Actuarial notation glossary

Every symbol and technical term used in the calculators’ working and on the Learn pages is defined here, in the same words as the pop-up definitions you can tap anywhere on the site. Entries are grouped by the page that explains them in full.

Actuarial notation packs a lot into one symbol. The main letter says what is measured ( $p$ survival, $q$ death, $e$ expectation of life, $a$ an annuity value). The subscript on the right is the age, or ages, it applies to. A subscript on the left is a time span, and a vertical bar there means the payments or the risk start only after that time. Marks above the letter change the version: two dots for payments in advance, a ring for complete rather than whole years, a bar over two ages for the last survivor. An angle around $n$ marks a fixed term instead of a lifetime.

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339 symbols and terms

Life tables

Explained on the page about life tables.

$x$ Age x: Your age today, in whole years.
$q_{y}$ Death probability q_y: The same one-year death probability for the second person, at age y.
$q_{x}$ Death probability qₓ: The chance that someone aged exactly x dies before reaching x + 1, read from the life table.
$d_{x}$ Deaths dₓ: In a life table, the number of the lₓ survivors who die between ages x and x + 1.
$\omega$ Limiting age ω: The age by which everyone in a life table is assumed to have died. Survival to ω or beyond is zero.
$p_{x}$ One-year survival pₓ: The chance that someone aged x survives one more year: 1 − qₓ.
Period life table: A table of one year's death rates at every age. It does not allow for death rates falling in future, so people alive today will probably live a little longer than it shows.
$l_{x}$ Survivors lₓ: In a life table, the number of people still alive at exact age x out of a starting group (100,000 at birth in SSA tables).

Survival probability

Explained on the page about survival probability.

${}_{A-x}p_{x}$ Chance of reaching age A: The chance that someone aged x is still alive A − x years from now, when they would turn A.
${}_{A-y}p_{y}$ Chance of reaching age A (spouse): The chance that the second person, aged y, lives to turn A.
$q_{x+k}$ Death probability in year k: The one-year death probability at age x + k, the year being part-lived.
$q_{y+k}$ Death probability in year k (spouse): The spouse's one-year death probability at age y + k.
${}_{t}q_{x}$ Death probability ₜqₓ: The chance that someone aged x dies within t years: 1 − ₜpₓ.
${}_{k|}q_{x}$ Deferred death probability: The chance that someone aged x survives k years and then dies in the year after: kpₓ times qₓ₊ₖ.
$s$ Fraction of a year s: The part of the next year, between 0 and 1, after the last whole year: t = k + s.
$p_{x+k}$ One-year survival at age x + k: The chance that someone who has reached age x + k survives one more year: 1 − q at that age.
${}_{k}p_{y}$ Survival probability ₖp_y: The same k-year survival chance for the second person, aged y.
${}_{k}p_{x}$ Survival probability ₖpₓ: The chance that someone aged x is still alive k whole years from now. It multiplies (1 − q) for each year in turn.
${}_{t}p_{y}$ Survival probability ₜp_y: The chance that the second person, aged y, is still alive t years from now.
${}_{t}p_{x}$ Survival probability ₜpₓ: The chance that someone aged x is still alive t years from now.
$t$ Time t: A length of time from now, in years. It can include a fraction of a year.
Uniform distribution of deaths: The assumption that deaths within a year of age happen evenly through that year. It fills in survival between birthdays.
$k$ Whole years k: A whole number of years from now.

Life expectancy and IRS tables

Explained on the page about life expectancy.

Applicable denominator: The IRS name for the distribution period: the number the year-end balance is divided by to get the required minimum distribution.
Cohort life table: A table that follows people born in the same year, using the death rates expected for them at each future age, including expected improvements.
$\mathring{e}_{y}$ Complete life expectancy (spouse): The average remaining lifetime of the second person, aged y.
$\mathring{e}_{x}$ Complete life expectancy e̊ₓ: The average number of years still to live for someone aged x, counting part years. The ring marks the complete version.
$K_{x}$ Curtate future lifetime Kₓ: The number of whole years someone aged x will go on to live: Tₓ rounded down.
$e_{x}$ Curtate life expectancy eₓ: The average number of whole years still to live for someone aged x. It is about half a year less than the complete figure.
$L$ Distribution period: The IRS life expectancy factor for your age. The required minimum distribution is the balance divided by it.
$x + \mathring{e}_{x}$ Expected age at death: Age today plus the complete life expectancy: the average age at death for people like you under the table.
$y + \mathring{e}_{y}$ Expected age at death (spouse): The second person's age today plus their complete life expectancy.
$T_{x}$ Future lifetime Tₓ: The number of years, including part years, that someone aged x will go on to live. Unknown in advance, so it is treated as a random quantity.
$A$ Milestone age A: An age the calculator asks about, such as 80, 90 or 100.
$\text{Balance}$ Prior year-end balance: The account balance on December 31 of the year before the distribution year.
$\text{RMD}$ Required minimum distribution: The smallest amount the IRS requires you to take out of a tax-deferred retirement account for the year.
Single Life Expectancy Table: IRS Table I. Beneficiaries who take distributions over their life expectancy use it.
$L_{0}$ Starting distribution period L₀: The Table I life expectancy at the beneficiary’s age in the year distributions begin.
Uniform Lifetime Table: IRS Table III. Most IRA owners use it for distributions during their lifetime.
$j$ Years since distributions began j: The number of years after the first distribution year. The Table I figure is reduced by one for each.

