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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Minimum of four exponential variables

Four accidents occur independently, with each accident following an exponential distribution with a mean of 22.5. What is the expected value of the minimum of the four accidents? Attempt: ...
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Calculate the variance in the total number of claimants - SOA Exam P question

The number of accidents follows a Poisson distribution with mean $12$. Each accident generates $1,2,$ or $3$ claimants with probabilities $\frac{1}{2},\frac{1}{3},\frac{1}{6}$ respectively. Calculate ...
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Conditional Probability of Two Poisson RV's

Question: During a given year for a circus performer, let X represent the number of minor accidents, and let Y represent the number of major accidents. The joint distribution is: $f(x,y) = \Large ...
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Conditional Probability with two subsets

Question: A man plans to ship six boxes. Two of the boxes are insured, while the other four aren't. Each package that is shipped has a 10% chance of being damaged. What is the probability that: ...
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Effective Rate of Discount and Continuous Interest

I have two problems that I am having trouble with Using an interest rate of 5%, find the present value of 5000 payable in 10 years and the effective rate of discount between the 7th and 8th year. ...
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Find $_4P_x$, the probability that a person of age $x$ survives the next $4$ years given the following information?

$_{k|}q_x = 0.02(k+1)$ for $k = 0, 1, 2, 3$ and $4$, where $_{k|}q_x$ means that a person of age $x$ will survive $k$ years and dies within $1$ year. So I have that: $_4P_x= ...
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Total Present Value of Multiple Cash Flows

I understand how to calculate the total accumulated and present values of multiple cash flows over n years, but I don't quite understand how this works when m of those n years aren't included. For ...
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A 39-year annuity-immediate will pay 13 in each of the first 3 years…

A 39-year annuity-immediate will pay 13 in each of the first 3 years, 12 in each of the next 3 years, etc., until payments of 1 are made in each of the last 3 years. The present value of the payments ...
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Calculus Population Growth Problem for Actuary Exam P

I am studying for the first actuary exam, and I came across this problem in the very first section (reviewing calculus, algebra, set theory, etc.) I have many questions of my own, so please bear with ...
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Calculating the standard deviation involving a moment generating function

An actuary determines that the claim size for a certain class of accidents is a random variable, X, with moment generation function: $$M_X(t)=\frac{1}{(1-2500t)^4}.$$ Calculate the standard deviation ...
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Eggs and Cranes, What is the probability that it is a whooping crane’s nest?

The problem said: In a certain region, blue cranes are twice as common as whooping cranes. Suppose that the number of eggs laid by a blue crane is a Poisson(! = 3) random variable and the ...
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Actuarial Problem. (Policyholder).What is the probability that a new policyholder will have an accident within a year of purchasinag a policy?

Problem said: Suppose people can be divided into two classes: those who are accident-prone and those who are not. The statistics show that an accident-prone person will have an accident at some ...
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Annuity and Loan Repayment Question. Show the amount of Loan.

A loan was taken out on 1 September 1998 and was repayable by the following scheme: The first repayment was made on 1 July 1999 and was £1000. Thereafter, repayments were made on 1 November 1999, 1 ...
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Nominal Rates/Effective rate computation, confusion.

Given a nominal rate of 6% per annum. Change it to an effective rate per month. What I do is: $$(1+\frac {0.06}{12})^{12}=(1+i)^{12}$$ where $i$ is the effective interest rate per month. Now what ...
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Annuity that pays $t^2$ at time $t$ in arrears annually.

I am asked to show that such an annuity for $n$ years will be expressed as, $$\frac{2(Ia)_{\bar n|} - a_{\bar n|}-n^2u^{n+1}}{1-u}$$ where $u=\frac{1}{1+i}$ and $i$ is the annual effective ...
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Confused with “nominal” “convertible” rates; How to calculate a rate for an $n^{th}$ of a year?

The terms and how they're calculated is very unclear to me. My understanding of "nominal" is that this is a rate which isn't in unit time. i.e. $5\%$ per annum "is" in unit time (year) but $5\%$ ...
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Actuarial problem. Amortization Problem in a 25 year period at 5% [duplicate]

From The problem statment: What is the monthly payment for a $800,000 mortgage for the first 119 payments that is due in 10 years, has a 25 year amortization, at 5% interest? What is the amount ...
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Calculating FV. Where is my mistake? Is my error in excel or in using BAII?

Problem statement: What is the FV of an investment of $10,000 which pays 7% interest, compounded monthly, for five years? What is the FV is it's compounded semi-annually for 5 years? Using excel I ...
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Loan calculation; what went wrong?

I am attempting a question given as follows A loan is payable over 20 years by level installments of $\$1000$ per annum made annually in arrear. Interest is charges at $5\%$ per annum effective ...
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Annuity calculation; what is wrong with my calculation to the following question?

First off, I will be honest that I am rather confused not to much with the concepts but more of the language used in questions in finance. So I must acknowledge the possibility that I have ...
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Actuarial : “ Amortization - mortage”

What is the monthly payment for a $800,000 mortgage for the first 119 payments that is due in 10 years, has a 25 year amortization, at 5% interest? What is the amount of the 120th payment? I ...
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Present Value of an annuity payable with $n^2$ at $t=n$

Find the Present Value of an annuity payable with $n^2$ at $t=n$ , $t\in [0,n]$ What I have is: PV=present value $PV=1u+2^2u^2+3^2u^3+\cdots+n^2u^n$ I don't seem to know how to simplify it to: ...
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Have I understood the question properly? Annuities in Actuarial math

I am wondering if I have interpreted the language correctly in the following question The force of interest at time $t$ is given by $\delta(t) = 0.05-0.005t$ for $\leq t < 5$ and ...
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MGF, Calculate the standard deviation of $X + Y$ . Method.

