Life Contingencies Study Note for CAS Eam S Tom Struppeck (Revised 9/19/2015) Introduction Life contingencies is a term used to describe surviva modes for human ives and resuting cash fows that start or stop contingent upon surviva. As such it is a centra topic for ife insurance actuaries. Whie there are many appications of these techniques to property and casuaty insurance, ife contingency cacuations in P&C work are typicay intermediate resuts used to estimate some other quantity of interest. This note is intended to iustrate terminoogy and techniques in straightforward contets, and to point out to the reader some paces where further refinement is possibe. A reader who needs more detais or is interested in eporing these refinements shoud consut any of the many fine ife contingencies tets that have been written over the years. Chapter 1: Life tabes and an etended eampe A ife tabe is simpy a way of presenting a famiy of conditiona surviva random variabes in a very concise way. Before giving a definition, et s ook at an eampe. Eampe 1: The SUX Vacuum Ceaner Company manufactures vacuums that ast up to four years. One third of the company s vacuums fai in the first year, another third in the second year, one sith in the third year, and the remaining sith in the fourth year. On 01/01/XX the company ses 300 vacuums. How many of these are epected to fai in each of the net four years? These quantities are easiy seen to be 100, 100, 50, and 50, respectivey. This type of data is very suitabe for presentation in a tabe: Year XX 01/01/XX Number Functioning Number Faiing During Year XX 00 300 100 01 200 100 02 100 50 03 50 50 1
After the first year, assuming that we saw the 100 faiures that we epected, we coud dispay the information about the remaining vacuums in another tabe: Year XX 01/01/XX Number Functioning Number Faiing During Year XX 01 200 100 02 100 50 03 50 50 Obviousy, the second tabe is eacty the ast three rows of the first tabe, but there is one subte difference. In the first tabe, the information is about a three hundred origina vacuums. Whereas, in the second tabe, the information is ony about those that survived the first year. Resuts for businesses are often epressed on an annua basis, so the number epected to fai in the coming year is a quantity of interest. Initiay, there are 300 functioning vacuums, and we epect one third of them (100) to fai during the first year and another third (100) to fai during the second year. Looking at the second of the two tabes above, we see that of the 200 vacuums functioning at the start of Year 1, one-haf of them are epected to fai during the year. This ratio, the number epected to fai during the coming year divided by the number sti functioning at the start of that year, is the conditiona one-period probabiity of faiure sometimes caed the mortaity rate or even more succincty the mortaity. The number functioning at the start of the year, the number faiing during the year and the mortaity are denoted by the etters,, d, and q, respectivey. Since these generay vary by year, a subscript is used to indicate which year, as in the foowing tabe: Age 0 300 100 0.333 1 200 100 0.500 2 100 50 0.500 3 50 50 1.000 d q Suppose that the company decides to offer a warranty that wi pay $100 at the end of the year in which the machine fais. Since a of the machines fai by the end of year 4 and no machine can fai more than once, this wi cost the company eacty $100/machine. Sti, the timing of the payment is uncertain and the discounted vaue of the payment can be thought of as a random variabe. In order to make sense of that, we need to know the rate(s) of interest for discounting. 2
In practice, yied curves are sedom eve, but the formuas for cassica ife contingencies are much simper in the case that there is a singe deterministic interest rate. Hence, that assumption is generay made when introducing ife contingencies, and we wi make it too. At the time of the vacuum s sae, the epected vaue of the warranty payment is: APV 3 j 1 100* ν + *Pr(K j) = = j= 0 where 1 ν =, i is the interest rate, and K is the number of fu years that the machine functions. 1 + i APV stands for actuaria present vaue, which is simpy the present vaue of a stream of cash fows that depend upon the vaue of a random variabe. If the eve interest rate is 5%, the reader shoud verify that the APV = $90.09. The company decides that this is too much to give away. Instead it offers a subscription service so that purchasers of a new vacuum can make annua payments to buy the warranty. The company woud ike the payments to be eve and, of course, customers wi stop subscribing once they have coected on the warranty. How much shoud the company charge for this if it wants to break even on it? The anaysis is very simiar to the above cacuation: 3 j APV = S * ν *Pr(K j) j= 0 Here S is the annua subscription payment. This payment is made at the beginning of each year that the machine functions, which epains the one ess year of discount from the earier cacuation. The reader shoud confirm that each doar of subscription charged has an APV of $2.08, so to break even the company shoud set the two APV amounts equa and charge $43.30/year for the warranty (amounts are rounded). The above eampe iustrates most of the major ideas in ife contingencies. The remainder of this note feshes them out a bit more. Chapter 2: Comments on Life tabes In this section we wi discuss ife tabes a bit more, this time using human ives as our eampe instead of machine faiure. 3
In the above eampe, a of our machines were manufactured on 01/01/XX and the tabe described their states on 01/01 of the foowing years. In other words, our machines were of eact age. Peope have different birthdates, but for practica reasons we wi use a tabe that assumes that their eact age in known. Even the notion of age requires a more carefu definition: is it age nearest birthday (so that age 60.9 is considered age 61) or is it age ast birthday (so that age 60.9 woud be age 60)? We wi not concern ourseves with these nuances, but you shoud be aware that they do eist. Mosty we wi be interested in individuas that are having their birthday today, so that they just turned age and we wi denote that person by (). For eampe, (40) is an individua whose eact age is 40. In the eampe we denoted the number of machines functioning at the start of year by. When this number is the number of individuas who are eact age (), we use the same notation. In the eampe, we denoted the number of machines that faied during the year by d. For individuas, this quantity is the number of peope that died between eact age and +1. In other words: We aso defined: d = + 1 q d = These we caed (annuaized) mortaities. They can each be interpreted as the probabiity that a person age dies before reaching age +1. In our eampe we started with 300 machines, but we coud just as we have started with some other number, say 600. The number of machines at the start of each year doubes as does the number faiing during the year, but the mortaity stays the same as it is a ratio of these two. Typicay we wi be interested ony in the moraity, so the choice of 0 does not matter any positive number wi do. That quantity (caed the radi) is usuay picked to be some convenient round number, such as 100,000. It is convenient to have a standard notation for surviving from age to +1 and we denote this by p. p = + 1 This foows from the above two reationships, as the reader shoud confirm. We wi aso want a standard notation for surviving from age to +n and we denote this by n p. If n=0, by convention we interpret this symbo as equa to one. Notice: 4
+ n p = pp p... p = n + 1 + 2 + n 1 This ast fact iustrates how convenient a ife tabe can be for performing this type of cacuation. As one might guess, the symbo n q denotes 1 n. Eercises: p Using the Iustrative Life Tabe for this eam, compute: 1) The probabiity that (50) ives to be at east 60 years od. 2) The probabiity that (50) dies at age 60 (meaning sometime during the year). 3) The probabiity that (40) ives to age (65) and an independent ife (30) dies before age (50). 4) The epected number of deaths during the net 10 years of 1,000 peope age (80). 5) The variance in 4) 6) Using the norma approimation, give a 95% confidence interva for the number of deaths in 4) Chapter 3: Contingent Cash fows Consider a contract that pays $1 at the end of the year of death of (). We are interested in computing the actuaria present vaue of this payment. We wi need to know the mortaity assumptions (from the ife tabe), the discount rate (we assume a eve term structure of rates), and the timing of the payment (we are tod that it occurs at the end of the year of death). Such contracts are caed ife insurance contracts. Some ife insurance contracts ony remain in effect for a fied number of years the resut being that if the insured dies during the contract period, the $1 is paid, otherwise nothing is paid. Such contracts are caed term ife insurance and the ength of the period is caed the term. Poicies with an unimited term are caed whoe ife poicies. Since a of the vacuums in our earier eampe faied within four years, the warranty the company offered was effectivey whoe ife insurance. It is convenient to have a notation for the actuaria present vaue of a whoe ife insurance poicy of $1 on (), and we wi introduce this shorty. This poicy pays $1 at the end of the year of death of (). Measuring years from the start of the poicy, ca that year k with the first year being k=1. Consider the end of the first year, one of two things has happened: either () has died --- this has probabiity q --- or () ived, which has probabiity p. So, the actuaria present vaue of the payment (if any) at the end of year 1 is ν q. If there is no payment, the contract continues for another year, but () is now one year oder. If we write A for the actuaria present vaue of a ife insurance of $1 on (), then we have: 5
