Survival Run

How it works

A cohort aged 65 collects $20,000 a year for up to 20 years, and you sold the annuity. The game animates a runner along a track of 20 hurdles - one per year of age from 65 to 84, each hurdle's height set by that year's mortality rate q from the table, which rises from 0.012 at 65 to 0.111 at 84. The distance the runner covers is the survival curve itself, shown as a live percentage.

The run halts at four checkpoints, each a numeric question with its formula and hints displayed: at year 1, the one-year survival probability p65 = 1 - q65; at year 3, the three-year survival, the product of the first three annual survival probabilities; at year 8, the expected present value of that year's payment, v^8 x 8p65 x 20,000 at a 4% annual effective rate; and at the end, the full annuity-due price, the sum over k = 0 to 19 of v^k x kp65 x 20,000. Tolerances widen with the size of the answer (0.0005, 0.0015, 60, and 900 respectively).

After the final checkpoint, news breaks: a new cardiac therapy cuts every q in the table by 25%. You priced the book on the old table, and the final question asks for the shortfall per policy - the improved-table EPV minus the priced EPV - within a tolerance of 900. Wrong checkpoint answers count as trips (tracked in the HUD) but the run continues either way; you can restart and price another cohort at the end.

How scoring works

Each of the four checkpoints pays +2 points for an answer within its tolerance; a miss adds one to the trip counter instead. The longevity-shock question pays +3. Maximum 11 per run.

Every answer locks after one check, and pressing Enter checks the current answer or continues the run once checked.

The survival-probability checkpoints record skill attempts under distributions; the present-value questions and the shock record under expected-value.

Survival compounds multiplicatively

The building blocks are q_x, the probability someone aged x dies within the year, and p_x = 1 - q_x. Surviving k whole years means clearing every hurdle in sequence, so kp_x is a product: p_x x p_(x+1) x ... x p_(x+k-1). Never add - the probabilities chain.

This compounding is the economic heart of the game: a small change in each annual q moves a long-dated survival probability far more than intuition suggests, because the change is applied 10 or 20 times over. It is exactly why the final longevity shock is expensive.

For hand computation at the early checkpoints, multiply the survival factors directly (0.988 x 0.987 x 0.985) rather than approximating - the tolerances are tight (0.0015 at year 3) and the table's q values are given in the hints.

Two discounts stack: time and mortality

A payment due k years from now is worth v^k x kp_x x amount today, where v^k = 1/(1.04)^k. One discount for the time value of money, one for the chance nobody is alive to collect. Both factors are below 1 and they multiply, which is why late payments in the schedule contribute so little to the price.

The whole annuity is just this term summed. Because it is an annuity-due, payments land at the start of each year: k runs from 0 to 19, and the k = 0 term is exactly 1 x payment - no discount, no mortality, the first payment is certain. Forgetting that certain first payment (or summing k = 1 to 20 as if payments came at year end) is the classic annuity-due versus annuity-immediate error.

For the year-8 checkpoint the game hands you both factors in the hint; the work is disciplined multiplication, keeping six decimals until the final product since the tolerance is 60 on a roughly 12,700 answer.

The annuity writer is short longevity

The shock scales every q by 0.75 - mortality falls 25% across the board. Since the liability is a sum of terms each containing kp_x, and every kp_x rises when every q falls, the EPV of the book can only go up. Reserves set on the old table are now short by the difference, per policy, and the game asks for exactly that: improved EPV minus priced EPV.

Practically, you compute the annuity-due sum twice, once per table, and subtract. The fastest reliable route is a running product for kp_x updated year by year alongside a running discount factor, accumulating the sum as you go - the same loop the game itself runs.

Notice the asymmetry the closing screen states: a 25% mortality improvement moves this 20-year liability by only a few percent, because discounting mutes distant payments and the damage concentrates in years the cohort was always likely to reach. On a whole-life annuity with no 20-year cutoff, the same improvement bites far harder - the payments it adds are exactly the ones never reserved for.

A worked example

Checkpoint 1 (age 65): p65 = 1 - 0.012 = 0.9880. Enter 0.9880.

Checkpoint 2 (three years): 3p65 = 0.988 x 0.987 x 0.985. First 0.988 x 0.987 = 0.975156, then x 0.985 = 0.960529. Enter 0.960529 - a 96.1% chance of clearing all three hurdles, and already visibly below any single year's survival.

Checkpoint 3 (year 8): v^8 = 1/1.04^8 = 0.730690 and 8p65 = 0.866447 (both given in the hint). EPV = 0.730690 x 0.866447 x 20,000 = about 12,662. Enter that (tolerance 60). The payment has lost about a third of its face value to the two stacked discounts.

Checkpoint 4: sum v^k x kp65 x 20,000 for k = 0 to 19, remembering the k = 0 term is exactly 20,000. The game confirms the exact figure; the shape to check is that each successive term shrinks, since both v^k and kp65 fall every year.

The shock: rebuild the sum with every q multiplied by 0.75 and subtract the original price. Survival to 84 jumps from about 51% to roughly 60%, the per-policy EPV rises by a few percent, and that difference - the shortfall per policy - is your answer (tolerance 900). Across the 1,000-policy book the screen translates it into a dollar-millions hole in the reserves.

Common mistakes

• Adding survival probabilities or averaging q values. kp_x is a product of one-year survivals; any additive shortcut fails the tight early tolerances.

• Dropping one of the two discounts at year 8. The EPV needs both v^8 and 8p65; using only the interest discount overstates the answer by about 15%.

• Starting the annuity sum at k = 1. Payments are due at the start of each year, so k = 0 to 19 and the first term is an undiscounted, mortality-free 20,000.

• Rounding intermediate survival products to three decimals. Compounded over many years the drift exceeds the tolerances; carry six decimals as the game's own hints do.

• Reasoning that better mortality is good news for the writer. The annuity writer is short longevity - every improvement in mortality raises the liability, and the shock question wants the increase, a positive shortfall.

Why interviews test this

This is core life-contingencies material - the q_x / p_x / kp_x notation, expected present values, and the annuity-due sum are the foundation of the actuarial exam sequence covering life contingencies, and interviewers for life and pensions roles expect them to be automatic. The annuity-due detail (first payment certain, sum starting at k = 0) is a favorite trap precisely because it is a one-index error with a visible price impact.

The longevity shock is the conceptual question that follows: who is short longevity, and what happens to reserves when mortality improves? Being able to say the annuity writer loses, quantify the shortfall as the difference of two EPVs, and explain why a capped-term annuity mutes the effect relative to whole-life, is the kind of answer that shows you can connect table mechanics to balance-sheet risk.