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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Am I correct in this calculation of return of pension plan?
Question: Suppose I decide to contribute $\$600$ at the beginning of each month for the next $30$ years to a pension plan that offers $5\%$ compounded monthly. When the plan matures, I will receive a ...
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Exposure curve for Pareto tail
I'm currently looking into exposure curves for the excess of loss reinsurance treaty. The idea is, that for a single risk (claim) $X$, we can decompose it into $X=D+R$, a deductible part $D$ and ...
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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 ...
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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 ...
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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:
...
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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 ...
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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:
$$
Z = \begin{...
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Coding scheme for an indicator variable with respect to a categorical predictor
I'm studying statistical learning for the SRM actuarial exam - I have a question regarding the multiple linear regression (MLR) model with qualitative (categorical) predictors (explanatory variables). ...
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Second Moment of an Increasing Term Annuity
I have an arithmetically increasing term annuity-due payable to a life aged $x$ for at most $n$ years under which the payment at time $t$ is $t+1$ for $t=0,1,\dots,n-1$. The actuarial present value ...
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What happens to the skewness as $\lambda \to \infty$ (the Poisson parameter) in a compound Poisson distribution?
In context of Compound Poisson disttibution, we have $N\sim \hbox{Poisson}(\lambda)$, a random variable $W$ with iid copies $W_1$, $W_2$, $W_3$ ... also independent of $N$. So:
$$C = \sum_{j=1}^N W_j$$...
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Definition of pensionable salary times past service in pension reserve valuation
I'm working through Booth's Modern Actuarial Theory and Practice and I can't wrap my head around what is meant by $ PSSAL_x $, defined only as the pensionable salary x past service summed for active ...
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What are these symbol? [closed]
Context:
I'm unfamiliar with some of the symbols depicted below - specifically what appears after (Da). That right angle above the n. Also, what are the two dots above the a?
Preceding text:
...
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Survival Models for Actuarial , npx
I was given:
$$\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} \...
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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 ...
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How would you calculate the lifetime risk of a specific cause of death for people living today, assuming annual rates are constant?
It's been about seven years since I took differential equations, and I'm trying to wrap my head around something. This is just for my curiosity, and it's absurdly oversimplified, but I just want to ...
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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:
$$ ...
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expected insurance value of the person
A person has a 0.16% chance of dying in the next year. A life insurance policy with a \$50,000 death benefit costs him \$125. What is the expected value of the insurance for the person?
a) -$45
b) $45
...
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Accumulated Value and Present Value of Perpetuities
So I've currently been learning about perpetuities in my actuarial math class and have come to a conceptual roadblock in the topic of infinite annuities. More specifically finding a present value for ...
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Death Probabilities with differing assumptions
So I've just started delving into survival analysis and its varying assumptions with regards to the probabilities of dying.
A question that I have stumbled upon and would like some assistance if ...
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Amortization Formula：$B_{t+1}=B_t\cdot (1+i)-R$ .Why is this established? [closed]
According to the book,The theory of interest,
at page 158, I found this formula:
$$B_{t+1}=B_t\cdot (1+i)-R,$$
while $B_t$ means the outstanding balance at time $t$ and $R$ means the installment ...
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*Simple* nominal interest rates
I have just finished learning nominal interest rates in my actuarial math class and I am curious on why the nominal rate is always tied with a compounding interest model. I tried to do some proving on ...
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Similar statistical test to Kupiec test
I am considering the Kupiec test for the proportion of failures, which I am considering for evaluating a statistical model that is used for estimating the proportion of claims from non-life insurance, ...
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2
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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:
...
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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 ...
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1
answer
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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$, ...
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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 ...
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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{...
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Show that $\mathbb{P}(Y\leq y|N^* =n)=G^{*n}(y)$
I'm studying actuarial mathematics and we're having the following problem:
Consider a benefit Y on the form:
$$Y=\sum_{k=1}^N I_k X_k$$
where $N, I_1, X_1, I_2, X_2,...$ are independnt, $I_1, I_2, ...$...
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Expected Value of Insurance Pmt with Policy Limit
I'm trying to answer the following SOA Practice Question:
Michael is a professional stuntman who performs dangerous motorcycle
jumps at extreme sports events around the world. The annual cost of
...
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Derive CDF of EPV of deferred whole life insurance
I'm studying actuarial mathmatics. Can you help me solving this question?
The PV random variable of $u$-year deferred whole life insurance benefit is
$Z = \begin{cases}
0, & \mbox{$T_x$ < $u$} ...