A = νq + ν pa + 1 In other words, a whoe ife poicy of $1 on () is equivaent to a doar paid in a year if () dies and if he doesn t die the present vaue of a repacement poicy deivered in one year on (+1). The reader is urged to match the words in the interpretation to the symbos in the formua --- as you wi see the order is sighty different. This recursion reationship can be appied repeatedy to produce the foowing formua: A n+ 1 n npq + n n= 0,1,2,... = Which says: the actuaria present vaue of an insurance of $1 on () is the discounted vaue of a payment made at the end of year n if () ives to age +n and then dies during the net year. So far we have ooked at whoe-ife insurance. Another commony sod product is term (ife) insurance. This product is just ike whoe-ife insurance ecept that at a certain age the poicy terminates (epires worthess); since some insureds are sti aive at poicy termination, there are fewer death caims and this resuts in a ower pure premium than that required for whoe-ife. A simiar product is caed endowment insurance, which is just ike term insurance ecept that at termination the poicy pays $1 instead of maturing worthess; ike whoe-ife, every such poicy eventuay pays $1, and for insureds that are sti aive at poicy maturity it pays earier than whoe-ife woud, so such poicies have a higher pure premium than whoe-ife. Finay, there is a pure endowment which is the difference between term insurance and endowment insurance, namey a payment at maturity if the insured is sti aive. For eampe, a ten-year term insurance on (40) pays $1 if (40) dies in the net ten years. A ten-year endowment insurance on (40) pays $1 if (40) dies in the net ten years and if (40) is sti aive at age 50, it pays $1 anyway. A pure ten-year endowment on (40) pays $1 if (40) ives to age 50. In these eampes a death benefits are assumed to be paid at the end of the year of death and maturity payments are made at maturity. In our eampe in the introduction, the company had the customers pay an annua fee in order to purchase the warranty. This is how ife insurance is typicay paid for. At the start of each year () pays a premium of P to the insurance company in echange for a payment of $1 at the end of the year of death of (). We need a notation for the actuaria present vaue of a stream of payments made at the beginning of each year whie () is aive. Such a stream of payments is caed a ife-annuity-due and its actuaria present vaue is denoted by a. The doube dot over the annuity symbo means that the payments are made at the start of the year. The reader may reca from interest theory that such an annuity is caed an annuity-due. If the payment is made at the end of the year (provided that () is sti aive then), it is caed a ife-annuity-immediate and its actuaria present vaue is denoted by a. 6
We woud ike to evauate actuaria present vaue of a ife-annuity on (). This can be done by directy writing down the recursive formua, just as we did for the whoe ife poicy on (). In words: a ifeannuity due of $1 on () is equivaent to an immediate payment of $1 and, if () is sti aive in one year a repacement annuity on (+1) deiverabe in one year. In symbos: a = 1+ν pa + 1 Epressing this as a sum, as we have done for insurance above, is simpe, and readers shoud do that for themseves as an eercise. There is an etremey usefu reationship between the vaue of a ife insurance poicy on () and the vaue of a ife-annuity on (). This reationship can be seen best through an arbitrage argument: an investor is ambivaent between having a doar today and receiving a combination of one year s interest on that doar and getting the doar itsef back at the end of one year. A moment s thought reveas that there is nothing specia about one year, it coud just as we have been three years or ten years or even a random number of years. Reca from interest theory that interest paid at the beginning of the year is caed discount and is denoted by d. (Not to be confused with d the number of deaths at age in a ife tabe from before.) Here is the reationship in symbos: 1 = da + A In words: an investor is ambivaent between (LHS 1 ) having $1 today and (RHS 2 ) receiving a stream of d doars at the start of each year whie () is aive and then at the end of the year of the death of () getting back the doar. You can think of the annua payment of d as the rent on the doar for the coming year. We immediatey obtain: a = 1 A d This ets us determine the eve annua premium for a whoe ife poicy on () by setting: A = Pa In other words, we want to find an annua premium payment, P, payabe at the start of each year that () is aive which is actuariay equivaent to a doar paid it the end of the year of death of (). This is easiy soved for P. 1 Left Hand Side of the equation 2 Right Hand Side of the equation 7