Two claimants place calls simultaneously to an insurers claims call center. The times X and Y , in minutes, that elapse before the respective claimants get to speak with call center ...
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Expected number of people who are hospitalized, given the total loss due to hospitalization

NOTE: I'm currently a high school senior planning to take the Actuary P-Exam next year. I've posted one question from a past P-Exam down below that I had trouble with. If anyone has any additional ...
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Probability of earnings from lottery

Question: A city's lottery works in the following way: An individual selects 6 numbers from the first 30 numbers. The city then selects 6 numbers from the first 30 numbers. If the individual selects ...
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Probability of waiting time

Question: At a railroad junction, a car and a truck arrive between 7:15 and 7:30. A train stops the traffic for five minutes from 7:20. What is the probability that the car and truck waited for ...
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Calculate the variance of the joint probability. {Actuarial Problem, Exam P. Help!}

I'm studing for exam P, and I found the following problem: I don't know how to solve it, and I will really apreciate if someone know how to do it, and learn from it. Thanks comunity again.
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Determination of Bond prices

Two 1000 dollar face value bonds are both redeemable at par, with the first having a redemption date 3 years prior to the redemption date of the second. Both are bought to yield 11 percent convertible ...
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Question concerning Lundberg-Exponent

My question is based on the beginning of Chapter 8.3.2 in the book "Modelling Extremal Events" by Embrechts,Klüppelberg and Mikosch. We consider a Cramer-Lundberg-Model and assume that the conditions ...
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Effective Rate of Interest and Inflation Problem

I'm trying to solve the following question from a past SOA FM exam, but there are no posted solutions so I'm not sure if I'm headed in the right direction. John deposits 10, 000 in a saving ...
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Deductible and Policy limit [closed]

I'm trying to figure out the solution to the following problem. I was working with the Adapt program for the p exam but I can't find the solution anywhere. Problem: Consider an insurance policy ...
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Finding pdf of X+Y - Finan 42.2

I am looking through Marcel Finan's 'A Probability Course for the Actuaries' and I am stuck on problem 42.2. It is as follows: Let X be an exponential random variable with parameter $\lambda$ and ...
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Annuity-Immediate Problem with Varying Payment (ASM FM Study Manual 10th Edition, Practice Exam 2 P.679 Q1)

The question asks: 'A 35-year annuity immediate pays $1.05^{35}$ in the first year, $1.05^{34}$ in the second year, etc., until 1.05 is paid in the 35th year. The PV of this annuity at 5% effective is ...
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Evaluating differing perpetuity payments

A perpetuity pays 1900 dollars on January 1 of 1980, 1982, 1984, ..., and pays X dollars on January 1 of 1981, 1983, 1985, ... If the present value on January 1, 1975 is 26500 dollars, and the ...
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Theory of Interest - Going from discount rate - annual effective and nominal to force of interest

[![Force of Interest & Discount Rate][1]][1] Hi there guys, I'm new to Theory of Interest and I might have started studying a bit too late, but I really need to just grasp this before the end of ...
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Proove the approximation for risk premium

Exercise comes from Bowers' "Actuarial Mathematics". I'm self-studying and have virtually no clue how to approach it. We make no assumptions about either utility function or distribution of X. Let ...
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Actuarial and Investment companies problem.

I am trying to study for the first actuarial exam, and I'm stuck with this problem. I tried to use Venn Diagrams to solve it, but I cannot reach the correct solution. The book said $135$ people have ...
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Simple discounting rate question and finding unknown time:

The thing is to solve for n. This is how far I got: FV=PV(1-d)^-t So, 1000(1-0.05)^-10 + 2000(1-0.05)^-5= 1000(1-0.1)^-(10-n)+2000(1-0.1)^-(10-2n) => 2584.71=1000(0.9)^-(10-n)+2000(0.9)^-(10-2n) ...
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Is there a more rigorous way to show the expected price of a stock?

I am studying for the Society of Actuaries' Financial Math exam and I found the solution to this practice problem to be not convincing enough: Is there a more mathematical / rigorous way of showing ...
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$E(Y)$ from $f(x,y)$ where $0<x<y<∞$

A device contains two circuits. The second circuit is a backup for the first, so the second is used only when the first has failed. The device fails when and only when the second circuit fails. Let ...
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Clarification on the set-up of this interest theory problem

The following problem and solution is taken from an actuarial exam (financial math) study manual: I would set this problem up completely different. I think it would be necessary to consider the ...
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Calculating conditional probability of discrete uniform r.v.

X is a discrete uniform random variable on $\{a, a+1, a+2, ... , b\}$ with mean 7 and variance 4. Find $Pr[X \leq 6| X > 4]$ I'm not familiar with the discrete uniform distribution. I was ...
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Determining bounds for change sum of continuous r.v.'s

I'm trying to understand how to determine the bounds when computing the sum of continuous random variables. