Funding the income target
How much needs to be invested today?
The certainty-equivalent discount rate works back from a future target to an opening investment, allowing for the member’s attitude to risk.
The worked pension example sets aside £1,973.04 at age 65 for its age-85 payment. This page explains how that amount is calculated, using the same investments, mortality assumptions and risk preference.
The certainty equivalent
A certainty equivalent is the certain amount a person values as highly as an uncertain outcome. For a person who dislikes risk, it is generally below the average projected outcome. How far below depends on the investment risk and their preferences.
Apply that idea to the growth of £1 over a rung’s investment horizon. The certainty-equivalent growth factor gives the amount to set aside:
Starting investment = target payment ÷ certainty-equivalent growth factor
A factor of 2.1348 gives a £1,000 target an investment-only starting cost of £468.44. Over this rung’s twenty-year horizon, that factor is equivalent to an annual compound rate of 3.8646%. A lower rate means setting aside more money today.
The rate describes the value of an uncertain outcome under the chosen preference. It does not fix the eventual investment return or the probability of reaching the target.
Definition and framework: expected utility and certainty equivalence.
Calculate the age-85 rung’s growth factor
The experiment assumes a 6.5% equity drift, 16% equity volatility and a fixed continuously compounded bond return of 3.5%. The age-85 rung starts with 44.8% in equities. That share falls monthly until it reaches zero five years before release at age 84.
The calculation uses constant relative risk aversion, with a coefficient of 3. Under the model’s lognormal investment assumptions, each month’s certainty-equivalent continuous rate is:
Portfolio drift − ½ × 3 × portfolio variance
At an equity weight w, the portfolio drift is 3.5% + w × (6.5% − 3.5%) and its volatility is w × 16%. Variance is the square of that volatility. The calculation sums the monthly rates over nineteen years, then adds the 3.5% return for the final distribution year.
The result is a cumulative continuous rate of 0.758352. Exponentiating it gives a growth factor of 2.134755. The horizon-average continuous rate is 3.7918%; its annual compound equivalent is 3.8646%.
Funding £1,000 at age 85
- Investment growth, allowing for risk
- £1 becomes £2.1348 in certainty-equivalent terms over twenty years.
- Starting cost without sharing
- £1,000 ÷ 2.1348 = £468.44
- Allow for survival to release at 84
- £468.44 × 56.7665% = £265.92
- Scale to the ladder’s annual target
- £265.92 × £7,419.80 ÷ £1,000 = £1,973.04
The growth factors and starting amounts for all forty rungs · The experiment’s method.
Then allow for longevity sharing
The age-85 payment releases at 84. The synthetic mortality model gives a 56.7665% chance of reaching that release date. In an ideal pool with a predictable survivor uplift, the member can therefore start with the same proportion of the investment-only capital.
Ideal pooled starting amount = survival to release × target payment ÷ investment CE factor
This is why £468.44 becomes £265.92 for each £1,000 target. The investment return assumption is unchanged. The reduction comes from exchanging rights over unreleased assets for the survivor allocations.
In a finite heterogeneous pool, the number, timing and size of those allocations vary. The payment results measure what the stated FTP policy actually produces. Their certainty equivalents already include the effects of sharing, so the survival adjustment must not be applied a second time.
Changing the preference changes the contribution
A coefficient of 3 is an assumption about how the member values risk. Greater aversion to low payments reduces the certainty-equivalent growth factor and increases the required investment. A minimum-income requirement would add a different constraint: enough assets or separately priced protection to meet that floor.
A constant-risk comparison, including a matching bond
This separate analytical example holds investment risk constant to make the comparison transparent. It uses a 6% annual drift, 10% volatility and relative risk aversion 3 for nineteen years, followed by a known 4% distribution-year return.
6% − ½ × 3 × (10%)² = 4.5%
For a £10,000 payment in twenty years, the growth factor is exp(0.045 × 19 + 0.04) = 2.4473, giving a starting amount of about £4,086.
Three starting amounts for a £10,000 target in year 20
Hypothetical investment assumptions; no pooling adjustment.
The mean-return calculation requires £3,073. The allowance for risk adds about £1,013. A matching nominal bond at this example’s 4% continuous yield costs £4,493.
With £4,086 invested, the model still has about a 33% probability of paying less than £10,000. Its fifth-percentile payment is £5,904. Raising the risk-aversion coefficient from 3 to 5 increases the required investment to about £4,941, above the £4,493 matching cost.
These assumptions differ from the forty-rung experiment. Amounts are nominal, with no fees or tax. Read the derivation and numerical checks · Calculation source.
Value the pension as a whole
The forty-rung experiment values each annual payment conditional on reaching release. The member also has preferences about income across different years, other pension assets and provision for beneficiaries. Those preferences can change the allocation.
The calculation also needs eventual payments for the paths where the current sharing policy stops. Until a settlement rule determines those outcomes, a complete lifetime valuation cannot be obtained from the recorded payments alone.
The next study will connect a specified continuation and payment policy to the full income distribution, then use that distribution to reassess the opening contributions. The worked example’s remaining decisions define that task.
Continue the worked example: from opening investment to payment