Just as we coud imit the contract period for whoe ife and obtain term ife, we can aso imit the term of an annuity. We coud pay unti () died or n years, whichever comes first. Another popuar structure is a ife annuity with a guarantee of paying at east n years. Such a contract woud pay unti () died or unti n-years had passed, whichever happens ast. Eercises: 1) What is the eve premium for a whoe ife of $1 on (40) using the Iustrative Life tabe? 2) Woud an increase in interest rates cause A to increase or decrease? How about 3) You are given that a $1 whoe ife poicy on () has an APV of $0.30 and that d =.035. What is the eve annua premium for this insurance? 4) On a given individua, rank in order of increasing APV: 10-year term, 10-year endowment insurance, and whoe ife. Epain. 5) Repeat eercise 4) for annuities with guarantee periods, annuities with maimum payment periods, and ife-annuities in pace of the three types of insurance. 6) (Variances of insurances) Reca that the variance of a random variabe (the second centra moment) can be computed by computing the second raw moment and subtracting from it the square of the first moment. Show that the second raw of a whoe ife insurance of $1 on () with a discount rate of υ is, in fact, the first moment of a whoe ife insurance of $1 on () with a 2 discount rate of υ. In other words, we obtain the second raw moment of an insurance if $1 by doubing the force of interest. Now produce a formua for the variance of such an insurance. 7) (Variance of annuities) Use the reationship between insurances and annuities aong with Eercise 6 (above) to produce a formua for the variance of a ife-annuity-due of 1 on (). a? Chapter 4: An eampe from workers compensation insurance An important ine of commercia insurance in the US is workers compensation insurance, which provides benefits to workers who are injured on the job. One appication of annuities is to estimate the future costs of paying an annuity to an injured worker who is unabe to return to work. The detais of workers compensation insurance are covered esewhere in the CAS Syabus; for our purposes it is enough to know that an injured worker coud be paid a monthy stipend and that this stipend stops when the worker returns to work. Depending upon the type of injury, the stipend might have a maimum number of months of payment or the stipend might have a guaranteed minimum number of payments. We wi ook at two simpified eampes. In both we wi make the unreaistic (but computationay easier) assumption of annua payments. We wi aso assume that the actuaria present vaues are discounted for the time vaue of money by a eve interest rate of 3%. (Under the accounting system used by insurance reguators in the US, most reserves for future oss payments are not discounted for time vaue of money, but certain reserves for workers compensation payments are.) 8
To compute the actuaria present vaues of the two annuities we wi use the foowing tabe: Age a 20 100,000 20.8952 25 95,000 19.8536 40 75,000 17.0563 65 37,750 11.4546 The mortaity (which in this case represents returning to work) was seected to make the cacuations more transparent. Ony the rows that we wi use are shown; a rea tabe woud have rows for every age. For our first eampe, we use the tabe to compute an annua ife annuity-due for a 20-year-od with a certain period of 5 years (i.e. it pays $1 at the start of each year unti (20) goes back to work or five years, whichever is onger.) The payments from this annuity are eacty the same as the payments from the foowing: a five-year annuity certain and a ife annuity on (25), deiverabe in five years, if (20) is sti aive. Recaing the notation for a five-year annuity-certain-due, a, we have in symbos: 5 a 5 + 25 20 (1.03) 5 a 25 The five-year annuity-certain at 3% = 4.7171 and the other quantities are taken from the tabe. For our second eampe, we wi compute the APV of an annuity on (40) which pays unti he returns to work or reach age (65), whichever comes first. Again, we find a repicating portfoio. Suppose that we gave (40) payments unti he went back to work, but made him start to give those payments back to us if he was sti not back to work at age 65. The net payments to him woud be eacty the payments we owe him. In other words, give a ife annuity today on (40) and take back in twenty-five years a ife 65 25 annuity on (65), if he has not returned to work. In symbos: a 40 (1.03) a 65 Here a of the numbers can be found in the tabe ecerpt. 40 Chapter 5: Why P&C companies don t se eve premium products 9
