Solutions Manual for Actuarial Mathematics for Life Contingent Risks This must- have manual provides solutions to all exercises in Dickson, Hardy and Waters'. This must-have manual provides detailed solutions to all of the + exercises in. Dickson, Hardy and Waters' Actuarial Mathematics for Life Contingent Risks. Solutions Manual for Actuarial Mathematics for Life Contingent Risks. Solutions PDF; Export citation. Contents. pp v-vi Solutions for Chapter 1. pp
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Request PDF on ResearchGate | Solutions manual for actuarial mathematics for life contingent risks. 2nd ed | Cambridge Core - Finance and Accountancy. Actuarial Mathematics for Life Contingent Risks, 2nd edition, is the sole required text for the Society of Actuaries Exam MLC Fall and. Actuarial Mathematics for Life Contingent Risks. How can actuaries best and solutions teach skills in simulation and projection through computational practice.
This is clearly an increasing function of x. Solutions for Chapter 3 3. The features at younger ages show up much better in Figure 3. Note that the near- linearity in Figure 3.
This reflects the pattern seen in Figure S3. Figure S3. The relatively high mortality in the first year of life shows clearly in Figure S3.
See the comments on Figure 3. The expected number of deaths increases gradually from around age 30 and sharply from around age It reaches a peak at around ages 77 males and 87 females even though the force of mortality continues to increase beyond these ages, as can be seen in Figure S3.
The peak occurs because the expected number of survivors to high ages, lx , is decreasing sharply see Figure S3. This follows directly from formula 3. See Section 3. See part d for an example.
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Note that this survival probability is considerably smaller than the corre- sponding probability for non-smokers calculated in Exercise 3. The required life table is as follows: Solutions for Chapter 3 27 3. Solutions for Chapter 4 4.
The formula in part a splits the benefit according to the possible payment times — death benefit at times 1, 2,. This argument applies whether the sum insured is payable at the moment of death, the end of the m1 th year of death, or at the end of the year of death. Solutions for Chapter 4 31 4. The contribution to the EPV resulting from death in the first policy year is v qx. This part of the benefit can be expressed as an increasing benefit of 1, 2,.
The amount is 1 in the first year, 2 in the second year, and so on. The higher the rate of interest we can earn, the smaller the premium required.
Solutions for Chapter 4 35 E 4. The reason for this is that UDD is fairly accurate when the mortality probabilities are low for adult ages, but UDD becomes rather less accurate at older ages. That is, if 50 survives the first 15 years, the accumulated premium will exceed the value of the benefit whenever it is paid.
Solutions for Chapter 4 39 E 4. An excerpt from the table is shown below. Solutions for Chapter 5 5. The annuity payments are guaranteed to be paid from the death of x until time n, and cease at time n. If x survives to time n, there are no payments. Because the benefit is payable continuously, the sum here is an inte- gral. The probability that the annuity is paid is the probability that x has died at that time, which is t qx. The discount factor is vt and the amount of benefit is dt.
As the term n must be an integer, the present value is less than 3 if at most 3 annuity payments are made i. Hence, the approximation overstates the true value. Solutions for Chapter 5 45 c The 95th percentile of the standard normal distribution is 1. However, the guar- antee reduces the variance of the present value as there is less variability in the payment terms — in part a the possible payment terms are 1, 2, 3,. Solutions for Chapter 5 47 5. Solutions for Chapter 6 6. When the cashflows are complicated, it is often convenient to value each element separately.
EPV of Policy Fees: EPV of Death Benefit: Solutions for Chapter 6 51 6. EPV of Other Expenses: EPV of Premiums: We can see from the answer that we were correct in our assumption that the premium would not increase by a large amount and that the extra death benefit would apply only for three years.
We calculate the select insurance and annuity functions by constructing the life table for . We have assumed uniform distribution of deaths; other fractional age assumptions would give very similar results. However, if the policyholder survives for 25 years, then 25 bonuses are added. Also, : Then the accumulation of premiums less expenses is 0. Thus, there is a profit at time t years if 0.
Hence there is a profit if the death benefit is payable at time Similarly, there is a profit if the policyholder survives 25 years, the amount of this profit being 0. Thus, there is a profit if the policyholder survives 24 years and pays the premium at the start of the 25th policy year. As the 99th percentile of the standard normal distribution is 2. Solutions for Chapter 7 7.
The distribution of L1 is as follows: The future cash flows for the term insurance are much more un- certain than those for the endowment insurance. Put more simply, for the endowment insurance the insurer knows that the sum insured will be paid at some time within the next two years, the only uncertainty is over the timing; for the term insurance it is not certain that the sum insured will ever be paid.
