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Questions tagged [actuarial-science]

Actuarial science is a discipline that uses mathematics and statistics to assess risk. The mathematics involved in actuarial science includes probability, statistics, finance, life insurance mathematics, and more.

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If integrated tail $F_I$ is subexponetial, is $F$ also subexponetial?

Im currently reading Non-Life Insurance Mathematics by Thomas Mikosch and in chapter 4 on Ruin theory there is constant distinction between light and heavy-tailed distributions (as there should be) ...
VlakecTomaz's user avatar
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Help me calculate the assumed interest rate $i'$ under $\frac{\bar{A}_x}{\mu_x}=\bar{a'}_x \neq \bar{a}_x$

Setup force of mortality $\mu_x = 0.005\cdot (1.01)^x$ original assumed interest rate $i = 0.05$ assumption: $$ \frac{\bar{A}_x}{\mu_x}=\bar{a'}_x (\neq \bar{a}_x) $$ At this point, we want to ...
ytnb's user avatar
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Probability of the Maximum of $6$ die rolls

Someone please explain to me why this is wrong. Here is the question: If I roll $6$ different fair $6$-sided dice, what is the probability that the maximum of the rolls is $5$? My approach was to ...
Haik Voskerchian's user avatar
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1 answer
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Why is $\varphi$ differentiable?

In the proof of a lemma (called the fundamental integral equation for the non-ruin probability) in the book Non-Life Insurance Mathematics by Thomas Mikosch (page 164) we get to the following ...
VlakecTomaz's user avatar
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1 answer
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Using the critical value from signs test to reject null hypothesis

Question 1 (a) A graduation using $20$ age groups has resulted in $6$ positive and $14$ negative deviations. Carry out the signs test on these data. (b) A graduation covers $20$ age groups. The number ...
Jessie's user avatar
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1 vote
1 answer
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Why is a factor of 24 used in this Central Limit Theorem Question?

I have a question from Example 12.3.1 from Marcel B. Finan book Lecture Notes in Actuarial Mathematics Question: Let $X_i$, $i = 1, 2, \dots, 48$ be independent random variables that are uniformly ...
mathexchangeisok's user avatar
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1 answer
75 views

Present value involving deferred annuities - is this a typo?

Catfish Hunter’s 1974 baseball contract with the Oakland Athletics called for half of his 100,000 salary to be paid to a life insurance company of his choice for the purchase of a deferred annuity. ...
Mitchell's user avatar
3 votes
1 answer
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Improper Integral $\int_{0}^{\infty} \log(t) t^{-\frac{1}{2}} \exp\left\{-t\right\} dt$

Background Hi. I am currently writing my undergraduate thesis which mainly revolves around the generalized log-Moyal distribution pioneered by Bhati and Ravi (see here). In the aforementioned article, ...
Karel's user avatar
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1 answer
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What is the difference in the wording of these questions?

Question 1: An account with an initial amount B earns compound interest at an annual effective interest rate $i$. The interest in the third year is $426$ and the discount in the seventh year is $812$. ...
Mitchell's user avatar
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Question about accumulated values and nominal rates of discount

A collection agency pays a doctor $\$5,000$ for invoices that the doctor hasn't been able to collect on. After two years, the collection agency has collected $\$6,000$ on the invoices. At what nominal ...
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435 views

Best books covering SOA Exam Fm

Looking for book recommendations to self learn material for SOA exam FM. (This exam is mostly equations involving compound interest). Particularly something that covers the underlying math intuitively ...
Older Amateur's user avatar
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Thiele differential equation for reserves

I am supposed to solve the below question using Thieles differential equation, and I am having a hard time formulating Thieles differential equation for this specific insurance case and would ...
idlatva's user avatar
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1 answer
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How to calculate the accumulated value of a simple interest annuity?