Joint life and last survivor

Explained on the page about joint and last-survivor probabilities.

$\Delta$ Age gap Δ: How many years younger your spouse is than you, using both ages on your birthdays this year.
$y$ Age y: The second person's age today (your spouse or partner), in whole years.
$P_{A}$ Chance at least one reaches A: The chance that at least one of two people lives to age A, each reaching A in their own time. It assumes their lifetimes are independent.
${}_{n|}\ddot{a}_{xy}$ Deferred joint-life annuity: The joint-life annuity, counting only payments after year n.
${}_{n|}\ddot{a}^{(12)}_{xy}$ Deferred monthly joint annuity: The monthly joint-life annuity, counting only payments after year n.
Independent lifetimes: The assumption that one person's death doesn't change the other's chances. Real couples' lifetimes are somewhat linked.
Joint and Last Survivor Table: IRS Table II. An owner uses it when the spouse is the sole beneficiary and more than 10 years younger.
$\ddot{a}_{xy}$ Joint-life annuity äₓᵧ: The present value of 1 a year paid while both people are alive. It stops at the first death.
$\mathring{e}_{xy}$ Joint-life expectancy: The average number of years until the first of two people dies.
${}_{t}p_{xy}$ Joint-life probability: The chance that both of two people are still alive t years from now.
$\ddot{a}_{\overline{xy}}$ Last-survivor annuity: The present value of 1 a year paid at the start of each year while at least one of two people is alive.
$\mathring{e}_{\overline{xy}}$ Last-survivor expectancy: The average number of years until the second of two people dies, that is, while at least one is alive.
${}_{t}p_{\overline{xy}}$ Last-survivor probability: The chance that at least one of two people is still alive t years from now. The bar over xy means "last survivor".
$\ddot{a}^{(12)}_{xy}$ Monthly joint-life annuity: Monthly payments of 1 a year in total while both people are alive.

Mortality laws and adjustments

Explained on the page about mortality laws.

$q'_{x}$ Adjusted death probability: The one-year death probability after the smoker adjustment: the table rate times the multiplier m, capped at 1.
$q'_{y}$ Adjusted death probability (spouse): The spouse's one-year death probability after the smoker adjustment.
$\mu_{x}$ Force of mortality μₓ: The instantaneous death rate at exact age x, per year. For small values it is close to qₓ.
Gompertz law: The observation that adult death rates rise roughly exponentially with age: each extra year of age multiplies the death rate by about the same factor.
$\alpha$ Gompertz level α: The scale of the Gompertz death rate: the force of mortality the fitted curve would give at age 0.
$\beta$ Gompertz slope β: How fast the force of mortality rises with age under the Gompertz law. Each year of age multiplies it by e to the power β.
$\lambda$ Makeham constant λ: The part of the death rate that does not depend on age, such as accidents, in the Makeham extension of the Gompertz law.
$m$ Smoker multiplier m: How many times higher the yearly death rate is taken to be for a smoker. A rough adjustment to population tables, not a smoker table.

Interest and present value

Explained on the page about present value.

