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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Conditional expected mean

I’m stucked on the following problem, and i just can’t seem to grasp what to. $$Let \: X_{1}, X_{2},… \: be\: i.i.d. \: random \: variable, \: and \: assume\: that: \\ \mathbb{P}(X_{1} =1)=1-\mathbb{P}...
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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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2 answers
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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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1 vote
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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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Second moment whole life insurance monthly

So, I got the assignment to compute the present value and the variance of a whole life annuity monthly due $\ddot{a}^{(12)}_x$. So, I already used different identities to relate the whole life annuity ...
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2 votes
1 answer
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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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Accumulated value of fund for defined contribution plan

Consider a defined contribution plan, we have: (i) A person enters the plan at age $30$ and earned $50,000$ in the previous year. (ii) The salary scale is $S_y = 1.025^y$. (iii) Contributions are $10$%...
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Insurance model transition intensity proof

Consider an insurance-with-withdrawals model. In this model, we have three states $0$ (alive), $1$ (Dead) and $2$ (Withdrawal), with one transition from $0$ to $1$ (with force of transition/transition ...
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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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Iterating over time with recursion

List of variables and formulas I'm looking for a way to simplify the value of the constant $A$. I have a data set that has explicitly identified values for the constant $r$, the constant $T$, $AAL$ in ...
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1 answer
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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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1 vote
1 answer
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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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3 votes
2 answers
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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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Distribution of $X(T)$ where $T$ is the time of ruin.

Given a ordinary renewal process $$ X\left(t\right)=u+ct-\sum^{N\left(t\right)}_{i=1}{Y_i},\ \ \ t\geq{0} $$ where $u\geq 0$ is the initial surplus, $c$ represents the insurer’s premium income per ...
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2 answers
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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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2 answers
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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 ...
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1 vote
1 answer
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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 ...
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2 votes
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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Payments structure in an increasing-payments perpetuity

I quote Life Insurance Mathematics (Gerber, 1997). A certain type of perpetuities with increasing payments is defined by two parameters, $m$ (the number of payments per year) and $q$ (the number of ...
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1 answer
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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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1 vote
1 answer
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Calculate present value at t=0 for a set C given the present value of all cash flow is equal for all sets at t=1

(*Take 𝑡 = 0 be the current time in this problem) We consider the following three sets of cashflows: Set A: It pays 200, 300 and 500 ( dollars) at time 1, time 2 and time 3 respectively. Set B: It ...
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2 votes
1 answer
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Understanding whole life Assurance

I asked suggestion for good book here. After reading the suggested book and others I want to ensure what I have learned is correct and want to clear some of my doubt. Whole life Annuity-due The ...
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0 answers
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Analyzing Reinsurance Effectiveness

Given data for a product (P&C) and a Reinsurance Treaty, how can I achieve the following: Determine claim distributions for frequency, severity and aggregate claims Perform Goodness-of-fit tests ...
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1 vote
1 answer
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Actuarial and Financial mathematics book suggestion

Recently, I was introduced with one of my course named "Introduction to Actuarial and Financial Mathematics". The book I followed was too dry and it's not contain a detail explanation of ...
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1 vote
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Evaluating the Present Value of a Perpetuity where each term is square [duplicate]

I'm currently studying for SOA Exam FM using Harold Cherry's study manual (11th edition). In practice exam 5, there is a question (#22) that asks the following: Determine the modified duration of a ...
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Compound Interest Differential Equation Question

I'm positive someone else has asked a nearly identical problem in the past but I can't find an example of this specific variation. Let's say initially you invest some principal amount ($I_P$) into a ...
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Accumulation functions and investments

Suppose we invest $\$50$ at time 0.5 and $\$200$ at time 1.5; the accumulation function that applies on the interval $[0,1]$ is $a(t) = 1+t^2$, and simple interest with $i = 5%$ applies on the ...
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1 vote
2 answers
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Integral of a sum under condition

I'm struggling with the following integral, would anyone have an idea on how to approach it? I solved it for N = 2 but the generalization to any positive integer is more difficult. $$\int_{(\omega_1, \...
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1 answer
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Question about the mean excess loss function.

