosen percentile outcome) and then use the same discount rate, regardless of asset allocation, to determine the present value of the fund contributions. This approach is likely to conclude that equities reduce cost, or that equities increase cost, depending on the selected percentile or risk premium. Regardless of the sophistication of the underlying stochastic model, either conclusion would be wrong, arising as it does from an inconsistency in the method used to calculate a present value (the most traditional and fundamental of all actuarial skills).
There are, however, two sophisticated economic approaches to valuation in a stochastic framework which are direct analogues of the simple approaches.
– The state price deflator approach, which uses different discount rates in different scenarios, is analogous to using a different average discount rate in the single-scenario case for equities and bonds. The method is, however, eminently practical for valuing combined asset and liability cashflows, since the same discount rate is used in the same scenario, regardless of investment policy.
– The risk neutral valuation approach is simply an extension of the ‘bodge’ which assumes that equities and bonds have the same expected returns for valuation purposes only. There is no suggestion that the actual average returns are the same, just that this approach can be shown to give the right answer.

The law of gravity
Discounting is not, of course, merely an arcane actuarial issue. Loose references to present values of future outcomes are commonplace among our clients, governments, and the general public. Just as it is a widely held belief that heavy objects fall faster than light ones (even in a vacuum), so it is widely said that equity investment is a savvy way to reduce the cost of a liability. But this general parlance skips over the issue of how the series of certain or uncertain future payments are compressed into a single item of cost. This is where the actuary should show his or her superior understanding. The answer, of course, is that we can only talk about cost today if we have a discount rate (a consistent discount rate) with which to convert a series of future payments into a present value today. Physicists know that the greater gravitational force on a heavy object is offset by its greater mass (leaving acceleration unchanged). Actuaries should know that the higher average returns on equities need to be offset by higher discount rates.

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