$\Delta B$ Account value given up: How much less the account holds at the end than it would have, from money taken out or borrowed.
$B_{\text{no loan}}$ Account with no loan: The vested balance grown at the assumed return to the comparison age.
$B_{\text{loan}}$ Account with the loan: The balance left after the loan grown at the assumed return, plus each repayment grown from the day it is paid.
$B^{\text{today}}_{\text{end}}$ Balance in today's dollars: The final balance divided by the growth in prices over the years, so it can be compared with prices now.
$V^{\text{today}}_{\text{Roth}}$ Balance in today’s dollars: The projected balance divided by the growth in prices, so it reads in today’s buying power.
$V^{*}_{529}$ Balance needed: The balance at the start of school that pays every school year in full, with the rest still earning the return.
$x_{b}$ Barista age: The first age at which savings are enough to stop contributing, cover the gap each year from savings, and still reach the FIRE number by your retirement age.
$N_{b}$ Barista FIRE savings needed: Savings needed to switch to part-time work: the FIRE number discounted to that age, plus the value of the yearly gap.
$W_{b}$ Barista gap: Spending that part-time income does not cover, drawn from savings at the start of each part-time year. Never below zero.
$x_{c}$ Coast age: The first age at which savings reach the coast number, so growth alone could reach the FIRE number by your retirement age.
$F_{\text{coast}}$ Coast FIRE number: What invested today would grow to the FIRE number by your retirement age at the real return, with no more contributions.
$F_{\text{coast}}(k)$ Coast FIRE number in year k: The coast number k years from now. It rises each year, because there is less time left to grow.
$j_{\text{edu}}$ Cost inflation: Your assumption for how fast school costs rise each year.
$v^{t}$ Discount factor for t years: What 1 paid t years from now is worth today at interest rate i: 1/(1 + i) multiplied by itself t times.
$v$ Discount factor v: What 1 paid one period from now is worth today: 1/(1 + i).
$F_{n}$ Final balance: The plan balance at the end of the last year, in future (nominal) dollars.
$F^{\text{real}}_{n}$ Final balance in today’s dollars: The final balance divided by the growth in prices over the years, so it reads in today’s dollars.
$x_{F}$ FIRE age: The first age at which savings reach the FIRE number if contributions continue every year.
$F_{\text{FIRE}}$ FIRE number: Spending a year divided by the withdrawal rate you assume, in today’s dollars: the savings that rate says would cover your spending.
$F_{\text{FIRE}}^{\text{nom}}$ FIRE number in future dollars: The FIRE number grown with inflation to your retirement year: the same buying power in the dollars of that year (nominal).
$K_{\text{first}}$ First-year cost: The cost of the first school year when it arrives, in future dollars.
$\delta$ Force of interest δ: The interest rate compounded continuously that matches i over a year: the natural log of (1 + i).
$FV$ Future value: What a set of payments grows to by the end, at a given interest rate.
$G$ Growth G: Investment growth in the year after fees.
$G_{\text{yr}}$ Growth in the year: Investment growth after fees on the starting balance and the year’s contributions.
$\pi_{\text{inf}}$ Inflation: The yearly rise in prices you assume, used to show future dollars in today’s money.
$\pi_{\text{CPI}}$ Inflation: Your assumption for the yearly rise in consumer prices.
$\pi$ Inflation π: The yearly rise in prices. Spending and income are assumed to rise at this rate.
$i$ Interest rate i: The interest rate per period, used to move money between dates.
$r_{\text{inv}}$ Investment return: The yearly return you assume for the investments, before fees and inflation (nominal).
$h_{\text{mid}}$ Mid-year factor: Contributions arrive through the year, so on average they are invested for about half of it: 0.5.
$I_{p}$ Part-time income: Income a year from part-time work in Barista FIRE, in today’s dollars.
$n_{b}$ Part-time years: Years of part-time work, from the Barista age to your retirement age.
$PV$ Present value: What a set of future payments is worth today, at a given interest rate.
$d$ Rate of discount d: Interest expressed as a share of the amount at the end of the period: i/(1 + i), or 1 − v.
$r_{\text{real}}$ Real return: The yearly return after inflation: (1 + r)/(1 + π) − 1. Money in today’s dollars grows at this rate.
$x_{R}$ Retirement age: The age by which the FIRE number is needed, when you would stop work altogether.
$r$ Return r: The yearly investment return, before inflation (nominal).
$V_{k}$ Savings after k years: Savings after k years of contributions and growth at the real return, in today’s dollars.
$h$ Timing factor h: The share of a year contributions are invested on average. Money paid in evenly through the year earns about half a year of return.
$W_{\text{real}}$ Withdrawal in today's dollars: The grown traditional balance in today's dollars, used to find the bracket it lands in.
$c_{\text{yr}}$ Yearly contribution: What you add to savings each year, in today’s dollars, paid at the end of the year and rising with inflation.
$i_{\text{yr}}$ Yearly interest rate: The interest rate for a whole year, compounded once a year (an effective rate).
$n_{\text{yrs}}$ Years: The number of years in the projection or until the money is used.
$n_{R}$ Years to retirement: Whole years from your age today to your retirement age.

Annuity formulas

Explained on the page about annuity formulas.

$V_{529}$ 529 balance at school start: The projected balance when the first school year begins, before paying for it.
$(Gs)_{\angl{n}}$ Accumulated growing annuity: What n payments that start at 1 and grow by g each period, paid at the end of each period, add up to by the end.
$(G\ddot{s})_{\angl{n}}$ Accumulated growing annuity-due: The same accumulated growing annuity with payments at the start of each period.
$s_{\angl{n}}$ Accumulated value sₙ: What 1 paid at the end of each period for n periods grows to at the end.
$\ddot{s}_{\angl{n}}$ Accumulated value s̈ₙ: What 1 paid at the start of each period for n periods grows to at the end.
$\text{factor}$ Annuity factor: The value of a payment of 1 each period. Multiply by the payment to get the value, or divide a value by it to get the payment.
$\ddot{a}_{\angl{n}}$ Annuity-due äₙ: The present value of 1 paid at the start of each period for n periods. The two dots mark payments in advance.
$\ddot{a}_{b}$ Annuity-due factor for the part-time years: The value today of 1 a year drawn at the start of each part-time year, at the real return.
$a_{\angl{n}}$ Annuity-immediate aₙ: The present value of 1 paid at the end of each period for n periods. The angle around n marks a fixed term.
$(Ga)_{\angl{n}}$ Growing annuity: The present value of n payments that start at 1 and grow by g each period, paid at the end of each period.
$(G\ddot{a})_{\angl{n}}$ Growing annuity-due: The same growing annuity with payments at the start of each period.
$(Ga)_{\infty}$ Growing perpetuity: The present value of a payment that starts at 1 and grows by g each period forever, paid at the end of each period: 1/(i − g). It needs g below i.
$(G\ddot{a})_{\infty}$ Growing perpetuity-due: The same growing perpetuity with payments at the start of each period: (1 + i)/(i − g).
$g$ Growth rate g: How fast each payment grows over the one before it, per period.
$I_{\text{loan}}$ Interest over the loan: All payments minus the amount borrowed. It goes back into your own account.
$A_{\text{pay}}$ Loan payment: The level amount paid each time, covering interest and part of the loan.
$N_{\text{pay}}$ Number of payments: Payments a year times the term in years.
$n$ Number of payments n: How many payments there are, or how many years a guarantee lasts.
$P$ Payment P: The amount of each payment.
$m_{\text{pay}}$ Payments a year: How many loan payments are made each year, usually one per paycheck. The law requires at least quarterly.
$f$ Payments a year f: How many payments are made each year: 1 for yearly, 4 for quarterly, 12 for monthly.
$a_{\infty}$ Perpetuity: The present value of 1 paid at the end of every period forever: 1/i.
$\ddot{a}_{\infty}$ Perpetuity-due: The present value of 1 paid at the start of every period forever: (1 + i)/i.
$j_{\text{loan}}$ Rate per payment: The yearly loan rate divided by the number of payments a year.
$V_{\text{Roth}}$ Roth IRA balance: The projected balance at the end, in future (nominal) dollars.
$C^{*}_{529}$ Saving needed each year: The level yearly contribution that builds the balance needed by the time school starts.
$N$ Term in years N: How many years the payments last.
$g_{\text{yr}}$ Yearly payment growth: How much the payment grows over a whole year, as a percentage.

