40.
Losses come from a mixture of an exponential distribution with mean 100 with probability p
and an exponential distribution with mean 10,000 with probability 1 p .
Losses of 100 and 2000 are observed.
Determine the likelihood function of p.
(A)
(B)
(C)
ACT452H1S - TEST 1 - FEBRUARY 11, 2015
Write name and student number on this page. Write your solution for each
question in the space provided. The only aid allowed is a calculator.
1. \" \# \$ is a random sample of size 3 from a distribution on the inter
ACT452H1S PROBLEM SET 4
1. You are given sample data of losses from a random variable \ in interval grouped form
as follows:
Interval
! "!
"! #!
#! %!
%! "!
Number of Observations
"!
#!
%!
$!
Assume uniform distribution within each interval.
(a) Find the
16.
You use a uniform kernel density estimator with b = 50 to smooth the following workers
compensation loss payments:
82
126
161
294
384
If F ( x) denotes the estimated distribution function and F5 x denotes the empirical
distribution function, determine
4.
For observation i of a survival study:
x di is the left truncation point
x xi is the observed value if not right censored
x ui is the observed value if right censored
You are given:
Observation (i)
1
2
3
4
5
6
7
8
9
10
di
0
0
0
0
0
0
0
1.3
1.5
1.6
xi
0
ACT452H1S PROBLEM SET 4
1. You are given sample data of losses from a random variable \ in interval grouped form
as follows:
Interval
! "!
"! #!
#! %!
%! "!
Number of Observations
"!
#!
%!
$!
Assume uniform distribution within each interval.
(a) Find the
21-22.
Use the following information for questions 21 and 22.
For a survival study with censored and truncated data, you are given:
Time (t)
1
2
3
4
5
21.
Number at Risk
at Time t
30
27
32
25
20
Failures at Time t
5
9
6
5
4
The probability of failing at o
ACT452H1S PROBLEM SET 5
1. \ has an exponential distribution with mean 1 ( pdf 0 B /B ).
Suppose that ] \ # .
(a) Find the exact mean and variance of ]
(b) Find the approximate mean and variance of ] using the delta method.
2. You are given the following
ACT452H1S PROBLEM SET 3
1. You are given the following for a sample of five observations from a bivariate distribution:
B
C
1
4
2
2
4
3
5
6
6
4
E is the covariance of the empirical distribution J/ as defined by these five observations. F is the
maximumpos
24.
You are given:
(i)
Losses are uniformly distributed on 0, T with T > 150.
(ii)
The policy limit is 150.
(iii)
A sample of payments is:
14, 33, 72, 94, 120, 135, 150, 150
Estimate T by matching the average sample payment to the expected payment per los
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University of Toronto - ACT370: Midterm 1
Name:
Student Number:
1. You are given the following for European options expiring in three months with strike price 50:
The options are modeled with a one-period binomial tree
For a call option, the replicating
ACT 370H1S W2017, Financial Principles for Actuarial Science II
(Jan 9, 2017 version )
Lecture Section
L0101/A&S and L2001/ASE
Lecture times
M11
W 11-1
Xuancheng (Bill) Hung
xuancheng.huang@mail.utoronto.ca
LM162
W 2:00-4:00
SS6011
Instructor
Office hours
Name:
ACT370: Term Test 1
Question 1
Question 1
The current exchange rate between USD and CAD is 1.20CAD/USD. The continuously compounded risk-free
rate for USD is 1%, and the risk-free rate for CAD is 2%.
(a) Find the forward price of a contract to purch
Question 1 (10 pts)
You are given:
The stock price is 40
The risk-free rate is 4%
A 3-month European call option on the stock with strike 40 costs 4.10
A 3-month European put option on the stock with strike 40 costs 3.91
(a) Assume that the stock pays
University of Toronto - ACT370: Midterm 1
Name:
Student Number:
1. You are given the following for European options expiring in three months with strike price 50:
The options are modeled with a one-period binomial tree
For a call option, the replicating
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University of Toronto
ACT348H1S - TERM TEST
2nd of November 2015
Instructor - Andrei Badescu
NOTES:
1. ONLY non-programmable calculators are allowed.
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3. Timing: 90 minutes.
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Example 22: For a group of lives aged 30'. containing an equal number of smokers and non-
smokers, you are given:
(i) For non-smokers= pi: = 0.08 , m 33 30
(ii) For smokers. p: = 0.16 , :e I: 30
Calculate qgu. for a life randome selected o1n those survivi
ACT247 Lec 2
January 12, 2017
10:08 AM
New Section 38 Page 1
New Section 38 Page 2
If a new borns survival function is e^(-0.02x), find Sussies
chance of getting paid if she buys (1) a 5-year deferred whole
life insurance, 2) a 5-year deferred 10-year ter
ACT247 Lecture 1
January 5, 2017
10:55 AM
Depends on
actual
survival
model
Time until death variable:
New Section 36 Page 1
New Section 36 Page 2
The chance of getting paid from life insurance:
n-year term insurance:
n-year pure endownment:
n-year deferre
ACT247
Introductory Life
Contingencies
Lecture 2
Prof. V. Zhang
Department of Statistical Science
Three ways to participate in polling:
1- Download the Poll Everywhere app on your phone and vote! - FREE
2- Open any web browser on your phone/tablet/laptop
ACT247 Lec 3
January 19, 2017
10:08 AM
Using Gompertzs Law of mortality - Sussies force of
mortality would grow faster than the general population
because of her diagnosis, lets make is 11% annual rate of
increase (as opposed to the 9.6% for the modern da
EXAHPLE 2A The survival function for newborns is
lmI
'33): W I 5 1m
fit-1m]
Calculate
1. The probability that a newborn survives to age Tr' but does not survive to age 34.
2. The probabilityT that [20 sunrives to age 15 but not to age 34.
3. 35:11:20]-
EX
ACT247 Lec 4
January 26, 2017
10:05 AM
New Section 45 Page 1
A new borns survival function S0(x) follows De Moires law of
mortality with a maximum age of 115. Sussie is currently 35
years old. Force of interest is 10%. If she purchases a whole life
policy
ACT247
Introductory Life
Contingencies
Lecture 3
Prof. V. Zhang
Department of Statistical Science
Using Parametric Survival Functions
Using Gompertzs Law of mortality - Sussies force of mortality would
grow faster than the general population because of he
ACT247
Introductory Life
Contingencies
Lecture 4
Prof. V. Zhang
Department of Statistical Science
Some possible definition of insurance cost/price
(ignoring expenses,profit, margin,etc.)
E(Z)
Perhaps the company wants to capture some variations in the p
ACT247
Introductory Life
Contingencies
Lecture 1
Prof. V. Zhang
Department of Statistical Science
Types of Life Insurance
Whole life insurance pays at the time of death (whenever that may be)
N-year term insurance (e.g. if Sussie enrolls in an experimen