Replacement Ratio

Study guides: Exam ALTAM, Exam RET-101

The Replacement Ratio R is a retiring member's pension income in the first year of retirement divided by her salary in the year before retirement. It measures how much of a member's working income a retirement plan replaces. A defined benefit plan's formula fixes it in advance, while a defined contribution plan can only aim at a target replacement ratio through its contribution rate.

R = \frac{B}{S_{x_r-1}}

F^{*} = R^{*}\,S_{x_r-1}\,\ddot{a}^{(12)}_{x_r}

c^{*} = \frac{F^{*} - F_0\,(1+i)^{m}}{\sum_{k=0}^{m-1} S_{x+k}\,(1+i)^{\,m-k}}

• Symbols: Following Dickson, Hardy & Waters (chapter 10), x_r is the retirement age, B is the annual pension in the first year of retirement and S_{x_r-1} is the salary earned in the year of age x_r - 1 to x_r. For a DC member aged x with m = x_r - x years to go, R^{*} is the target ratio, F^{*} is the fund that buys the target pension at the annuity price \ddot{a}^{(12)}_{x_r}, F_0 is her account today, i is the assumed return, and c^{*} is the contribution rate, with contributions paid at the start of each year.
• DB replacement ratio: In a plan whose pensionable salary is the final year's salary, R = n\alpha, so 35 years at 2% replace 70%. A final average salary, a career average, or a revaluation index that grows more slowly than salary gives less, because the pensionable salary then falls below S_{x_r-1}.
• DC target: The account at retirement is linear in the contribution rate, so a deterministic projection, with one assumed return, one salary path and one annuity price, gives c^{*} from a single linear equation. The member bears any shortfall if returns, salary growth or annuity prices turn out worse than assumed.
• Adequacy: A target is often set for total retirement income, so the plan's share is the target less the ratio that government benefits (Social Insurance) provide, as in plan design.

Final Salary versus Career Average Revalued Earnings A member joins at 30 and retires at exact age 65 with 35 years of service. Her salary rises 3.5% a year, of which 2% is price inflation. The accrual rate is 2%. Find her replacement ratio (a) in a final-salary plan based on the salary in the year before retirement, (b) in a CARE plan that revalues each year's salary by 2% a year from the end of that year to retirement, and (c) in a CARE plan that revalues by 3.5% a year.

Answer (a) The pensionable salary is S_{64}, so R = 35(0.02) = 70\%.

(b) The salary of the year of age 30 + k is S_{64}(1.035)^{-(34-k)}, and it is revalued by (1.02)^{34-k}. Writing j = 34 - k for the years from that salary to retirement:

\begin{align*}
R &= 0.02 \sum_{j=0}^{34} \left(\frac{1.02}{1.035}\right)^{j} \\
&= 0.02 \times \frac{1 - (1.02/1.035)^{35}}{1 - 1.02/1.035} \\
&= 0.02 \times 27.6055 \\
&= 55.2\%
\end{align*}

(c) Revaluing by 3.5% a year turns every year's salary into S_{64}, so the ratio is 70% again. Price revaluation keeps each year's earnings at their purchasing power, but it does not credit the 1.5% a year of real growth in her pay, which costs her 14.8 points of replacement. A CARE plan revalued with her own salary growth is the same as a final-salary plan for a member whose pay rises steadily.

Required DC Contribution Rate with an Existing Balance A DC member aged 45 has an account of $150,000. Her salary in the year of age 45 to 46 is $80,000 and rises 3% a year. Contributions of c times salary are paid at the start of each year from 45 to 64, the account earns 5% a year, and at 65 it buys a monthly life annuity at \ddot{a}^{(12)}_{65} = 13.0870 (SULT at 5%). Find the contribution rate that gives a replacement ratio of 60%.

Answer The target pension is 60% of the salary in the year of age 64 to 65.

\begin{align*}
S_{64} &= 80{,}000(1.03)^{19} \\
&= 140{,}280.48 \\
F^{*} &= 0.60(140{,}280.48)(13.0870) \\
&= 84{,}168.29(13.0870) \\
&= 1{,}101{,}510.41
\end{align*}

The existing account grows to 150{,}000(1.05)^{20} = 397{,}994.66, so contributions must provide 703{,}515.75. A contribution rate of 1 would accumulate to \sum_{k=0}^{19} 80{,}000(1.03)^k(1.05)^{20-k} = 80{,}000(44.47729) = 3{,}558{,}183.20.

\begin{align*}
c^{*} &= \frac{703{,}515.75}{3{,}558{,}183.20} \\
&= 19.77\%
\end{align*}

Without the existing balance she would need 1{,}101{,}510.41/3{,}558{,}183.20 = 30.96\% of salary. Twenty years of contributions are a short time to build a pension, and the money she saved earlier does more than a third of the work.

How it appears on the exam

• ALTAM asks candidates to calculate replacement ratios for DB plans with final average salary, career average earnings and career average revalued earnings formulas, using a salary scale, and for DC members from an account projection and an annuity factor.
• Required contribution rate: The syllabus also asks for the contribution rate that meets a target replacement ratio for a DC participant, by a deterministic approach: project the salary, find the fund that buys the target pension, and solve for c.
• Comparisons: Interpretation parts compare the same member's ratio under different formulas or plan types, or ask how a change in the return, salary growth or annuity price moves the ratio or the required rate.

Exam traps

• The wrong denominator: The ratio divides by the salary in the year before retirement, S_{x_r-1}. Dividing by the final average salary, a salary projected to age x_r, or today's salary gives a different answer.
• The wrong annuity factor: A monthly pension is priced with \ddot{a}^{(12)}_{65} = 13.0870, not \ddot{a}_{65} = 13.5498. The annual factor overstates the fund needed by 3.5% and so overstates the required contribution rate.
• The existing balance: An account already built up keeps earning interest until retirement. Subtracting F_0 without accumulating it, here $150,000 instead of $397,994.66, overstates the contribution rate.
• Contribution timing: A contribution at the start of year k earns (1+i)^{m-k}. Contributions paid monthly or at mid-year earn about half a year less, and need a slightly higher rate for the same target.