TERM INSURANCE CALCULATION ILLUSTRATED. This is the U.S. Social Security Life Table, based on year 2007.

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1 This is the U.S. Socia Security Life Tabe, based on year This is avaiabe at The ife eperiences of maes and femaes are different, and we usuay do separate anayses for men and for women. The coumns have the foowing meanings. Eact Age Probabiity This corresponds to birthday. The numeric information in the ine for Eact Age = 6 appies from the 6 th birthday to the 7 th birthday. This is the probabiity of death before the net birthday. For instance, a si-year-od mae has an estimated probabiity of of dying before his seventh birthday. This starts out at 100,000 for each se. The data vaues have been scaed to this arbitrary (but convenient) baseine. For instance, of 100,000 newborn femaes, it is estimated that 99,0 wi reach the age of eight. This is amost the obvious epected vaue cacuation, as noted beow. There is common notation for these. The eampes wi be for femaes. We wi use the symbo for age. The number aive at each age is denoted. For eampe 0 = 100,000, 1 = 99,390, 2 = 99,347, and so on. The vaues are often presented as integers, as they are here. These coud reasonaby be given to greater precision. The number dying between age and age + 1 is denoted d. Of course, d = For instance, d 3 = 3-4 = 99,322 99,303 = 19. The probabiity of dying between age and age + 1, conditiona of course on d + 1 reaching age, is denote q. Note that q = =. Thus 19 q 3 = 99, (As the vaues are given as integers, there wi be sight rounding discrepancies.) The symbo ω represents the maimum age. In this tabe, ω = 119. It was formery the case that ife tabes stopped at ω = 100. Medica care has enabed peope to ive onger, and government panning for Socia Security and Medicare needs more information on advanced age. The description odest od now is used to refer to peope over age 85. 1

2 Nonetheess, the use of ω = 119 seems a bit of a stretch. We see here that very, very few peope are epected to get beyond age 110. The symbo T is the tota years of ife for a peope after age. The specific form is T = ω. For eampe, T 20 = 98, , , , = 6,107,716. This cacuation suffers from the fact that the vaues are presented as integers. Actuaries have somewhat more sophisticated methods to cacuate T ; deaths are spread (amost) uniformy through each year, rather than a occurring at the end of the year. The symbo e o is the epectation of future ife, conditiona on having reached age. This is found as e o T =. Thus e o 6,107, = ,983 The vaue printed in the tabe is the sighty ower 61.20, which was based on a method that spreads deaths (amost) uniformy through each year. The discrepancy gets severe at etreme od ages, as the rounding of the vaues is very crude. Eact Age Probabiity M A L E Probabiity F E M A L E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

3 Eact Age Probabiity M A L E Probabiity F E M A L E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

4 Eact Age Probabiity M A L E Probabiity F E M A L E , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

5 Eact Age Probabiity M A L E Probabiity F E M A L E Note: The period ife epectancy at a given age for 2007 represents the average number of years of ife remaining if a group of persons at that age were to eperience the mortaity rates for 2007 over the course of their remaining ife. The Socia Security area popuation is comprised of i) residents of the 50 States and the District of Coumbia (adjusted for net census under-count); ii) civiian residents of Puerto Rico, the Virgin Isands, Guam, American Samoa and the Northern Mariana Isands; iii) Federa civiian empoyees and persons in the U.S. Armed Forces abroad and their dependents; iv) crew members of merchant vesses; and v) a other U.S. citizens abroad. This is identified as a period ife tabe. This means that the probabiities are based on ony a short period of time, not an entire human ifetime. This tabe is based on eperience for year The ogic is epained in the Note at the end of the tabe. So what eacty does this mean? The tabe has used data from year 2007 to estimate the q vaues, and then these estimated q are strung out to estimate the other parts of the tabe. Thus, iustrating this for maes, 0 = 100,000 (arbitrary) and so on. 1 = 0 (1 q 0 ) = 100,000 ( ) = 99, = 1 (1 q 1 ) = 99,262.1 ( ) 99,

6 The tabe therefore does not represent the eperience of any singe rea person over a ifetime. After a, q 0 was estimated on babies in 2007, q 1 was estimated on one-year-ods in 2007, q 2 was estimated on two-year-ods in This is the essence of the note that appears right after the tabe. Let s use this tabe to ask about a ife insurance poicy. Let s suppose that a -year-od mae, Michae, wants to buy a 30-year eve term poicy with the face amount $100,000. (If he wants, say, $500,000 of coverage, we just scae everything up by a factor of 5.) Let s find the actuaria fair vaue premium P. But... et s be very cear about what the words mean. * The 30-year arrangement means that Michae wi pay the amount P now (year 0), wi pay the amount P one year from now (year 1), wi again pay the amount P two years from now (year 2), and wi continue to year 29. Atogether Michae wi make 30 payments. (In some poicies, the payments wi be made semi-annuay or even monthy.) * The word eve means that the premium payment is P throughout. Over time, infation wi make the payment P a bit ess stressfu. Aso, Michae s saary is ikey to rise. Michae might find the payment painfu now, but he find it easier as time goes by. * If Michae dies before the 30 th anniversary of the poicy, his named beneficiary (usuay spouse) wi coect $100,000. * If Michae is aive on the 30 th anniversary of the poicy, Michae and his beneficiary receive nothing. It woud somewhat off the mark to say that Michae is a oser in such case. He put out a fair sum of money, but he did get 30 years of protection. * The actuaria fair premium is that vaue of P for which the insurer s epected vaue is zero. Michae wi of course be asked to pay more, perhaps by the factor (1 + λ). Here λ is caed the oading factor, and it coud be something ike 20%. The insurer cannot force Michae to keep this poicy going unti he reaches age 55. There is a rea possibiity that Michae wi keep the poicy to, say, age 31 and then et it apse. The insurers make money on poicies that apse. They have very good data on apse rates, and they can use this to reduce the oading factor. We do the cacuation without considering poicies that might apse. 6