Life annuities

Explained on the page about life annuities.

${}_{n|}\ddot{a}_{x}$ Deferred life annuity: A life annuity whose payments start only after n years. The bar after n marks the deferral.
${}_{n|}\ddot{a}_{y}$ Deferred life annuity (spouse): The spouse’s life annuity, counting only payments after year n.
${}_{n|}\ddot{a}^{(12)}_{y}$ Deferred monthly annuity (spouse): The spouse’s monthly life annuity, counting only payments after year n.
$e$ Escalation e: The yearly increase in annuity payments: 0 for a level annuity, or expected inflation for a CPI-linked one.
$\ddot{a}_{\overline{x:\angl{n}}}$ Guaranteed life annuity: Payments for n years whether or not the person is alive, then for as long as they live.
$\ddot{a}_{y}$ Life annuity-due (spouse): The same life annuity value for the second person, aged y.
$\ddot{a}_{x}$ Life annuity-due äₓ: The present value of 1 a year paid at the start of each year for as long as someone aged x lives.
$a_{x}$ Life annuity-immediate aₓ: The present value of 1 a year paid at the end of each year for as long as someone aged x lives. It equals äₓ minus 1.
$\ddot{a}^{(12)}_{x}$ Monthly life annuity: The present value of 1 a year paid in 12 monthly instalments at the start of each month, for life.
$\ddot{a}^{(12)}_{y}$ Monthly life annuity (spouse): The same monthly life annuity for the second person.
$P_{\text{month}}$ Monthly payment: Each monthly payment: the yearly income divided by 12.
$\text{Payout rate}$ Payout rate: The first year’s income as a percentage of the premium. It is not an interest rate: part of each payment is your own money coming back.
$\text{Premium}$ Premium: The lump sum paid to buy the income.
$\rho$ Reversion ρ: The share of the income that continues to the spouse after the main annuitant dies, e.g. 50%.
$\ddot{a}$ Total annuity factor: The value of 1 a year of starting income, allowing for every feature chosen: guarantee, reversion and increases.

Market returns and simulation

Explained on the page about sequence-of-returns risk.

$a$ Age a: An age to check against: the calculation counts the histories in which savings last past it.
$\bar{r}$ Average return r̄: The average (arithmetic mean) yearly return in the normal model. By default it is the return you entered.
$R^{b}_{1}$ Bond return Rᵇ: The 10-year US Treasury bond return in the calendar year drawn for year 1 of a simulated history.
$\bar{P}$ Chance of outliving the money: The chance someone is still alive when the money runs out, averaged over every simulated history. Histories where the money lasts count as 0.
$D$ Depletion age D: The age at which the savings run out in one simulated history.
$n_{a}$ Histories lasting past a: The number of simulated histories in which the savings last past age a.
$n_{F}$ Histories reaching the FIRE number: The number of simulated histories in which savings reach the FIRE number by your retirement age.
$\pi_{1}$ Inflation in year 1: The inflation a simulated history uses in its first year: your inflation figure in the normal model, or that calendar year’s CPI change in the historical model.
$\sigma_{L}$ Log-return volatility σ_L: The volatility of the logarithm of one plus the return, set so that yearly returns have the volatility σ.
$D_{50}$ Median depletion age: The age the money runs out in the middle simulated history: half of the histories run out earlier and half later.
$V_{50}$ Middle-history savings: Savings at your retirement age in the middle simulated history: half of the histories end higher and half lower.
$N_{\mathrm{sim}}$ Number of histories: How many simulated market histories the calculation runs, for example 10,000.
$Z$ Random draw Z: A random number from the standard bell curve (mean 0, standard deviation 1) that sets one year’s return in the normal model.
$r_{1}$ Return in year 1, r₁: The investment return a simulated history uses in its first year, before inflation (nominal).
$V_{1}$ Savings after year 1: Savings at the end of the first year of a simulated history, in today’s dollars.
$w_{s}$ Share in stocks wₛ: The part of the savings held in US stocks; the rest is in 10-year US Treasury bonds. Rebalanced every year.
$\Pr(D > a)$ Share of histories lasting past a: The share of simulated market histories in which savings last past age a. It describes the simulation, not a forecast.
$\Pr(V_{n_{R}} \ge F_{\text{FIRE}})$ Share of histories reaching the FIRE number: The share of simulated market histories in which savings reach the FIRE number by your retirement age. It describes the simulation, not a forecast.
$R^{s}_{1}$ Stock return Rˢ: The S&P 500 total return, dividends included, in the calendar year drawn for year 1 of a simulated history.
$\Sigma_{p}$ Sum of survival chances: The chance someone is alive at the age the money runs out, added up over all histories (0 for a history where it never runs out).
$\sigma$ Volatility σ: How much yearly returns swing around their average: their standard deviation. About two years in three fall within one σ of the average.

