The magazine of the Institute & Faculty of Actuaries
.

# Smoother ways of smoothing

The concept of smoothing of investment returns has recently been having a rough ride. The aim of this article is to outline an objective smoothing algorithm that may be fairer than current discretionary methods.
The key is to view smoothing as the realisation of a set of forward contracts of different durations. Let us use a maturing life policy as an example. In the case of a linked policy, the value at maturity would depend on the market price of the units attached. The value could be smoothed if the maturing policyholder were to enter into series of forward contracts (each for part of the maturity value) in the months prior to maturity.

A simple theory
The theory for determining forward prices is relatively simple. If I enter into a contract to buy an asset at a particular price in one year, it would be equivalent to borrowing money to buy the asset now, and repaying the loan at the end of the year. Similarly, the counterparty could sell the asset now and put the money on deposit. The forward price of the asset should therefore be the current market price plus one year’s interest. The rate of interest to use will depend on the bargaining power of the participants, tax, expenses, credit risks, and market conditions. The interest rates can be derived from similar future contracts, but the market in these is small at present.
Suppose we start to smooth n years before maturity (n may be smaller than the term of the contract). The smoothed value at maturity would then be determined by the following formula assuming dividends were accumulated, and formed part of the price at maturity.
UtPt(1+it)n-t

Where: Ut is the number of units committed to forward contracts at time t
Pt is the market price of the units at time t
it is the spot rate of interest of term n-t at time t
Insurers could choose an appropriate value for the smoothing period n, and a formula for Ut. The smoothed return would be equivalent to that obtained by disinvesting from equities, and buying fixed-interest assets as maturity approaches.

There are three main reasons for using a smoothing formula rather than switching assets or entering into future contracts, even if these were available for the required terms (quite apart from the possible unavailability of such contracts for the required terms).
u There are no dealing or margin costs.
u Incoming policyholders are able to gear themselves cheaply.
– Depending on the market, less tax may be normally payable on equities than on fixed-interest income.
This approach also allows for the determination of fair, market-consistent, surrender values without anti-selection risks. The forward prices can be worked backwards to determine their current market value for purposes of surrender, or to allow for contracts with a term shorter than the smoothing term to participate in the smoothing. The surrender value at time 0 UtPt(1+it)n-t/(1+ik)n-k + [TUk-Ut”*Pk

where TUk is the total number of units allocated to the policy at time k.

What might have been
I have looked to see how this approach would have applied in South Africa over the past 30 years. Given South Africa’s relatively high and volatile rates of inflation, I was reluctant to use nominal spot rates, and so I would rather determine the price of the forwards using the consumer price index (CPI) and use real interest rates for pricing. The first formula above then changes to:
UtPt(1+rt)n-t(CPIn/CPIt)

Where: rt is the real spot rate of interest of term n-t at time t
CPIt the consumer price index at time t
Because we do not have real interest rates going back more than a few years, I took the dividend yield on the all-share index as an estimate. I used a value of
n = 60 months for the smoothing period, and
Ut =1/(n-t+1)*value of uncommitted units allocated at time t.
The results shown in figure 1 across are encouraging. The graph shows the maturity payouts for a ten-year policy with regular premiums of 1 per month. The large payouts in the 1980s reflect two decades of inflation in excess of 15%pa. Maturing smoothed policyholders would have received much smoother returns than linked policyholders. They would have been protected from much of the share market slump of the 1970s and other stockmarket crashes, but would not have benefited from the euphoria of the late 1970s or the mid-1980s and early 1990s. The results do not appear to be particularly sensitive to adjustments in the smoothing term and formula.
The same approach can obviously be applied to defined contribution retirement funds, and to with-profits annuities and pensions. The record-keeping may get fairly complicated, but it is not insurmountable. However, there seems no point in applying it to defined benefit funds as the employer would in effect be on both sides of the forward contracts.

Essence of the approach
One question that arises is the relationship between with-profits contracts and this approach to smoothing. If the essence of the with-profits approach is participation in pooled experience with a minimum of guarantees, then this approach is entirely consistent. It can be seen as a pooling of investment experience over time.
It also significantly reduces the value and cost of investment guarantees. Given my experience with South African inflation, I regard guarantees as gimmicks where they are issued responsibly, and foolish otherwise. It would, however, be an interesting exercise to see the impact of this approach on the cost of guarantees.
The other aspect of this question is the extent to which this smoothing is consistent with the actuarial approach that discounts future dividends to determine a smoothed value of investments. I was surprised twice in following up this train of thought.

The surprises
First, there was the pleasant surprise in finding a relationship between my smoothing formula and the discounting of dividends. This arose from my use of dividend yields as a proxy for real interest rates. What would happen if the CPI were to be replaced by a dividend index? We get:
UtPt(1+dyt)n-t(Dividend indexn/Dividend indext)

where dyt = Dividend indext/Pt.
If the dividends are paid rather than accumulated in the unit price, this becomes:
UtPt(Dividend indexn/Dividend indext)

= Ut(Dividend indexn/dyt)

= Pn*dyn* Ut/dyt

ª Pn*dyn/(Averaged dy)
ª Pn*dyn/(Long term dy) or actuarial value

The second surprise was less pleasant. The second graph shows the real values of the all-share index, and the actuarial value of the assets if the assets are valued by treating the real dividend index as a perpetuity at 4%. The actuarial value may be made more or less consistent with forward pricing. Smooth it is not. It may be smoother in more diversified markets, and there may be ways of ad hoc adjustment, but it does not appear suitable for smoothing in South Africa. My preferred approach would therefore be to use a real rate of interest and inflation to determine smoothed forward prices.

In conclusion
It seems to me that any smoothing method that cannot be fitted into this framework would be unfair to either the policyholder concerned or to the insurance company and/or remaining policyholders.
If I am right, we have available a method of smoothing the investment returns of life companies and defined contribution retirement funds that is objective, fair, and flexible and offers participation in the equity risk premium without the associated volatility. Actuaries can be smooth!

02_09_03.pdf