An account is credited interest using 6% simple interest rate from the date of each deposit into > the account. Annual payments of 100 are deposited into the account. Calculate the accumulated ...
j.jerrod.taylor's user avatar
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1 answer
52 views

Calculating the withdrawal amount from a fund as an annuity-immediate incorrectly

I have the following problem. Consider an investment of $5,000 at 6% convertible semiannually. How much can be withdrawn each half-year to use up the fund exactly at the end of 20 years? I can tell ...
j.jerrod.taylor's user avatar
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1 answer
88 views

Negative binomial distribution with non integer parameter

I am solving an actuarial math problem. Suppose that the number of claims, $N$, in a year follows a Poisson distribution $\mathrm{Po}(\Theta)$, where $\Theta$ is a gamma distribution $\Gamma(5, 1/2)$, ...
linkup's user avatar
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1 answer
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Macaulay Duration?

I have a problem with a question, I don't know if the question is well worded, it reads as follows Having the following information about a loan: Interest rate: 11.5% per annum, compounded monthly. ...
thelast12e1's user avatar
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Determine the probability for a pair (x, y) that:

(a) y dies after the year of death of x (b) y dies after x, assuming that times of death over the year of death are equally distributed. Tabulate and compare the results obtained for a few concrete ...
lenamagdalena's user avatar
1 vote
1 answer
69 views

Proof about Insurance benefits [closed]

Prove this statement is the cash value of a death insurance policy on the second life. $$ A_{\overline{xy}} = A_x + A_y - A_{xy} = 1 - d \cdot \ddot{a}_{\overline{xy}} $$ I can't ...
lenamagdalena's user avatar
0 votes
1 answer
74 views

How to determine an annuity is due or immediate in the FM exam

There is an classic SOA problem I found confused about. College tuition is 6000 for the current school year, payable in full at the beginning of the school year. College tuition will grow at an annual ...
Cooper's user avatar
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1 answer
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Chain Ladder Model: Unbiasedness of estimators for the variance parameters

I don't understand the equality of the middle term: from page 40 of "Wüthrich, M. V., & Merz, M. (2008). Stochastic claims reserving methods in insurance. John Wiley & Sons." I see ...
Frodo361's user avatar
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1 answer
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Using $a_{\overline {n} \rceil i}$ from Exam FM.

I am trying to solve the following problem Olga buys a 5-yr increasing annuity for $X$. Olga will receive $2$ at the end of the first month, $4$ at the end of the second month, and for each month ...
Cooper's user avatar
  • 183
5 votes
0 answers
68 views

Expected value of surplus process at the moment of ruin

This problem comes from actuarial exams. We consider a continuous surplus process $$ U(t) = ct - \sum_{k=1}^{N(t)} X_k,$$ where $N(t)$ is Poisson process such that $\mathbb{E}N(t)=\lambda t$, and $X_k$...
Jacek's user avatar
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2 votes
0 answers
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Derivation of least squares solution using SVD for Lee-Carter mortality model

The Lee-carter model aims to model central mortality rates using the following $$ log(m_{x,t}) = a_{x} + b_{x}k_{t} + \epsilon_{x,t} $$ where $\epsilon_{x,t} \sim N(0,\sigma^{2})$ The following ...
plan's user avatar
  • 21
0 votes
1 answer
39 views

Determine the seller's payment expectations for the warranty [closed]

Suppose the value of an instrument (v) is based on the number of years since purchase (t), thus $v(t)=e^{7-0.2t}$ If the tool is damaged in the first 7 years since the tool was purchased, then the ...
Winny's user avatar
  • 13
0 votes
1 answer
58 views

Determine the probability that the sample will contain two shirts B and two shirts C [closed]

A factory makes three different types of clothes: clothes A, clothes B, clothes C. The factory produces hundreds of clothes each year, with twice as many clothes B as clothes A. The number of clothes ...
Winny's user avatar
  • 13
0 votes
2 answers
95 views

Exercise 1-28 A high school lottery uses two sets of numbered balls...

