discussion by Sze (2001), this approach does not appear to be fair, particularly if investment returns are poor and the fund is declining.
Blake et al. (2003) investigate the benefits of a smoothing algorithm that truncates higher and lower returns so that bonuses lie within a limited range.
Blake et al. do not, however, provide a justification for the method. All three methods are vulnerable to anti-selection.
of the policies being sufficiently large to serve as margins. Given the illiquid nature of the forwards, the margin required would be significantly higher than that required on futures contracts regularly marked to market. Determining the size of the margins required is a problem not addressed here. The objective nature of the algorithm nevertheless allows for it to be addressed explicitly, unlike the implicit algorithms of discretionary smoothing.
The payouts on policies maturing (or likely to claim) in the next five years may become large relative to the size of the fund for a particularly aged work- force. In such a case, it may be necessary to enter into future contracts in a suitable futures market, reduce the fund’s equity exposure or reduce the smoothing. Contract design may be dominated by tax considerations, which may also lead to participation in the forward contracts by the insurance company’s shareholders.
Pricing Forwards and Futures
Determining forward prices is relatively simple. Entering into a contract to buy an asset at a particular price at a future date is equivalent to borrowing money to buy the asset now, and repaying the loan at that date. 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 interest for the period.
The rate of interest to use will depend on market conditions, the bargaining power of the participants, tax, expenses and credit risks. The terms on similar listed future contracts will provide a first estimate for the interest rates to use.
Because the market in long-term future contracts (over a year or two) is rela- tively thin, short-term rates will have to be used to give an indication of the important considerations. The starting point in determining a fair rate would normally be the rate of return on riskless government stock – the standard assumption of financial economics. Those who take the short position in future contracts are able to participate in the market at this rate, so should be given a higher rate for participating in these illiquid forward contracts. They are also taking some counterparty risk, suggesting that a further margin should be added.
Policyholders taking the long position would probably be subject to borrowing rates significantly higher than the risk-free-rate – even though their position would be secured by the surrender values of their policies. This would suggest that the pre-tax interest rates should perhaps be somewhat higher than those for good-quality corporate debt. The rate on home loans might be considered as a retail analogy for a secured loan, and thus function as a ceiling.
Smoothed Maturity Value
The smoothed payout at maturity of those units committed to forward contracts would be determined by the following formula
Maturity value =
Σ
t=1n U–tP–t(1 + i–t)t (8.1)where:
maturity takes place at t = 0 nis the smoothing term
U–tis the number of units committed to forward contracts at time –t
P–t is the market price of the units at time –t, which includes reinvested dividends.
i–tis the spot rate of interest of term tat time –t.
Insurers, or the policyholders, could choose an appropriate value for the smoothing term n, and formula for U–t. An obvious formula would be
U–t= 1/(t+ 1)*
[
TU–k–Σ
t=kn U–t]
(8.2)where:
TUkis the total number of units allocated to the policy at time k.
For policies with no premiums during the smoothing term, this would provide for an equal number of units to be committed to future contracts in each of the nperiods. Where premiums are still being paid during the smooth- ing term, this formula would produce an increase in the number of units as the policies near maturity. If a more equal weighting over the period were wanted in such cases, an appropriate adjustment to Equation (8.2) would need to be made for the units that were expected to be bought by the unpaid premiums.
Anti-selection
The approach described here allows for the determination of fair, market- consistent, surrender values without the anti-selection risk that some policy- holders will be able to trade their units on terms disadvantageous to others.
This is because the forward prices can be worked backwards to determine their current market value for purposes of surrender. The surrender value at time k before maturity will be
Smoothing investment returns 153
SVk= t=k
Σ
n U–tP–t(1 + i–t)t/(1 + i–k)k+[
TU–k– t=kΣ
n U–t]
*Pk (8.3)where:
SVkis the surrender value at time k.
Time 0 would still be the planned maturity date, 0 <k <n
Another type of anti-selection problem arises if new policyholders wish to take out a policy with a term of less than n, the smoothing term. A similar problem exists if an existing policyholder wishes to pay an additional premium in the last nyears of the policy’s term. These cases are analogous to a reverse surrender, and Equation (8.3) would then form the basis for the number of units to allocate to the policy. The additional premium would be equivalent to the surrender value. There would be any number of ways of determining U–t, the number of units that would be committed to forward contracts at the time the additional premium was paid. One approach would be for no immediate allocation, that is U–t=0 for all values of tless thank. An alternative would be for a proportionate allocation; that is U–t= TU–k/nfor all values of tless than k. TU–khas to be determined as it is the number of new units to allocate in respect of the additional premium.
Effects on Investment Returns
It can be seen that the smoothed return is equivalent to that obtained by
‘lifestyle’ disinvesting from equities, and buying zero-coupon fixed interest assets as maturity approaches. As discussed above, this foregoes the equity premium as retirement approaches. The difference is that this algorithm provides for gearing in the initial years of the policy that allows for the recap- ture of the equity premium that will be foregone later. It is unlikely that the two will exactly counterbalance each other, but they should be of broadly the same value ex ante.
The proposed algorithm has other advantages over the lifestyle approach because most of the transactions should be internal (between generations of policyholders). This reduces the administration and its costs, saves the costs of brokerage on dealing, and saves the difference between the interest rates on borrowing and investments.