金融数学代写

程序代写 CS代考

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University College Dublin
An Col ́aiste Ollscoile, Baile A ́tha Cliath

SEMESTER 1 STAT 40920

ACTUARIAL & FINANCIAL MATHS ASSIGNMENT #1

Head of School: Brendan Murphy

Lecturer: Andrew D Smith

Instructions for Candidates

Answer all three questions. To obtain full credit, working and intermediate steps must be shown.
.
This assignment is due by 11 am on Tuesday 16 October. You can either put your script in the pigeon-hole by the maths office in Science North, or you can bring it to the 11am lecture. Marks will be deducted for late submission.
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Please put your student number and not your name at the top of each page of your answer, and fasten pages with a staple or paperclip.
Scripts will be returned during the tutorial at 11am on Tuesday 23 October. If you want to see the solutions and understand any places where you might have gone wrong, you need to turn up to this tutorial.
.

You may quote results from your lecture notes when you attempt these questions.

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In the questions for this assignment, let ζ be the number between 0 and 999 formed by the last three digits of your student number.
For example, if your student number is 1234567 then ζ = 567.

  1. In a given investment market, the value of an investment does one of the following every year:

    • Falls by 20%, with probability (500 + ζ)/5000
    • Rises by 20%, with probability (600 + 2ζ)/5000 • Is unchanged, with probability (3900 − 3ζ)/5000

    This investment has no investment income for the period considered, so all returns are in the form of capital gains (or losses). Returns in different years are indepen- dent.

    1. (a)  For the future value s5 of five equal instalments of e1 each, calculate:
      (i) The mean Es5 [4]

      (ii) The standard deviation Stdev(s5 ) [4]

    2. (b)  Calculate the expected present value Ea5 of a five-year annuity certain payable

      at the end of each year. [4]

    State your answers to four decimal places.

  2. For this question, you should refer to the following tables of Irish inflation. To save your typing, you can also download the data from:

    https://www.cso.ie/en/statistics/prices/consumerpriceindex/.

    The online tables are floating point numbers; to answer this question you should round the index values down to the nearest integer, for example using Excel’s INT function. You should extract a 70-year subset of annual inflation data as follows.

• The range of years should be determined by ζ modulo 5:

ζ mod 5 0
1
2
3
4

Start Year 1930 1931 1932 1933 1934

End Year 2000 2001 2002 2003 2004

• The month should be determined by ζ mod 12. Then 0 = December, 1 = January, 2 = February etc.

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• This should give you 71 data points for the inflation index, which you should express as 70 annual observations of the log inflation rate.

  1. (a)  From your 70-year inflation series, estimate:

    (i) the mean of the log rate of inflation Xt. [2] (ii) the (unconditional) standard deviation of Xt. [2] (iii) the unconditional correlation between Xt−1 and Xt. [2]

  2. (b)  State the parameters (c,σ,φ) of a first order auto-regressive (AR1) model consistent with the the statistics you have just observed. [6]
  3. (c)  Estimate the unconditional correlation between Xt−2 and Xt. Compare this to the prediction of your fitted autoregressive model. [3]
  4. (d)  Suggest, with reasoning, a modification to the AR1 model which could capture your estimated unconditional correlation between Xt−2 and Xt. [3]

    State your answers to four decimal places.

3. The surplus of an insurer follows a Lundberg process. Every year, premiums of e1 are received. A claim of size e3 occurs with probability (1500+ζ)/10000 each year; otherwise there is no claim that year.

  1. (a)  Calculate the probability of ruin within the first four years, starting with an initial surplus of e2. In this calculation, assume that ruin occurs if the surplus falls strictly below zero at one (or more) of the time points {0, 1, 2, 3, 4}. [8]
  2. (b)  Let ψ(u) denote the infinite-horizon ruin probability with initial surplus u. Derive a relationship expressing ψ(u + 1) in terms of ψ(u) and ψ(u − 2). [4]
  3. (c)  In what follows, you may assume that ψ(u) takes the following form for integers

u ≥ 0:

2

(ii) Find the smallest value of u for which ψ(u) < 1%.

ψ(u)= 12 where−1 <λ1 <0<λ2 <1areconstants.

21

(1−λ )λu+2 −(1−λ )λu+2

λ2 − λ1 (i) Find the numerical values of λ1 and λ2

[4] [4]

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Irish Consumer Price Index 1930-1949

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1930 179 179 179 168 168 168 168 168 168 168 168 168 1931 164 164 164 156 156 156 157 157 157 165 165 165 1932 162 162 162 159 159 159 153 153 153 155 155 155 1933 151 151 151 148 148 148 149 149 149 156 156 156 1934 152 152 152 149 149 149 152 152 152 157 157 157 1935 153 153 153 151 151 151 156 156 156 162 162 162 1936 159 159 159 157 157 157 159 159 159 166 166 166 1937 167 167 167 167 167 167 170 170 170 177 177 177 1938 173 173 173 171 171 171 173 173 173 176 176 176 1939 174 174 174 172 172 172 173 173 173 192 192 192 1940 197 197 197 204 204 204 206 206 206 214 214 214 1941 218 218 218 220 220 220 228 228 228 237 237 237 1942 237 237 237 240 240 240 250 250 250 273 273 273 1943 273 273 273 275 275 275 284 284 284 294 294 294 1944 296 296 296 292 292 292 296 296 296 296 296 296 1945 295 295 295 292 292 292 293 293 293 298 298 298 1946 294 294 294 287 287 287 288 288 288 293 293 293 1947 295 295 295 305 305 305 319 319 319 309 309 309 1948 316 316 316 319 319 319 316 316 316 316 316 316 1949 316 316 316 316 316 316 319 319 319 319 319 319