Exercise 1-28 A high school lottery uses two sets of numbered balls. One set consists of ten white balls numbered 1-10 and the second set contains twenty blue balls numbered 1-20. To play, you select ...
ihavenoidea's user avatar
0 votes
1 answer
254 views

How to price a contract in this scenario?

Here is the question: You want to sell a painting you inherited from a grandparent. There is a 10% chance it is painted by a famous artist, in which case it's worth 100k. There is a 30% chance it is ...
CountDOOKU's user avatar
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0 votes
1 answer
81 views

Curtate Expectation of life aged x question

Curtate Expectation of life aged x So I did @k=0, P(60)= 0.9 @k=1, P(61)= 0.9*[0.9*(1-(0.1*1)) = 0.729 @k=2, P(62)= 0.90.729 [0.9*(1-(0.1*2)) = 0.47239 Not sure if that's right, or where to go from ...
user22222969's user avatar
3 votes
1 answer
159 views

Is it possible for interest to = principal for all and each installments?

In practice, there are two main amortization methods used by and large for retail mortgages: The classical amortization formula where all installments have the same value, however the interest % and ...
oculator's user avatar
0 votes
1 answer
102 views

Posterior Mean and Distribution of a Binomial and Beta Probability

Let $X\sim Bin(m,\theta)$ and $X = \{X_1, ..., X_n \}$ be a random sample of $X$. Assume $\Theta$ follows a beta distribution with hyperparameters $\alpha$ and $\beta$. I have proven that the ...
a9302c's user avatar
  • 349
0 votes
1 answer
234 views

How to comprehend the formula of life expectancy?

I want to know how to comprehend the $e_x$. In my view, $T_x$ is the total number of people lived at and over age $x$. Dividing $T_x$ by $l_x$ represents the concept of "times". Why dividing ...
justin's user avatar
  • 3
0 votes
1 answer
48 views

Showing equilibrium probability density function is a density function

Given the equilibrium probability density function: $$ g_Y(x)=\frac{1}{\mu}\int_x^\infty\!f_Y(y)\,\mathrm{d}y $$ check that $g_Y(x)$ is indeed a probability density function. Solution attempt: ...
Free Man's user avatar
0 votes
1 answer
48 views

Why is the graph of this solution like this?

The aggregate claim is defined as $P(S \le x) = \sum_{i=0}^n P(N=n) P(S \le x | N=n) $. The actual graph shows this instead: of $f(s) = 0.1s$ and $f(s) = 0.2-0.01s$. Why? Did they integrate it? I ...
Kid Cudi's user avatar
-1 votes
1 answer
33 views

Finding density of piece-wise with 2 random variables

Question: Let $X$ be a uniformly distributed random variable over $[0, \pi/2)$ and let $Y$ be uniformly distributed over $[0,1)$. We assume that $X$ and $Y$ are independent. Define: $$ Z = \begin{...
LemNon's user avatar
  • 69
0 votes
1 answer
190 views

Second Moment of an Increasing Term Annuity

I have an arithmetically increasing term annuity-due payable to a life aged $x$ for at most $n$ years under which the payment at time $t$ is $t+1$ for $t=0,1,\dots,n-1$. The actuarial present value ...
Gretchen's user avatar
2 votes
1 answer
111 views

What are these symbol? [closed]

Context: I'm unfamiliar with some of the symbols depicted below - specifically what appears after (Da). That right angle above the n. Also, what are the two dots above the a? Preceding text: ...
prismo's user avatar
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0 votes
1 answer
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Survival Models for Actuarial , npx

I was given: $$\mathring e_x = 12.38763065, \quad \mathring e_{x+3} = 11.65757292$$ and $${}_2 q_{x+1} = 0.1201466209, \quad q_x = 0.05917676022$$ and I was asked to find: $$\require{enclose} \...
MathStudent127's user avatar
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0 answers
65 views

Why is $n+j-i$ a even number?