Look back at the end of the first eampe where the company sod the warranty to the customers for a eve annua premium. The first year, the warranty premium was $43.29. What does the customer get for this? They get coverage for the first year and they aso get the option to buy coverage for future years at that same fied price. If they had ony wanted to buy coverage for the first year, the APV of this coverage is $31.75. The reader shoud check this and shoud compute the one-year term premium for each of the remaining three years. (Coverage for each year coud be caed one-year term insurance.) We said that the purchaser got an option to purchase coverage in future years. True, our contract specified that they had to buy coverage for each year, but in reaity there is no way to enforce that they actuay purchase it. A purchaser that does not renew a poicy is said to have apsed. In our eampe, if a purchaser apses, the company has the etra $11.54. This amount is non-negative because the mortaity in our eampe is non-decreasing over the ife of the poicy. Human mortaity beyond very eary ages tends to be non-decreasing, so these amounts (sometimes caed poicy vaues or premium reserves) are positive for ife insurance. In fact, these amounts can be so significant that state aws often require that some portion of it be shared with a apsing poicyhoder. On the other hand, suppose that mortaity were decreasing over the coverage period. Now if we try to price a eve premium poicy the same way we find a probem. When the moraity was increasing, as above, the insured paid etra premium in year 1 which went to subsidize the insufficient premium in ater years, but if mortaity is decreasing the insured woud pay too itte the first year and we woud rey on the etra premium from subsequent years to make up the shortfa. Unfortunatey, we can t force the insureds to renew and the rationa thing for them to do is apse (after obtaining as much subsidy as possibe) and then purchase term insurance for future years to obtain the same coverage at ower cost. That scenario is not as far-fetched as it sounds, because property and casuaty risks tend to have decreasing hazard rates (a synonym for mortaity rates) over time. This happens because the worst risks tend to have their accidents earier and over time ony better risks remain in the cohort. Chapter 6) What happens mid-year? So far we have ony concerned ourseves with annua timing of events and payments. The ife tabe tes us the probabiity of () dying in the net year, but it does not te us when during the year that that might occur. Suppose that we wanted to do everything on a monthy basis. We coud simpy refine our ife tabe be a monthy tabe with monthy mortaities. We woud need to obtain the entries from somewhere. Typicay this is done by some form of interpoation between the annua mortaities. There are many ways to interpoate, each with its own advantages and disadvantages. Methods used in practice are inear interpoation, eponentia interpoation, and hyperboic interpoation. The ast one appears in practice because it is we-suited for the data that one sometimes gets from cinica trias. It is aso possibe to do things in continuous time. In that setting we think of mortaity in terms of surviva functions and hazard rates. These are covered esewhere in the CAS Syabus, so we won t 10
deveop them here. The reader shoud know that the hazard rate when used to describe mortaity is caed the force of mortaity. There is one situation where the timing of events during the year might matter to us. Suppose that we have sod worker s compensation insurance. One of the coverages requires us to pay the saary (or a portion of it) to a worker that was injured on the job. We wi pay this amount every month unti the worker returns to work, reaches age 65, or dies, whichever comes first. In evauating the actuaria present vaue of these payments, we need to consider the ikeihood of the three possibe events that coud make us stop paying, namey the probabiity that the worker returns to work, the probabiity that the worker reaches age 65, and the worker s mortaity. Notice that these are not independent of one another. The more ikey it is that the worker returns to work, the ess ikey it is that he reaches age 65 or dies whie we are sti paying the caim. We think of the caim as being in a paying status and that status has mutipe forces acting on it. These forces are competing in the sense that ony one of them wi cause the caim to stop being paid. An increase in the ikeihood that a worker returns to work wi decrease the probabiity that the worker is sti on caim at age 65. Sometimes an injured worker has injuries so severe that the usua ife tabe is not appropriate. This is often described as a percentage increase in mortaity, athough saying that the force of mortaity has increased or that the odds-ratio of surviva has decreased is often more accurate. For instance, if the probabiity of () dying in the net year is 10% and we are tod that his mortaity tripes, we coud use 30% as the probabiity of () dying in the net year. Of course that woud not work if his probabiity of dying were aready 50% as probabiities are imited to 100%. Sti that phraseoogy is often used and since mortaities (fortunatey) tend to be sma we wi foow custom and just mutipy the mortaities when we are tod that they are increased 3. Finay, it is possibe to have an annuity that pays based on the surviva of more than one person (joint ives). Typicay, there wi be two peope () and (y) and in practice they wi often be a married coupe. The annuity may pay unti one of the two dies (denoted a ) or it may pay unti both persons have died (denoted a y y ). There are reationships between annuities on singe ives and annuities on joint ives. One such reationship is easiy seen through an arbitrage argument: A coupe is ambivaent between (LHS) each receiving $1 whie they individuay are aive and (RHS) receiving $1 whie either of them is aive and a second doar whie they are both aive: 3 Surviva one year can be thought of surviving the cumuative hazard rate for an additiona year. Increasing this k hazard rate from λ ( t) to kλ ( t) changes the surviva probabiity from p to p and the probabiity of death k k from 1 p to 1 p. If p is near 1, then1 p (which is 1 (1 q ) k ) wi be near kq. 11