Retirement projection

Used by the how long will my money last calculator.

$C_{15}$ 403(b) 15-year catch-up: Extra 403(b) deferrals for employees with at least 15 years at a qualifying employer such as a school, hospital or church: the least of three amounts the IRS sets.
$L_{402(g)}$ 403(b) or 401(k) limit: The deferral limit for a 403(b) or 401(k) at your age, which is separate from the 457(b) limit.
$L_{457}$ 457(b) limit: Your 457(b) deferral limit for the year.
$C_{457}$ 457(b) special catch-up: Extra 457(b) deferrals in the three years before the plan’s normal retirement age, using limit you did not use in earlier years. It cannot be combined with the age catch-up in the same year.
$C_{529}$ 529 contribution each year: What goes into the 529 plan at the end of each year until school starts.
$P_{10}$ Additional tax (the "penalty"): The 10% additional tax on an early distribution.
$r_{\text{add}}$ Additional tax rate: The 10% additional tax on early distributions taken before 59½ unless an exception applies (IRC §72(t)).
$r_{p}$ Additional tax rate: The additional tax on early distributions under section 72(t) of the tax code.
$\text{age}$ Age at separation: Your age when you leave federal service to retire.
$C_{a}$ Age catch-up: Extra deferrals the IRS allows once you reach catch-up age, with a higher amount in some years of your early 60s.
$W_{\text{pen}}$ Amount the additional tax applies to: The taxable part of the withdrawal not covered by an exception.
$i_{C}$ Assumed C Fund return: Your own assumption for the C Fund’s yearly return. Past returns don’t predict future ones.
$i_{F}$ Assumed F Fund return: Your own assumption for the F Fund’s yearly return. Past returns don’t predict future ones.
$i_{G}$ Assumed G Fund return: Your own assumption for the G Fund’s yearly return. Past returns don’t predict future ones.
$i_{I}$ Assumed I Fund return: Your own assumption for the I Fund’s yearly return. Past returns don’t predict future ones.
$i_{S}$ Assumed S Fund return: Your own assumption for the S Fund’s yearly return. Past returns don’t predict future ones.
$F_{1}$ Balance after one year F₁: The balance at the end of the first year, after the withdrawal and growth.
$B_{\text{end}}$ Balance at the end of the year: The starting balance plus the year’s contributions and growth.
$B_{\text{start}}$ Balance at the start of the year: The account balance on January 1 of the year.
$F_{\text{last}}$ Balance before the last year: The HSA balance at the start of the final year of the projection.
$F_{k}$ Balance Fₖ: The balance at the start of year k.
$A_{0}$ Basic annuity: The FERS annuity before any reduction: multiplier × high-3 × years of service, a year.
$L_{b}$ Basic limit: The IRS limit on elective deferrals for the year before any catch-up: the 402(g) limit for 401(k), 403(b) and TSP, the 457(e)(15) limit for 457(b).
$Y_{b}$ Birth year: The year you were born, which sets your Minimum Retirement Age.
$L_{\text{both}}$ Both plans together: The most you can defer in a year with a 457(b) and a 403(b) or 401(k), because the two limits are separate.
$t^{*}_{\text{ret}}$ Break-even tax rate: The retirement tax rate at which Roth and traditional give the same after-tax amount: the tax rate now.
$L_{\text{catch}}$ Catch-up limit: The extra amount allowed if you are 50 or older by December 31, with a higher amount for ages 60 to 63.
$D^{\text{Roth}}_{\text{catch}}$ Catch-ups that must be Roth: From 2026, if your FICA wages from the employer last year were over the IRS threshold, your age catch-up contributions must be Roth.
$C_{\text{pre}}$ Contribution (pre-tax pay): The pay set aside: all of it goes in as a traditional contribution, or it goes in as Roth after income tax is paid on it.
$p_{\text{eff}}$ Contribution rate the match sees: Your contributions for the year divided by your pay. It can be below your chosen rate if you reach the limit.
$C$ Contributions C: Everything paid into the plan in the year: your deferrals and employer money.
$C_{\text{yr}}$ Contributions in the year: Your contributions plus employer money for the year.
$K_{\text{today}}$ Cost of a school year today: Tuition, fees, housing and other costs for one school year at today’s prices, as you entered them.
$\text{CPI}$ CPI increase: The yearly rise in the Consumer Price Index that sets the cost-of-living adjustment. Here it is your own assumption.
$p$ Deferral rate p: The share of each paycheck you put into the plan.
$L_{\text{base}}$ Elective deferral limit: The most you can put into a 401(k) from your pay in a year before catch-ups, set by the IRS each year (§402(g)). Traditional and Roth contributions count together.
$E$ Employer balance E: Money in the account from employer contributions and its growth.
$C_{\text{emp}}$ Employer contributions: Match and other employer money added to your account in the year.
$C_{r}$ Employer contributions: Match plus contributions the employer makes whatever you defer.
$E_{\text{HSA}}$ Employer HSA contribution: What your employer puts into your HSA in the year. It counts toward the same limit.
$X$ Exempt amount X: Social Security’s earnings test exempt amount for the year: earnings up to it don’t reduce the supplement.
