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Final Test MATH 3281
Creation date：2024-04-08 16:31:40
Belonging category：金融Finance
Final Test

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Final Test MATH 3281 

Given name and surname:
Student No:
Signature:
INSTRUCTIONS:
1. Please write everything in ink.
2. This exam is a ‘closed book’ exam, duration 180 minutes.
3. Only non-programmable calculators are permitted.
4. The text has 21 pages, and it contains 20 questions. Read the ques-
tions carefully. Fill in answers in designated spaces. Your work must
justify the answer you give. Answers without supporting work will
not be given credit.
5. Hand in your examination answer booklets.
Useful formulas:
1.) For t ∈ [0, 1), and non-negative integer k, we have that
P[T (u) ≥ k + t] UDD= (1− t)P[T (u) ≥ k] + tP[T (u) ≥ k + 1].
2.) For ∆f(k) := f(k + 1)− f(k), k ≥ 0, it holds that
n∑
k=m
f(k)∆g(k) =
[
f(k)g(k)|n+1m
]− n∑
k=m
g(k + 1)∆f(k).
GOOD LUCK!
Final Test MATH 3281 3.00 Page 2 of 21
Question 1 The random variable Z is the present value random variable for an n-year endowment
insurance. The insurance is issued to (x), and its n-year term part is payable upon
death. Let n = 20. If the force of interest is δ = 0.05 and the force of mortality is
µ(x + t) ≡ 0.01. Formulate the cumulative distribution function of Z and graph it.
Calculate Ax:n := E[Z] using the cumulative distribution function of Z.
Cont.
Final Test MATH 3281 3.00 Page 3 of 21
Question 2 Let Z1 and Z2 denote, respectively, the n-year term and the pure endowment insurance
coverages bearing $1 face amounts and issued to (x). Assume that the n-year term
insurance is payable upon death. Show that
lim
n→0
Cov[Z1, Z2](n) = lim
n→∞
Cov[Z1, Z2](n) = 0.
Explain.
Cont.
Final Test MATH 3281 3.00 Page 4 of 21
Question 3 For t > 0, let b(t) = t, µ(x + t) ≡ µ and the force of interest be constant δ > 0, say.
Also, let T (x) denote the future life-time random variable of (x). Derive expressions
for (IA)x := E[b(T )v
T ] as well as the variance of the random variable b(T )vT ,
Cont.
Final Test MATH 3281 3.00 Page 5 of 21
Question 4 Recall that the Makeham law of mortality is given by the following special form of the
force of mortality
µ(x) = A+Bcx, x > 0, (1)
where A ≥ −B, B > 0, c > 1. Assume independence of the random future life-times of
(x) and (y). Also, assume that the corresponding forces of mortality follow Makeham
equation (1). Show that ax:y is equal to the actuarial present value of an annuity on a
single life (w) such that cw = cx + cy and the force of interest δ′ = δ + A. Use these
findings to demonstrate that
Ax:y = A
′
w + Aa
′
w,
where the primed functions are evaluated at the force of interest δ′.
Cont.
Final Test MATH 3281 3.00 Page 6 of 21
Question 5 Let I = 1 represent death by accidental means and I = 2 represent death by other
means. You are given that
• δ = 0.05;
• the force of decrement for death by accidental means is µ(1)(x + t) = 0.005 for
t ≥ 0;
• the force of decrement for death by other means is µ(2)(x+ t) = 0.02 for t ≥ 0.
A 20-year term insurance policy payable at the moment of death is issued to (x)
providing a benefit of $2 if the death is by accidental means, and providing a benefit
of $1 if the death is by other means. Find the expectation and the variance of the
present value of the benefits random variable.
Cont.
Final Test MATH 3281 3.00 Page 7 of 21
Question 6 Verify the formula:
δ(Ia)T (x) + T (x)v
T (x) = aT (x) ,
where T (x) represents the future life-time random variable of (x). Use it to prove that
δ(Ia)x + (IA)x = ax.