7 Let s see what P wi buy. You might find it amusing to make a guess now as to what premium P woud get $100,000 coverage for Michae! The interest rate is reevant in doing this math. We use r = 0.03 = 3%. Let M denote the death benefit; here M = $100,000. We wi make a the cacuations reative to present vaue now, t = 0. We wi aso make a convenience assumption that Michae buys the poicy on his th birthday. Thus, Michae s th birthday is t = 0, his 26 th birthday is t = 1, and son on. We wi aso make the assumption that the insurance company wi pay benefits at the net poicy anniversary. (Yes, there are mid-year corrections that can be made, but for now we make the math easy.) If Michae dies between age and age + 1, the beneficiary wi receive M on what woud have been Michae s ( + 1) st birthday. The payment that Michae makes now (meaning t = 0) is worth P. Suppose that Michae dies before his 26 th birthday. The insurer coects P at t = 0 and pays $100,000 at t = 1. The insurer counts this as receiving P and paying M e - r. d According to the ife tabe, the probabiity that this wi happen for Michae is 26 97,834 97,693 = = = q. 97,834 If Michae dies between his 26 th birthday and his 27 th birthday, then the insurer coects P at t = 0 coects P e - r at t = 1 [vaued at t = 0] pays out M e - 2r at t = 2 [vaued at t = 0] The probabiity that this wi happen for Michae shoud be cacuated as P[ die between 26 and 27 aive at ] = P[ aive at 26 aive at ] P[ die between 26 and 27 aive at 26 ] 26 Use = 97,693 97, as the estimate of P[ aive at 26 aive at ]. 7

8 27 Simiary, use 1 97,555 = as the estimate of 26 97,693 P[ die between 26 and 27 aive at 26 ]. This is of course q 26. We write P[ die between 26 and 27 aive at ] as = d = 26 If Michae dies between his 27 th birthday and his 28 th birthday, then the insurer coects P at t = 0 coects P e - r at t = 1 [vaued at t = 0] coects P e - 2r at t = 2 [vaued at t = 0] pays out M e - 3r at t = 3 [vaued at t = 0] The probabiity that this wi happen for Michae shoud be cacuated as d27 P[ die between 27 and 28 aive at ] = There are many strategic ways to organize the math. We show here a method that works we with Ece. Let s consider the present vaue of the premium stream. Michae s initia payment at age, is P. His payment at age 26 is worth P e - r, his payment at age 27 is worth P e - 2r, and so on. 26 The probabiity that he makes the payment at age 26 is just probabiity that he s aive at age 26., which is just the 27 The probabiity that he makes the payment at age 27 is epected vaue of the premium stream; it s made at age 54, so the sum stops at = P r( ) e. This aows us to find the. The ast payment is 8

9 The events aive at 26, aive at 27, aive at 28, and so on, are certainy not statisticay independent. That s not a probem for this epected vaue cacuation. If Michae dies between age and age + 1, the insurer wi pay out at age + 1, and the r ( 1 ) present vaue of that payment woud be M e + r( 24) = M e. The probabiity that Michae wi die in that interva is d e 54 r( 24) M. = d. The epected payout by the insurer is then We can set up Ece to compute both and 54 r( ) P e epected premium stream = d e 54 r( 24) M epected payout (benefit) = The first is proportiona to P, and the second is proportiona to M. It wi be very easy to reate these. In particuar, these two are equa when P M d e d e r( 24) r( 24) = = r( ) r( ) e e = = = = We can set up the two sums easiy with Ece. Using r = 0.03, these are 4, and 1,918, Then P M = 4, ,918, This says that the annua premium shoud be about 0.% of the face vaue of the poicy. Thus, for a $100,000 poicy, the yeary premium shoud be about $0. 9

10 Here is an interesting way to get to a quick-and-easy approimate answer. (1) Use r = 0 (so that we don t have to do any time-vaue math). (2) Note that the number of deaths is 97,384 89,270 = 8,114. And assume that these a die hafway through the 30 years (so that each wi make 15 premium payments). We are wiing to make this hafway approimation because the death probabiities are fairy eve from age to age 55. (3) The proportion who ive through the 30-year period wi be 89,270 97, Let s ca this 92%. With probabiity 0.92, an insured person wi make 30P in premium payments and coect nothing. With probabiity 0.08, an insured person wi make 15P in premium payments and coect M. The insurer has an epected vaue of 0.92 (30P) (15P) M This is equa to zero when 0.92 (30P) (15P) = 0.08 M or 28.8 P = 0.08 M. This is our carefu answer P M = This is within about 12% of So. if we use interest rate is r = 0.03 and if Michae wants $100,000 of coverage, the actuaria fair vaue for the annua premium wi come out to about $0. That s about $22 per month. 10

11 The insurance company wi of course charge more than this. After a, the company has to make a profit and it aso has to keep a reserve. Reserve? What s that a about? Insurance companies are required by aw to have a big positive net worth. This insures that they can cover their poicies in case of disasters. For ife insurance, a disease epidemic woud count as a disaster. Do not be angry that insurance companies are reay rich; it s appropriate. The insurance company benefits if * a poicy is aowed to apse * the interest rate turns about to be higher than the r = 0.03 used in the cacuations * heath conditions improve 11

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