a + a = a + a y y y Eercise: 1) A ife annuity of $1 on (40) costs $15 and one on (50) costs $13. If an annuity that pays $1 whie either of them are aive costs $20, how much does one that pays $1 whie they are both aive cost? Chapter 7: Life epectancy For some purposes it is usefu to be abe to estimate the number of years an individua wi ive. Of course we can ony answer this on average. We ca this average the ife epectancy. We might be interested in the number of fu years or in the actua number of years. The epected number of fu years is caed the curtate ife epectancy. It can be computed from the ife tabe. If we are interested in the actua ife epectancy (not just fu years), it is caed the compete epectation of ife and it wi be onger. To compute it we need to make an assumption about the timing of death during the year. If we assume that a deaths are uniformy distributed throughout the year, then we can get the compete epectation by adding 0.5 to the curtate epectation. It shoud be mentioned that the curtate epectation itsef need not be an integer; rather it is an average of integers. Property & Casuaty insurers pay caims for ines other than workers compensation. For some of these ines, the payment may be made severa years in the future. There is a cost to simpy having a fie open, in the past these were caed unaocated oss adjustment epenses. Suppose that 55% of caims get cosed in the first year, 25% in the second year, 10% in the third year and 10% in the fourth year. A new caim has just come in. How many fu years do we epect the caim to be open? We know that 100%-55%=45% of the caims remain open after one fu year. Aso, 20% remain open after an additiona fu year. Finay, 10% remain open after a third fu year. So, the curtate epectation of a caim being open is 0.45+0.20+0.10 = 0.75 years. If we assume that when a caim coses, it on average coses in the midde of the year, we can compute the compete epectation of a caim being open as 0.75 + 0.50 = 1.25 years. The cacuation above iustrates how to compute the curtate epectation of ife of () from a ife tabe, namey compute n p (defined at the end of Chapter 1) and sum this over a n starting with n=1. <End> 12
Appendi: Soutions for seected probems from Life Contingencies Study Note for CAS Eam S Chapter 2 Eercises: Using the Iustrative Life Tabe for this eam, compute: 1) The probabiity that (50) ives to be at east 60 years od. 50 = 8,950,901 and 60 = 8,188,074 so the desired probabiity is: 8,188,074 0.9148 8,950,901 = 2) The probabiity that (50) dies at age 60 (meaning sometime during the year). He needs to ive to 60, and then die in the net year. q 60 = 0.01376 so the desired probabiity is: 0.9148*0.01376 = 0.01259 3) The probabiity that (40) ives to age (65) and an independent ife (30) dies before age (50). The ives are independent, so we can compute the probabiities of the events separatey and mutipy 65 50 them: * 1 = 0.04687 40 30 4) The epected number of deaths during the net 10 years of 1,000 peope age (80). This is 100010 q80 = 1000( 80 90) / 80 = 729.6 directy from the tabes. 5) The variance in 4) Using a binomia mode the variance is npq = 197.29 6) Using the norma approimation, give a 95% confidence interva for the number of deaths in 4) Epected +/- 1.96*standard deviation = 729.6 +/- 1.96*sqrt(197.29) = (702.1,757.1) Chapter 3 Eercises: 13
8) What is the eve premium for a whoe ife of $1 on (40) using the Iustrative Life tabe? A 40 = 0.16132 and a 40 = 14.8166 from the tabe. Since Pa 40 = A40, P = 0.010888 9) Woud an increase in interest rates cause A to increase or decrease? How about An interest rate increase decreases the present vaue of a future payments, contingent or not. 4) On a given individua, rank in order of increasing APV: 10-year term, 10-year endowment insurance, and whoe ife. Epain. Assuming that there is some chance that the individua ives more than ten years, 10-year term is the smaest, since it might not pay, and 10-year endowment is the argest since it might pay earier than whoe ife, which is in the midde. a? Chapter 6) Eercise: 2) A ife annuity of $1 on (40) costs $15 and one on (50) costs $13. If an annuity that pays $1 whie either of them are aive costs $20, how much does one that pays $1 whie they are both aive cost? An easy arbitrage argument shows that this annuity must cost $15 + $13 - $20 = $8. <End> 14