Here (Ia)x is the actuarial present value of a life annuity to (x) under which payments
are being made continuously at the rate of $t per annum at time t > 0.
Cont.
Final Test MATH 3281 3.00 Page 8 of 21
Question 7 If g(X) is a non-negative function, and X is a non-negative random variable with
probability density function f(x), and if the expectation below is well-defined and
finite, justify the inequality
E[g(X)] =
∫ ∞
−∞
g(x)f(x)dx ≥ kP[g(X) ≥ k], k > 0.
Then use the above to show that
ax ≥ a◦
ex
P
[
aT (x) ≥ a◦ex
]
= a◦
ex
P
[
T ≥ ◦ex
]
,
where T (x) denotes as always the future life-time random variable of (x).
Cont.
Final Test MATH 3281 3.00 Page 9 of 21
Question 8 Consider the following portfolio of annuities-due currently being paid by a pension
fund
Age Number of annuitants
65 30
75 20
85 10
Each annuity has an annual payment of $1 as long as the annuitant survives. As-
sume an earned interest rate of 0.06 and the Illustrative Life Table mortality. For
the present value of these obligations of the pension fund, calculate the expectation,
the variance, and the 95% percentile of the distribution. Assume that the lives have
mutually independent future life-times whenever required.
Cont.
Final Test MATH 3281 3.00 Page 10 of 21
Question 9 Assume that T (x) and T (y) are positively quadrant dependent future life-time random
variables, that is we have that
P[T (x) ≤ u, T (y) ≤ v] ≥ P[T (x) ≤ u]P[T (y) ≤ v]
for any u, v ≥ 0. Denote by ax:y the joint whole life annuity that is calculated under the
assumption of positively quadrant dependent future life-times of (x) and (y). Denote by
a⊥x:y the joint whole life annuity that is calculated under the assumption of independent
future life-times of (x) and (y). Check that
ax:y ≥ a⊥x:y.
Cont.
Final Test MATH 3281 3.00 Page 11 of 21
Question 10 If the force of mortality strictly increases with age, show that P (Ax) > µ(x), that is
the benefit premium is greater than the force of mortality evaluated at zero.
Cont.
Final Test MATH 3281 3.00 Page 12 of 21
Question 11 Define the notion of actuarial premium calculation principle. Show (using either the
insurer’s premium defining equation, or the insured’s premium defining equation) that
every actuarial premium must be at least as large as the mathematical expectation of
the underlying risk random variable. State the condition for the feasibility of insurance
contracts.
Cont.
Final Test MATH 3281 3.00 Page 13 of 21
Question 12 A 20 payment life policy is designed to return, in the event of death, $10, 000 plus
all contract premiums without interest. The return-of-premium feature applied both
during the premium paying period and after. Premiums are annual, and death benefits
are paid at the end of the year of death. For a policy issued to (x), the annual contract
premium is to be 110% of the benefit premium plus $25. Express in terms of the
actuarial present value symbols the annual contract premium.
Cont.
Final Test MATH 3281 3.00 Page 14 of 21
Question 13 Let T (x) ∼ Exp(µ), µ > 0. Derive the probability density function of the Loss-at-Issue
random variable L in terms of the probability density function of the random variable
T (x). Show that E[L] = 0 when the benefit premium is chosen to price the whole life
insurance on (x). Determine the premium pi such that P[L > 0] = 0.5.
Cont.
Final Test MATH 3281 3.00 Page 15 of 21
Question 14 A whole life insurance issued to (25) pays a unit benefit at the end of the year of death.
Premiums are paid annually to the age 65. The benefit premium for the first 10 years
is P25 followed by an increased level annual benefit premium for the rest 30 years. Use
the Illustrative Life Table to find the following
• The level annual benefit premium paid at ages 35 through 64.
• The tenth year benefit reserve.
