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Ryoooh · 2020年03月05日

问一道题：NO.PZ2020011901000032 [ FRM I ]

  1. FRM I
  2. Financial Markets And Products

问题如下：

What is the minimum USD annual premium that an insurance company should charge for a two-year term life insurance policy with face value of USD 1 million when the policyholder is a woman aged 71? (Use Table 2.1 and assume an interest rate of 3% compounded annually.)


选项：

A.

18,153

B.

17,874

C.

17,996

D.

17,767

解释：

B.

The probability of a payout in the first year (time 0.5 years) is 0.017275. The probability of a payout in the second year (time 1.5 years) is

(1 - 0.017275) * 0.019047 = 0.018718

The PV of the expected cost of the policy is therefore:

17,275/(1.030.5)+18,718/(1.031.5)=34,92817,275/(1.03^{0.5}) + 18,718/(1.03^{1.5}) = 34,928

The first premium is at time zero. The second premium, at time one year, has a probability of 1 - 0.017275 = 0.982725 of being made. If the premium is X, the expected present value is

X + 0.982725X/1.03 = 1.954102X

The minimum premium is given by solving:

1.954102X = 34,928

It is 17,874.

请问老师，我计算第一年活着第二年死亡的概率是（1）用第二年活着的概率除以第一年活着的概率，得到这个人从第一年活到第二年的概率。（2）然后再用1减去（1）得到的概率，这样计算问题在哪呢？
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袁园_品职助教 · 2020年03月05日

同学你好！

你说的（1）里面是乘法不是除法吧？我猜你想用第一年活着的概率乘以第二年活着的概率得到这个人第一年和第二年都活着的概率对吗？

但是不管（1）你是怎么算的，当你用1减去（1）的时候，你计算的是这个人“第一年和第二年都活着”的反面，这个反面包含两种情况：

第一种情况是第一年活着第二年死了（这是你想计算的结果）；

第二种情况是这个人第一年就死了。

所以你的计算结果会把第二种情况也包含在里面了。

Ryoooh · 2020年03月08日

谢谢老师解答，我这里的思路其实是想用课件第36页的思路，计算出这个71岁的人活到72岁的概率。不过这样用1减前面得出的概率好像也有你提出的问题，就是包括了第一年死掉的概率。但是追加想问一下，从71岁活到72岁的概率按上面的算法好像跟mortality table里面直接给的72岁存活的概率不同，具体区别在哪呢？

袁园_品职助教 · 2020年03月08日

同学你好！

第一个问题：课件上计算的是条件概率，即在一个人已经活到70岁的条件下，他还能活到90岁的概率。这里并没有考虑这个人没有活到70岁这种情况~

第二个问题：我算了一下，应该是没问题的。用 活到71岁的概率 乘以 从71岁开始一年内不死亡的概率 得到 活到72岁的概率，即0.81525 x (1-0.017275) = 0.80117

Ryoooh · 2020年03月15日

懂啦！十分感谢！

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