I'm new to this risk thing. I am trying to obtain the mean excess loss function evaluated at a point for the following Pareto distribution: $F(x)=1-(\frac{\theta}{x})^{\alpha}$ The excess loss ...
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0 votes
1 answer
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Find expressions for the single sums $X$ with simple annuities

Find expressions for the single sums $X$ equivalent to the set of seven payments of Fig. 5-21 at times $(a)1,(b)5,(c)12$ and $(d)12$, assuming a rate $i$ per period. Solution: $(a)$ At $1$, $X$ is ...
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Calculate discounted cash flows for a fixed net present value with some restrictions

I was wondering what my maximal investment in year zero can be, given certain ratio that is pre-specified between cash flows, for a fixed net present value: This solution was offered by Bob: If you ...
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0 votes
1 answer
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expected value of geometric distribution

The following question has me extremely confused: At a certain university, registrations for courses have to be made over the telephone. There are so many calls that $90$% of the time you will get a ...
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2 answers
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Negative Binomial and Geometric Distributions

So this is the problem I'm dealing with. An actuary has determined that the number of claims per month can take any number 0, 1, 2, 3,... and follows a negative binomial distribution with mean 3 and ...
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Redington immunization with a bond

A company pays two liabilities: $100$ in $2$ years and $50$ in $4$ years. The company is attempting to protect itself through immunization by purchasing a $3$-year, par-valued bond with annual coupons,...
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-1 votes
1 answer
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Annuity & Perpetuity problem

Suppose Joe has been paying $600$ from his monthly salary at the end of every month for the past $n$ years. After $n$ years of payments, he retires having purchased a perpetuity-due plan that begins ...
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0 votes
1 answer
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Accumulated Value of Two investments

Suppose the force of interest is: $$F(t) = \frac{0.02 + 0.01t}{1 + 0.02t + 0.005t^2}$$ , where $t$ is the number of years beginning March 31, 2001. An investment of $100$ is made on March 31, 2002, ...
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2 votes
1 answer
41 views

Approximating effective interest rate on a bond

If you have a bond with the following properties: Issued at price $X$; Redeemed at price $Y$; Has a coupon rate of $p$ ; Has a term of $n$ years; then I am told, in my accounting studies, that the ...
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1 vote
1 answer
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Calculating the present value of bond when face value is not equal to value at maturity

I need to calculate PV of the following bond: Face value is $1000$. It pays $50$ every $6$ months. Annual rate is $8\%$. Value at maturity is $1050$. The bond lasts $20$ years. I know how to ...
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  • 460
0 votes
1 answer
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Difference between yield rate and interest rate (solution verification)

An insurer enters into a four-year contract today. The contract requires the insured to deposit $500$ into a fund that earns an annual effective rate of $5.0\%$, and from which all claims will be paid....
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1 vote
1 answer
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Bonds, Exact Matching Payments, Bank Payments

A bank accepts a 20,000 deposit from a customer on which it guarantees to pay an annual effective interest rate of 10% for two years. The customer needs to withdraw half of the accumulated value at ...
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1 vote
1 answer
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Finding the present value of a continuously varying perpertuity

A perpetuity provides for continuous payments. The annual rate of payment at time $t$ is $1$ if $0\le t<10$ and $(1.03)^{t-10}$ if $t\ge 10$. Using an annual effective interest rate of $6\%$, the ...
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0 votes
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Modified and Macaulay Duration

Repost but with a proposed solution.* I am practicing for an upcoming examination and came across the following problem: You are given the following information about an asset: (1) The present value ...
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80 views

Swap rate with coupons

Suppose you were given the following information about the prices of zero-bond coupons, all with a maturity of $1$: The price of a $1$ year zero-coupon bond is $0.943$. The price of a $2$ year zero-...
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Macaulay, Modified Duration

I am practicing for an upcoming examination and came across the following problem: You are given the following information about an asset: (1) The present value of the cash flows at an annual ...
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