$T_{\text{fed}}$ Federal income tax on the withdrawal: The extra federal income tax the withdrawal causes.
$r_{\text{fee}}$ Fees: Yearly fund and plan costs as a share of the balance, taken off the return.
$c$ FERS COLA c: The yearly cost-of-living adjustment to a FERS annuity. It follows the CPI rise in full only when the rise is small; the formula shows the bands.
$A_{s}$ FERS supplement: A monthly payment until 62 for people who retire on an immediate annuity before then. It stands in for the Social Security earned by FERS service.
$W_{1}$ First-year withdrawal W₁: The amount taken from savings in the first year.
$Y_{c}$ Full career: The number of years the FERS supplement treats as a full career.
$\text{Match}^{\max}$ Full match: The most the match formula pays in a year, earned by contributing at least the top of the last tier.
$G_{1}$ Growth G₁: Investment growth in the first year, on the balance left after the withdrawal.
$H_{3}$ High-3 average salary: Your highest average basic pay over any three consecutive years of service.
$H_{12}$ Highest loan balance in 12 months: The largest amount you owed the plan at any time in the 12 months before the new loan.
$c_{\text{HSA}}$ HSA additional contribution: The extra HSA contribution allowed if you are 55 or older by the end of the year.
$V_{\text{HSA}}$ HSA balance: The projected HSA balance at the end, in future (nominal) dollars.
$b_{\text{HSA}}$ HSA limit for the coverage: The yearly HSA limit for self-only or family high deductible health plan coverage.
$\Delta V_{\text{HSA}}$ HSA tax difference: The HSA balance minus the taxable account balance: what the HSA’s tax treatment adds under these assumptions.
$B$ Income B: Social Security, pensions and other regular income in a year, in today’s dollars.
$c_{\text{IRA}}$ IRA catch-up: The extra IRA contribution allowed if you are 50 or older by the end of the year.
$\ell_{\text{IRA}}$ IRA contribution limit: The most anyone can put into traditional and Roth IRAs together for the year, before any catch-up.
$X_{\max}$ Largest new loan: The legal limit, not more than the vested balance left and not below zero.
$X_{\text{law}}$ Legal loan limit: The largest new loan the tax law allows (IRC §72(p)). A plan can allow less, or no loans.
$c_{2}$ Lifetime cap c₂: The most the 403(b) 15-year catch-up can add over a whole career.
$\ell_{\text{Roth}}$ Limit after the phase-out: The Roth IRA limit once the income phase-out is applied, rounded up to the next $10.
$b_{\text{IRA}}$ Limit before the phase-out: The lesser of the IRA limit (with any catch-up) and your taxable compensation.
$L_{\text{loan}}$ Loan amount: The amount borrowed from the plan.
$A_{\text{cap}}$ Loan dollar cap: The $50,000 maximum on plan loans, reduced by recent loan balances.
$r_{\text{loan}}$ Loan interest rate: The yearly rate the plan charges. Plans set it; the interest is paid back into your own account.
$O_{\text{now}}$ Loans outstanding: What you owe the plan today on earlier loans.
$\text{Match}_{\text{due}}$ Match before the annual additions limit: The match the formula pays before the §415(c) cap on total yearly additions is applied.
$\text{Match}$ Match by the formula: The employer match for the year worked out on the year's contribution rate.
$\text{Match}_{1}$ Match from tier 1: The employer match earned on the part of your contributions in the first tier of the formula.
$\text{Match}_{2}$ Match from tier 2: The employer match earned on the part of your contributions in the second tier of the formula.
$\text{Match}_{k}$ Match from tier k: The employer match earned in one tier of the formula.
$\text{Missed}$ Match left on the table: The full match minus the match paid: employer money not received in the year.
$\Delta M$ Match left on the table: The match the formula offers that you did not receive this year.
$M$ Match M: The employer matching contribution for the year.
$\text{Match}_{\text{paid}}$ Match paid: The match actually paid in. A plan that matches each paycheck stops matching when your contributions stop, unless it makes a year-end true-up.
$m_{k}$ Match rate in a tier: Cents the employer adds for each dollar you put in within that tier: 100% is dollar for dollar, 50% is 50 cents.
$M_{1}$ Match, tier 1: The employer match earned in the first tier of the match formula.
$M_{2}$ Match, tier 2: The employer match earned in the second tier of the match formula.
$A_{\text{floor}}$ Minimum loan allowance: The $10,000 the law allows even when half the vested balance is less, if the plan permits it.
$\ell_{\min}$ Minimum reduced limit: A reduced Roth IRA limit above zero but below this amount is raised to it (Publication 590-A).
$\text{MRA}$ Minimum Retirement Age: The earliest age a FERS employee can retire on an immediate annuity. OPM sets it by the year you were born.
$\text{MAGI}$ Modified AGI: Adjusted gross income with certain items added back or taken out, as Publication 590-A defines it for Roth IRA purposes. Income from a Roth conversion is left out.
$u$ Months under 62: Full months you are under age 62 when the annuity starts.
$M^{*}$ Most the match pays: The match you would get by deferring at the top of the match formula all year.
$N_{\text{last}}$ Net deposits in the last year: Contributions from you and your employer in the final year, less medical costs paid from the HSA. They earn half a year’s growth.
