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Consider how to determine the annual premium of a five-year term policy, for $10,000, issued to a man who has just turned 60
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 -10,000 +P D S Year 1 The insured pays the first premium at the beginning of Year 1. During Year 1, one of two things will happen. Either the insured dies (D), in which case the company pays $10,000 at the end of the year, or the insured survives (S), in which case he pays the second premium at the start of Year 2.
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 -10,000 +P D D S S Year 2 During Year 2, again one of two things will happen. Either the insured dies, in which case the company pays $10,000, or the insured survives, in which case he pays the third premium at the start of Year 3.
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 +P -10,000 +P D D D S S S Year 3... and so on during Year 3...
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 +P -10,000 +P D D D D S S S S Year 4...and Year 4...
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 +P 0 -10,000 +P D D D D D S S S S S Year 5 The contract ends at the end of Year 5.
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 D D D D D S S S S S 77,861 1,558 1,667 1,777 1,885 1,990 70,794 According to the mortality table, of 77,861 men aged 60...... 1,558 are expected to die at age 60...... 1,667 at age 61...... and so on...... while 70,794 are expected to survive age 64.
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 D D D D D S S S S S 77,861 1,558 1,667 1,777 1,885 1,990 70,794 Therefore, the probability of death at age 60 is 1,558/77,861, or 0.0200. 0.0200
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 D D D D D S S S S S 77,861 1,558 1,667 1,777 1,885 1,990 70,794... the probability of death at age 61 is 1,667/77,861, or 0.0214. 0.0200 0.0214
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 D D D D D S S S S S 77,861 1,558 1,667 1,777 1,885 1,990 70,794... and so on... 0.0200 0.0214 0.0228 0.0242 0.0256
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 D D D D D S S S S S 77,861 1,558 1,667 1,777 1,885 1,990 70,794 The probability of surviving age 64 is 70,794/77,861, or 0.8860. 0.0200 0.0214 0.0228 0.0242 0.0256 0.8860
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Summarizing...
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 +P 0 -10,000 +P D D D D D S S S S S (0.0200) (0.0214) (0.0228) (0.0242) (0.0256) (0.8860)
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 +P 0 -10,000 +P D D D D D S S S S S (0.0200) (0.0214) (0.0228) (0.0242) (0.0256) (0.8860) If the insured dies at age 60 (and the probability of this event is 0.0200)...... the companys net revenue is P-10,000.
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 +P 0 -10,000 +P D D D D D S S S S S (0.0200) (0.0214) (0.0228) (0.0242) (0.0256) (0.8860) If the insured dies at age 61 (and the probability of this event is 0.0214)...... the companys net revenue is 2P-10,000.
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 +P 0 -10,000 +P D D D D D S S S S S (0.0200) (0.0214) (0.0228) (0.0242) (0.0256) (0.8860) If the insured dies at age 62 (and the probability of this event is 0.0228)...... the companys net revenue is 3P-10,000.
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 +P 0 -10,000 +P D D D D D S S S S S (0.0200) (0.0214) (0.0228) (0.0242) (0.0256) (0.8860) If the insured dies at age 63 (and the probability of this event is 0.0242)...... the companys net revenue is 4P-10,000.
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 +P 0 -10,000 +P D D D D D S S S S S (0.0200) (0.0214) (0.0228) (0.0242) (0.0256) (0.8860) If the insured dies at age 64 (and the probability of this event is 0.0256)...... the companys net revenue is 5P-10,000.
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Year 1Year 2Year 3Year 4Year 5 Age 60Age 61Age 62Age 63Age 64 Year 6 Age 65 +P -10,000 +P D D D D D S S S S S (0.0200) (0.0214) (0.0228) (0.0242) (0.0256) (0.8860) 0 While if the insured survives age 64 (and the probability of this event is 0.8860)...... the companys net revenue is 5P.
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