• At the end of 10 years the policyholder has the option to continue with the benefit
premium P25 until age 65 in return for reducing the death benefit to $y for the
death after age 35. Calculate y.
Cont.
Final Test MATH 3281 3.00 Page 16 of 21
Question 15 A fully discrete whole life insurance with a unit benefit issued to (x) has its first year’s
benefit premium equal to the actuarial present value of the first year’s benefit, and
the remaining benefit premiums are level and determined by the equivalence princi-
ple. Determine formulas for (a) the first year’s benefit premium, (b) the level benefit
premium after the first year, and (c) the benefit reserve at the first duration.
Cont.
Final Test MATH 3281 3.00 Page 17 of 21
Question 16 Consider the life insurance policy from Question 16. Calculate Cov[C0, Ch]. Repeat
your calculations for Cov[Cj, Ch], where 0 < j < h.
Cont.
Final Test MATH 3281 3.00 Page 18 of 21
Question 17 Let hLx(K(x);Px) represent the prospective loss random variable at time h = 0, 1, . . .
for a whole life insurance with unit benefit, premiums payable due if the insured is
alive, and the benefit payable upon death. Show that there exists random variable hCx
such that the identity
hLx =
∞∑
j=h
vj−hjCx
holds.
Cont.
Final Test MATH 3281 3.00 Page 19 of 21
Question 18 Under the assumption of the uniform distribution of death in each year of age, prove
or disprove
kv(Ax:n ) ≈ i
δ
kv(Ax:n ).
Cont.
Final Test MATH 3281 3.00 Page 20 of 21
Question 19 If 10v35 = 0.150 and 20v35 = 0.354, then what is the value of 10v45?
Cont.
Final Test MATH 3281 3.00 Page 21 of 21
Question 20 Bonus. LetX represent a random variable with finite second moment. It is well-known
that the variance of the random variable X is given, for µX := E[X], by
Var[X] = E[X2]− µ2X ,
and so the variance depends on the mean of X. This is somewhat misleading because
it is not clear why a good measure of variability should depend on the mean. Show
and prove an additional formulation of the variance that emphasizes that Var[X] in
fact does not depend on the mean of the random variable X.
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COMSM0093 COMP281 AFM 333 IBUS1001 CP2501 Math 105A VE320 TEC101 SOCY2166 COM240 EFIM20033 ENGI4467 EFIM20034 EC 979 BE432 7EJ502 CSSE6400 COMP90025 Econ 312 ENG2031 ECO0006 EXERSCI 207 CAN209 ECO 137W FT312 ECO 2199 MATH32052 BDW4213 APS 360 SAS 6 CHME0017 PO92Q Economics 01AV MATH39512 MATH 5486 COMP8620 ISE102 ECON235 ECON 206 FI 345 PHAY0020 MATH0099 MATH0085 EFB106 PHY 134 MAE115 circuits Chemistry CSE12 MATH-UA 233 PHAS0038 COMP2140 CONMGNT 7049 COMP151101 ECON 485 ACX5903 ENGN6536 ARCH1162 Law PHAS0042 CCF1C03 MATH 351 Phys 139 AG942 MGT001371 APH106 ECON1051 MA4AM MA3Z7 NUMBER THEORY REST0006 SESS0026 STAT0023 TABL 2741 CMP4008Y EFIM10015 Acct 560 PSYC735 GEOG 5 ACCT3104 CSEC5616 PHIL1012 ITLS6111 ECON8022 5CCM226A compX123 MA4XJ R001 EWB LLP120 COMP643 ENG 103 ECON 23950 ACCT 207 FINC 616 Bio99 Math 137A Psychology BCPM0054 MATH20212 EC3450 CENV6141 PSY 205 CE335 CS365 Calculus ACSD INFT6800 MCIT INFO6030 PHYS 111LV H63EEJ ECON2151 ENGI 2181 MTRX1701 PLIT08009 CS240 Math 270 CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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