$X_{415}$ Over the annual additions limit: Employer money above the §415(c) limit on what can go into your account in a year (catch-ups are not counted toward it).
$O_{\text{file}}$ Owed at filing: Federal tax and additional tax due when you file, less what was withheld. Negative means a refund.
$S_{\text{pay}}$ Pay: Your salary for the year, before tax.
$S_{\text{elig}}$ Pay counted for the match: Your pay up to the IRS annual compensation limit (§401(a)(17)); pay above it is not matched.
$c_{3}$ Per year of service c₃: The amount per year of service in the third 15-year catch-up test, before taking off earlier deferrals.
$A_{\text{Roth}}$ Phase-out start: The modified AGI above which your Roth IRA limit starts to shrink, for your filing status.
$W_{\text{Roth}}$ Phase-out width: The span of modified AGI over which the Roth IRA limit falls to zero.
$C^{*}_{\text{HSA}}$ Planned HSA contribution: The yearly amount you entered, before checking it against the limit.
$C^{*}_{\text{Roth}}$ Planned Roth contribution: The yearly amount you entered, before checking it against your limit.
$r^{\prime}$ Return after tax: The yearly return less tax on it at your income tax rate: what the taxable account keeps of each year’s growth.
$R$ Roth balance R: Money from Roth deferrals and its growth. Qualified withdrawals are tax-free.
$C_{\text{Roth}}$ Roth contribution each year: The yearly contribution used in the projection: your planned amount, capped at your limit and held level.
$D_{\text{Roth}}$ Roth contributions: The part of your contributions made as Roth.
$\Delta V$ Roth minus traditional: The difference in after-tax value. Positive when Roth comes out ahead under the assumptions; negative when traditional does.
$s_{\text{Roth}}$ Roth share: The part of your contributions you choose to make as Roth (taxed now, tax-free later if qualified).
$V^{\text{net}}_{\text{Roth}}$ Roth value after tax: What the Roth contribution is worth when withdrawn tax-free, after the tax paid on it now.
$F_{0}$ Savings today F₀: The balance of your savings now.
$c_{529}$ Share covered: The share of the total cost the 529 plan pays, drawing the balance down each school year.
$w_{C}$ Share in the C Fund: The share of your TSP account in the C Fund.
$w_{F}$ Share in the F Fund: The share of your TSP account in the F Fund.
$w_{G}$ Share in the G Fund: The share of your TSP account in the G Fund.
$w_{I}$ Share in the I Fund: The share of your TSP account in the I Fund.
$w_{S}$ Share in the S Fund: The share of your TSP account in the S Fund.
$q_{\text{Roth}}$ Share phased out: How far your modified AGI is into the phase-out range, as a fraction rounded to three places. That share of the limit is removed.
$SS_{62}$ Social Security estimate at 62: Your estimated Social Security benefit at 62 for a full career, a month. The FERS supplement starts from it.
$S - B$ Spending gap: Spending minus income: the amount savings must cover each year, in today’s dollars.
$S$ Spending S: What you spend in a year, in today’s dollars.
$X_{\text{state}}$ State deduction cap: The most your state lets you deduct for 529 contributions in a year, as you entered it.
$T_{\text{state}}$ State income tax: State income tax on the taxable withdrawal at the rate you entered.
$t_{\text{state}}$ State tax rate: The state income tax rate you enter for the withdrawal. No state rules are built in.
$D_{\text{state}}$ State tax saved: State income tax saved by deducting contributions, added up over the years until school starts.
$A'_{s}$ Supplement after the earnings test: The yearly supplement after the earnings test takes off part of any earnings above Social Security’s exempt amount.
$T_{U}$ Tax on an unpaid loan: Federal and state income tax and any additional tax on a loan balance that is treated as a distribution.
$\text{Tax}_{\text{hi}}$ Tax on the higher income: Federal income tax on the income that includes the slice being measured, from the 2026 brackets and standard deduction.
$\text{Tax}_{\text{lo}}$ Tax on the lower income: Federal income tax on the income without that slice.
$t_{\text{ret}}$ Tax rate in retirement: The tax paid per dollar of the traditional withdrawal, on top of your other retirement income.
$t_{\text{now}}$ Tax rate now: The federal tax saved per dollar of a traditional contribution this year: the rate on that slice of your income.
$\tau$ Tax rate on withdrawals τ: Your own assumption for the tax rate on pre-tax money when it comes out of the plan.
$\Delta T_{\text{HSA}}$ Tax saved on the contribution: Income tax, and payroll tax through payroll, not paid because the contribution is deducted or excluded from pay.
$\text{Tax}_{\text{with}}$ Tax with the withdrawal: Federal income tax for the year on your other income plus the taxable withdrawal.
$\text{Tax}_{\text{without}}$ Tax without the withdrawal: Federal income tax for the year on your other income alone.
$V_{\text{taxable}}$ Taxable account balance: What the same pre-tax pay grows to if it is taxed as wages and invested in an account whose growth is taxed each year at your income tax rate.
$F^{\prime}$ Taxable balance before the last year: The comparison taxable account’s balance at the start of the final year.