The gross premium policy value is less than the net premium policy value when the expected present value of future expenses is less than the expected present value of future expense loadings.
This is generally greater than zero as the expense loadings include the amortized initial expenses. That is, the gross premium policy value is less than the net premium policy value because it allows for the recovery of the initial expenses from future premiums. Solutions for Chapter 7 63 This is precisely 1V , as we know it must be. This gives the actual inter- est minus the expected interest as 0. The contribution from expenses, allowing now for the actual interest earned, is 1.
Inserting this value for P into formula 7.
We have 2. Using the equivalence principle, we have 0. Figure S7.
Note that this uses the assumed, rather than the actual, expenses. In particular, the change in the mortality basis changes 16 p44 from 0. Hence, the benefit on survival to the end of the term is the most significant contribution to the EPV of the benefits.
Since t V repre- sents the value of the investments the insurer should be holding at duration t, adopting the suggestion of a proportionate paid-up sum insured would give the insurer a small profit for each policy becoming paid-up, assuming that experience exactly follows the assumptions. It is generally considered reasonable for the insurer to retain a small profit, on average, as the policy- holder has adjusted the terms of the contract.
We need the following val- ues, calculated using numerical integration: Using these values, we have S. The values of S.
This is not surprising.
Solutions Manual for Actuarial Mathematics for Life Contingent Risks
As t increases from 0, the time until the sum insured is likely to be paid decreases and so the present value of the loss increases. This will increase the standard deviation of the present value of the loss provided there is still considerable uncertainty about when the policyholder is likely to die, as will be the case for the range of values of t being considered here. An excerpt from the resulting table of values is shown in part c below.
Then, equating policy values before and after the alteration, we have 1 3 Since this term is zero, it has been omitted. For time 0, the explanation is the same except that the amount we accumu- late for one year is the premium rather than the policy value.
An excerpt from the table of values is shown below. The left-hand side of the formula, t V i , is the EPV of all future net cash flows from the insurer from time t, given that the life is in state i at time t.
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Alternative solution: Solutions for Chapter 8 85 8. We can re- consider the model in Figure S8. The life is alive when the process is in state 0 or state 2. The two state model is the same as the three state model under which the two alive states are merged. The model is now: The integral expression for the probability can be evaluated numerically to give 0.
We will use that figure to reference the states involved in the annuity payments. For t p02 02 x: Let m denote male mortality, f m denote female married mortality and f w denote female widowed mortality, then 00 fm 22 fw t px: Let superscript f denote the female sur- vival model, and superscript m denote the male survival model.
The equation of value for the premium is 0. Hence, the first term is likely to be much larger than the second term. Hence, the formula holds whatever values the random variables take.
This will be useful in part d. Then, for example, 0. Then Solutions for Chapter 9 9. Dividing X by the salary earned between ages 59 and 60 gives the replacement ratio as If the member survives to age 26, the value at that time of the accrued benefits is 0. As there are no exits other than by death and there is no death benefit , the funding equation gives the contribution as this amount, and hence the contri- bution is 9. The amount of the pension does not depend on the financial performance of the underlying assets.
If, for example, the underlying assets provide a lower level of accumulation than expected, the replacement ratio would be reduced under a defined contribution plan, but not under a defined benefit plan.
For example, if the underlying assets do not perform strongly, the employer may have to increase its contributions to the fund to ensure that benefits can be paid. The pension is payable from age 65 without actuarial reduction, or at age x with the reduction factor applied.
Her salary in the year prior to retirement is s On retirement at age exact 62, the pension is 0.
Life tables and selection; 4. Insurance benefits; 5. Annuities; 6. Premium calculation; 7. Policy values; 8. Multiple state models; 9. Joint life and last survivor benefits; Pension mathematics; Yield curves and non-diversifiable risk; SearchWorks Catalog Stanford Libraries.
Send to text email RefWorks EndNote printer. Responsibility David C. Dickson, Mary R. Hardy, Howard R. Publication Cambridge: Cambridge University Press, Physical description 1 online resource pages: Series International series on actuarial science. Online Available online. Cambridge Core Full view.The amount of the pension does not depend on the financial performance of the underlying assets.
This argument applies whether the sum insured is payable at the moment of death, the end of the m1 th year of death, or at the end of the year of death. Name of resource. This will increase the standard deviation of the present value of the loss provided there is still considerable uncertainty about when the policyholder is likely to die, as will be the case for the range of values of t being considered here. As t increases from 0, the time until the sum insured is likely to be paid decreases and so the present value of the loss increases.
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