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Irish Consumer Price Index 1950-1969

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1950 319 319 319 325 325 325 319 319 319 325 325 325 1951 329 329 329 348 348 348 354 354 354 360 360 360 1952 364 364 364 367 367 367 389 389 389 392 392 392 1953 392 392 392 402 402 402 399 399 399 399 399 399 1954 396 396 396 396 396 396 403 403 403 400 400 400 1955 403 403 403 406 406 406 409 409 409 418 418 418 1956 420 420 420 428 428 428 429 429 429 428 428 428 1957 429 429 429 440 440 440 455 455 455 453 453 453 1958 460 460 460 464 464 464 466 466 466 466 466 466 1959 469 469 469 468 468 468 461 461 461 458 458 458 1960 460 460 460 467 467 467 467 467 467 470 470 470 1961 474 474 474 479 479 479 480 480 480 482 482 482 1962 491 491 491 504 504 504 502 502 502 500 500 500 1963 509 509 509 508 508 508 507 507 507 523 523 523 1964 526 526 526 546 546 546 553 553 553 559 559 559 1965 565 565 565 575 575 575 577 577 577 577 577 577 1966 577 577 577 588 588 588 598 598 598 599 599 599 1967 600 600 600 610 610 610 611 611 611 615 615 615 1968 628 628 628 638 638 638 639 639 639 648 648 648 1969 670 670 670 681 681 681 692 692 692 698 698 698

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Year Jan Feb 1970 709 709 1971 780 780 1972 853 853 1973 938 938 1974 1065 1065 1975 1318 1318 1976 1530 1530 1977 1786 1786 1978 1933 1933 1979 2142 2142 1980 2475 2475 1981 2994 2994 1982 3599 3599 1983 4003 4003 1984 4410 4410 1985 4683 4683 1986 4898 4898 1987 5066 5066 1988 5163 5163 1989 5335 5335

Irish Consumer Price Index 1970-1989

Mar Apr May Jun Jul Aug Sep Oct Nov Dec 709 738 738 738 751 751 751 768 768 768 780 801 801 801 817 817 817 834 834 834 853 866 866 866 890 890 890 903 903 903 938 967 967 967 990 990 990 1017 1017 1017

1065 1124 1124 1124 1167 1167 1167 1221 1221 1221 1318 1399 1399 1399 1388 1388 1388 1426 1426 1426 1530 1626 1626 1626 1650 1650 1650 1720 1720 1720 1786 1853 1853 1853 1873 1873 1873 1906 1906 1906 1933 1967 1967 1967 2027 2027 2027 2057 2057 2057 2142 2211 2211 2211 2302 2302 2302 2385 2385 2385 2475 2657 2657 2657 2736 2736 2736 2820 2820 2820 2994 3111 3111 3111 3287 3287 3287 3478 3478 3478 3599 3764 3764 3764 3844 3844 3844 3906 3906 3906 4003 4113 4113 4113 4230 4230 4230 4308 4308 4308 4410 4511 4511 4511 4566 4566 4566 4597 4597 4597 4683 4746 4746 4746 4816 4816 4816 4824 4824 4824 4898 4956 4956 4956 4964 4964 4964 4976 4976 4976 5066 5097 5097 5097 5124 5124 5124 5128 5128 5128 5163 5191 5191 5191 5234 5234 5234 5265 5265 5265 5335 5390 5390 5390 5468 5468 5468 5511 5511 5511

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Year Jan Feb 1990 5561 5561 1991 5704 5704 1992 5913 5913 1993 6024 6024 1994 6128 6128 1995 6283 6283 1996 6409 6409 1997 6471 6510 1998 6588 6620 1999 6685 6724 2000 6952 7010 2001 7316 7381 2002 7671 7725 2003 8040 8117 2004 8187 8256 2005 8371 8440 2006 8625 8717 2007 9061 9134 2008 9452 9569 2009 9442 9406 2010 9071 9107

Irish Consumer Price Index 1990-2010

Mar Apr May Jun Jul Aug Sep Oct Nov Dec 5561 5577 5577 5577 5627 5627 5627 5660 5660 5660 5704 5748 5748 5748 5825 5825 5825 5864 5864 5864 5913 5958 5958 5958 5991 5991 5991 6002 6002 6002 6024 6013 6013 6013 6073 6073 6073 6090 6090 6090 6128 6178 6178 6178 6228 6228 6228 6233 6233 6233 6283 6349 6349 6349 6376 6376 6376 6382 6382 6382 6409 6437 6437 6437 6470 6470 6470 6503 6503 6503 6516 6523 6536 6555 6549 6536 6568 6581 6607 6627 6653 6685 6711 6744 6724 6744 6763 6770 6744 6737 6744 6776 6809 6828 6802 6841 6867 6874 6887 6965 7056 7108 7160 7206 7225 7264 7290 7342 7368 7375 7440 7505 7550 7589 7570 7596 7628 7654 7648 7687 7794 7864 7902 7917 7887 7933 7971 8010 8017 8071 8179 8202 8194 8194 8133 8187 8202 8194 8194 8225 8286 8317 8333 8379 8356 8402 8409 8417 8433 8440 8463 8502 8532 8556 8556 8594 8663 8671 8655 8648 8755 8825 8863 8886 8917 8978 9009 9009 9040 9071 9198 9270 9306 9325 9352 9397 9424 9433 9488 9497 9660 9669 9742 9787 9760 9805 9833 9814 9724 9606 9406 9334 9288 9261 9188 9225 9188 9170 9170 9125 9116 9134 9188 9179 9179 9243 9234 9234 9225 9243

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