$E_{\text{comp}}$ Taxable compensation: Wages, salary, self-employment income and similar pay. IRA contributions for the year can’t be more than this.
$N^{\prime}$ Taxable net deposits in the last year: The same pre-tax dollars after income tax (and payroll tax through payroll), less the same medical costs.
$W_{\text{tax}}$ Taxable part of the withdrawal: The withdrawal less any part that is not taxable, such as your own Roth contributions.
$c_{k}$ Tier end: The contribution rate, as a share of pay, where a tier of the match formula ends.
$b$ Tier end b: Where a match tier ends, as a share of pay.
$a_{k}$ Tier start: The contribution rate, as a share of pay, where a tier of the match formula begins.
$K_{\text{total}}$ Total cost: The costs of all the school years added up, each in the dollars of the year it is paid.
$O_{\text{IRA}}$ Traditional IRA contributions: What you put into traditional IRAs for the same year. It uses up part of the same yearly limit.
$V^{\text{net}}_{\text{trad}}$ Traditional value after tax: What the traditional contribution is worth when withdrawn, after the tax due then.
$S_{529}$ Uncovered cost: The part of the school costs the account runs short of, added up over the school years.
$U_{457}$ Unused 457(b) limit: Basic 457(b) limit from earlier years of the plan that you did not defer.
$U$ Used before U: 403(b) 15-year catch-up contributions made in earlier years.
$V$ Value V: A balance after an adjustment: the vested share, or the value after tax at your assumed rate.
$V_{\text{vest}}$ Vested balance: The part of the account you own outright: your contributions plus vested employer money.
$\text{Match}_{\text{vested}}$ Vested match: The part of the match you would keep if you left the employer.
$v_{\text{share}}$ Vested share: The part of employer money you would keep if you left now, set by the plan’s vesting schedule and your years of service.
$K$ What you keep: The withdrawal less federal and state income tax and the additional tax.
$W_{\text{gross}}$ Withdrawal: The amount taken out of the account, before any tax.
$W_{t}$ Withdrawal needed: The spending gap t years from now, grown with inflation.
$W_{k}$ Withdrawal needed in year k: The spending gap in year k, grown with inflation.
$w$ Withdrawal rate w: The first-year withdrawal as a share of savings.
$W$ Withdrawal W: The amount taken out of the plan, or the part of it the additional tax applies to.
$H_{\text{wh}}$ Withheld: Federal tax taken out of the payment before you receive it. It counts toward your tax bill when you file.
$r_{\text{wh}}$ Withholding rate: The share of the payment sent to the IRS up front: 20% for a 401(k) payment made to you, 10% by default for an IRA.
$c_{1}$ Yearly cap c₁: The most the 403(b) 15-year catch-up can add in one year.
$y_{\text{sch}}$ Years in school: How many school years the account pays for.
$Y_{s}$ Years of service: Years of service with the employer that sponsors the 403(b).
$Y$ Years of service Y: Creditable federal service in years. OPM drops any part of a month.
$T$ Years the money lasts T: How long savings cover the spending gap, including the part of the final year they cover.
$L_{\text{you}}$ Your contribution limit: The elective deferral limit plus any catch-up you use, for the year.
$p_{\text{you}}$ Your contribution rate: The share of each paycheck you put into the plan.
$D_{\text{you}}$ Your contributions: What you put into the plan from your pay in the year, traditional and Roth together.
$C_{e}$ Your deferrals: What you put into the plan in the year: your rate of pay, up to the limit for your age.
$C_{\text{HSA}}$ Your HSA contribution: Your own contribution for the year, capped so that it and the employer’s fit within the limit.
$\ell_{\text{HSA}}$ Your HSA limit: The most that can go into your HSA for the year from you and your employer together.
$t_{\text{inc}}$ Your income tax rate: The federal plus state income tax rate on your last dollars of income, as you entered it.
$t_{\text{pay}}$ Your payroll tax rate: The payroll tax rate you entered. It counts only for contributions made through payroll (a cafeteria plan); otherwise it is 0.
$L_{\text{Roth}}$ Your Roth IRA limit: The most you can contribute to Roth IRAs for the year, after the phase-out and any traditional IRA contributions.
$\tau_{\text{state}}$ Your state income tax rate: The state income tax rate the deduction saves, as you entered it.

Questions

How do you read actuarial notation like tpx?

Read the main letter first, then the subscripts. In ₜpₓ, p means a probability of surviving, the x on the right is the age now, and the t on the left is the number of years ahead. So ₜpₓ is the chance that someone aged x is alive t years from now. A q in the same place means dying instead.

What do the dots, rings, bars and angles mean in actuarial symbols?

Two dots over a, as in äₓ, mean payments at the start of each period. A ring over e, as in e̊ₓ, means the complete expectation of life, counting part years. A bar over two ages means the last survivor. An angle around n means a fixed term of n periods rather than for life.

Sources

Social Security Administration, Actuarial Life Table (period life table)

Glossary of symbols and terms