Basically it says that for this binomial probability for the simple random walk, that the number of steps made in the upward direction, i.e. by $+1$ is $u = 0.5(n+j-i)$ why is that this value is ...
Kid Cudi's user avatar
2 votes
1 answer
293 views

What is the expectation of a summation where the number of terms itself is a random variable?

I have to calculate the following $$ \mathbb{E} \sum_{i=1}^{N(t)} (t-T_i) $$ where $N(t)$ is a random variable that follows a Poisson process and $T_i$ is the arrival times. Can I do the following: $$ ...
coffee-raid's user avatar
0 votes
1 answer
493 views

expected insurance value of the person

A person has a 0.16% chance of dying in the next year. A life insurance policy with a \$50,000 death benefit costs him \$125. What is the expected value of the insurance for the person? a) -$45 b) $45 ...
dona's user avatar
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1 vote
1 answer
177 views

Accumulated Value and Present Value of Perpetuities

So I've currently been learning about perpetuities in my actuarial math class and have come to a conceptual roadblock in the topic of infinite annuities. More specifically finding a present value for ...
aort01's user avatar
  • 361
0 votes
0 answers
121 views

Death Probabilities with differing assumptions

So I've just started delving into survival analysis and its varying assumptions with regards to the probabilities of dying. A question that I have stumbled upon and would like some assistance if ...
Jack123's user avatar
  • 21
1 vote
1 answer
57 views

Amortization Formula:$B_{t+1}=B_t\cdot (1+i)-R$ .Why is this established? [closed]

According to the book,The theory of interest, at page 158, I found this formula: $$B_{t+1}=B_t\cdot (1+i)-R,$$ while $B_t$ means the outstanding balance at time $t$ and $R$ means the installment ...
zoey_413's user avatar
2 votes
2 answers
148 views

Using $(I^{(m)}a)^{(m)}_{\bar{n}|}$ to solve for the present value of an annuity where payments increase monthly

I've seen this answer and I understand the methodology, but I am wondering why my original solution using a different method did not work. This is the sample problem in my study guide for Exam FM: ...
cjwcz's user avatar
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-2 votes
1 answer
65 views

Uniform Distribution question..

Suppose an uniform distribution, with 0 and 80 as parameters, that explains the ‘future time of life’ of a person with age 20. Find: $f_T(30)$ $F_T(30)$ $S_T(30)$ $μ_T(30)$ I think he meant 'a ...
gabrielleAmbrosio's user avatar
2 votes
1 answer
106 views

Deriving an upper bound for an exponential function

I am studying risk theory this semester and we are currently covering the concept of adjustment coefficients, $R$. Essentially, the result we derived in class was that, for a random claim of size $x$, ...
Ethan Mark's user avatar
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0 votes
1 answer
227 views

Financial Mathematics. Specific Nominal Discount Rate Question [closed]

The question is from Marcel P Finan's "A Basic Course in the Theory of Interest and Derivatives Markets: Preparation for the Actuarial Exam FM/2". Suppose that $100$ is deposited into a ...
ArbInv's user avatar
  • 13
3 votes
2 answers
350 views

Transformation of Cumulative Distribution Function Involving Deductibles

The question is: The cumulative distribution function for health care costs experienced by a policyholder is modeled by the function \begin{align*} F(x)=\left\{\begin{matrix} 1-\mathit{e}^{-\frac{...
numericalorange's user avatar
0 votes
1 answer
145 views

Expected Value of Insurance Pmt with Policy Limit

I'm trying to answer the following SOA Practice Question: Michael is a professional stuntman who performs dangerous motorcycle jumps at extreme sports events around the world. The annual cost of ...
jeremy909's user avatar
  • 1,028
2 votes
0 answers
72 views

Derive CDF of EPV of deferred whole life insurance

I'm studying actuarial mathmatics. Can you help me solving this question? The PV random variable of $u$-year deferred whole life insurance benefit is $Z = \begin{cases} 0, & \mbox{$T_x$ < $u$} ...
dueun's user avatar
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