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Show the fair payment stream $PV(0-)=c(1+i)\frac{1-\frac{1}{(1+i)^k}}{i}$
I'm struck on the following question because i have difficulties fully understanding how to apply the following formula:
$$PV(t)=\int_{t}^\infty v(t,s) b(s) ds + \sum_{n=1} \mathbb{1}_{\{t<t_n\}} v(...
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Perpetuities with differing payment and compounding periods in practice
I've been studying annuities and perpetuities recently and have tried to solve a simple problem using them.
I want to find the lump sum I'd need now so that I could withdraw $\$10$ each month forever. ...
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Difference between annual net premium and face amount.
I am doing the revision, and confusing about the face amount, net single premium and annual net premium. As I know the net single premium is the present value of the lump sum pay at time 0. How about ...
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How to prove that $\mathring{e}_x \le \mathring{e}_{x+1} + 1$ - AMLCR Exercise 2.10a [duplicate]
I'm going through Actuarial Mathematics for Life Contingent Risks, 3rd. ed Exercise 2.10.
Part a. is the following question:
Show that $\mathring{e}_x \le \mathring{e}_{x+1} + 1$.
Based on the ...
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Survival Analysis - integrating the survival probability
recently I am doing 1 question
A life aged (40) is subject to an extra risk for the next year only. Suppose the normal
probability of death is 0.003, and that the extra risk may be expressed by adding ...
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Approximating CDF for an aggregate loss model using FFT
For a class assignment, I was asked to calculate the VaR at levels 95% and 99.9% for an aggregate loss random variable $S=\sum_{i=1}^{N}{X_i}$, where $N \sim \mathrm{NB}(r=5,\beta=\frac{1}{5})$, and $...
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Optimal Reinsurance for a diffusion approximation
What is the relation between the solutions of the optimal reinsurance problem for a diffusion approximation and the optimal reinsurance problem for the classical risk model?
Apparently this result in ...
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Bayesian Prediction
It is Question #35 from this https://math.illinoisstate.edu/actuary/ExamC/ExamCMay2005.pdf.
I understand all other parts except the first line of the answer key. For the geometric distribution, $Pr(...
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"Factorizing" a sum of GLMs
I was looking a little bit into insurance mathematics, which left me puzzling about a question. I'll give you some context first:
When estimating the total claim amount for, say, a motor insurance, it ...
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Comparing two mortage rates.
Say I buy a house and take a loan for amount, $a$. The monthly interest rate the bank charges me is $r$ (banks generally quote yearly rates; divide those by $12$ to get $r$), compounded monthly and ...
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Loss Development Factors
Suppose each quarter, we calculate the change (or factor) from the previous quarter. After a period of time, you will have a list that shows the change from quarter to quarter for however many years. ...
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Force of transition and probabilities
Suppose that a model has four states: $0, 1, 2,$ and $3$, and the only possible transitions between these states is $0\rightarrow 1$, $0\rightarrow 2$, and $0\rightarrow 3$. For $t\geq 0$, $\mu_{x+t}^{...
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Life insurance transition matrix (actuarial)
Suppose that a life insurance coverage waives premium upon disability of the insured. You model
the coverage as a homogeneous Markov chain with three states: active, disabled, and dead.
The annual ...
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Calculate the percentage reduction on the variance of the claim payment
The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of d, the insurance company reduces the expected claim payment by 10%. ...
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Relationship between insurance premium and portfolio size
How would actuarial science refer to the idea that the premium should increase as number of insured entities decreases.
In other words, what is the technical term for the intuition that I would charge ...
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Need some help in basic insurance math and calculus. Had a problem in doing expansion Annuity-immediate. [closed]
I am now learning math to be a actuary in korea.
I don't understand how this expansion can happen in this equation.
I've looked on my calculus book and I couldn't find anything useful.
https://i.stack....
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Sinking Fund Method - Loan Repayment
I am currently faced with a practice problem for a financial mathematics course, and I would like to verify that I am approaching, the problem correctly (and ultimately that my solution is indeed ...
1
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1
answer
67
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Doubt in derivative computation
I quote Life Insurance Mathematics (Gerber, 1997).
Let us consider a person aged $x$ years, denoted by $(x)$. We denote his/her future lifetime by $T$ or, more explicitly, by $T(x)$. Thus $x+T$ will ...
2
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1
answer
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Finding PV of a monthly increasing annuity given annual effective interest rate
Objective: To find present value of a 10 year annuity immediate with level payments of 2 for the first year, 4 for the second year and 6 for the third year. Given annual effective interest rate of 5%.
...
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Increasing annuities monthly and convertible rates
This goes on for 10 years
If you have to find a present value of payments:
$30 being paid at the end of each month for the first year
$40 being paid at the end of each month for the Second year
$50 ...
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