代做ETC2430 Actuarial Cash Flow Modeling Final exam Part 2代写留学生Matlab语言

ETC2430

Actuarial Cash Flow Modeling

Final exam

Part 2 (60%/100%)

Part D (Application): [12 marks]

An investor resides in a tax jurisdiction where the income tax rate is 20% pa. and no capital gains tax is applicable. The investor is considering two possible investments, one in a share of a company (an equity), and another in a fixed interest asset (a bond).  The investor’s desired money yield is 5% pa.

i). The bond under consideration has a term of 5 years, a redemption amount of $100, pays coupons annually in arrears, with income tax applied at a rate of 20%.  The bond is priced at $90. Calculate the coupon rate for the bond to achieve the desired yield? (4 marks)

ii). Suppose an ex-dividend equity after a dividend amount of D, has an assumed annual dividend growth rate of g in perpetuity, an expected annual yield of i, and an income tax rate of t is applicable to dividend payments.  Assuming that i > g, write down an expression for the price P in terms of D , g , t and i. (2 marks)

iii). The equity under consideration is also priced at $90. If the investor’s outlook on the company’s growth prospects is conservative and believes the dividend amounts will remain stable and not grow, use your expression from part ii). to calculate the required annual dividend amount to achieve the desired yield for this equity investment? (3 marks)

iv). If the investor revises their goal and now seeks to achieve a desired real  yield of 5% pa, and believes inflation will be greater than 0 over the coming years.  Discuss how the required coupon rate calculated in part i) and the required annual dividend amount calculated in part iii) would change? (3 marks)

Part E (Application): [18 marks]

A life insurance company has liabilities over the next ten years to meet its obligations to policy- holders through payments of $100 at the end of each year. Presently, the insurer’s asset portfolio holds 5-year government bonds to meet its liabilities; and these bonds pay an annual coupon in arrears at a rate of 4% of nominal value. Assume interest rates are 3% pa. for all maturities.

i). What is the present value of the liabilities? (2 marks)

ii). Calculate the total nominal value of the bonds so that the present value of the assets is equal to the present value of the liabilities. (3 marks)

The company is now considering its interest rate risk position:

iii). Calculate the Macaulay Duration of the liabilities. (2 marks)

iv). Calculate the Macaulay Duration of the assets. (2 marks)

v). Using your answers to parts iii) and iv), estimate by how much more the present value of the liabilities will increase than the present value of the assets if interest rates change from their current level at 3% to 2%. (2 marks)

vi). Briefly explain whether the company is immunised against movements in interest rates?  (1 mark)

To better hedge interest rate risk, the company now wishes to add 20 year zero coupon bonds to its asset portfolio; and will also re-evaluate the amount of nominal held in the 5 year bonds.

vii). Calculate the nominal amounts for the 5-year coupon bearing bonds and the 20-year zero- coupon bonds so that the asset and liability portfolios satisfies Redington’s first two conditions of equal present value and equal duration at the current interest rate of 3% pa. (4 marks)

viii). By testing Redington’s third condition on the resulting portfolios from your answer to part vii), confirm that the surplus portfolio (assets less liabilities) is immunised against small movements in interest rates. (2 marks)

Part F (Application): [14 marks]

A newly discovered organism, referred to as Organism A, has been studied and the following life table has been constructed to model the survival statistics of the organism:

i). Evaluate the survival probabilities 5p0 , 5p5  and 5p10 . (1.5 marks)

ii). From your answers to part i), describe the key feature of the underlying survival model governing mortality for organism A. (1 mark)

A diferent organism, Organism B, was also discovered, and scientistis believe that the two diferent organism types have populations governed by diferent survival models with force of mortality functions μA (x) and μB (x) for organisms A and B respectively.  The form of these functions is theorised to be, for some constant k > 0:

μA (x)  =  k

μB (x)  =  kx

iii). Derive an expression for the survival probability, tpx  under the survival model μA (x).  (1 mark)

iv). Derive an expression for the survival probability, tpx  under the survival model μB (x).  (2 marks)

v). How does the survival probability tpx change with increasing age x in the two models μA  and μB? (1.5 mark)

vi). For a new-born (x = 0), write down the range of complete future lifespans t for which survival is more likely in the model μA  than in μB? (2 marks)

vii). Using the life table data for Organism A, estimate the value of the constant k  (to 1 decimal place), and brie且y comment on how well the model μA (x) fits the emprical data. (3 marks)

viii). Using the value of k found in part vii), estimate the age at which only one half of a population of new-born (age 0) Organism B would remain? (2 marks)

Part G (Application): [16 Marks]

In the distant future, conditions on Earth have changed such that people can live much longer through advances in medical science, but then there are also other factors (wars and famine for example) that have resulted in people dying more frequently through middle age.  The insurance companies of the future have taken note of this, and have developed the following life table function which models the population mortality rates for ages 50 and above:

i). Derive expressions for the survival probability npx  and the curtate future lifespan probability function P (Kx = k). (2 marks)

ii). Derive expressions for the whole of life assurance function Ax  and the whole life annuity in advance function x. (4 marks)

iii). From your answer to part ii), comment on how the functions Ax  and x  depend on x.  (1 mark)

An individual aged 50 is looking to buy a whole life annuity payable in advance which pays $30, 000 annually at the start of each year, and is willing to consider buying a deferred annuity depending on what’son oferin the market. Interest rates are 5% pa. An insurance company is advertising a range of whole life annuity in advance products ranging from immediate commencement (0 deferment term), through to 15 years of deferment. The current costs are as follows:

iv). Using your answer to part ii), calculate the numerical value of 50   (to 3 decimal places).  (1 mark)

v). Calculate the expected present value of the choices ofered by the insurance company, and discuss which deferment term represents the best value to the customer? (5 marks)

vi). The same company is also ofering a whole life assurance policy with a sum assured of $500, 000 at an upfront cost-price of $200, 000.  Using your answer to part ii), evaluate the expected present value of this contract.  Comment on whether the cost-price is fair and whether it represnts a bigger saving than the whole life annuity-in-advance choices. (3 marks)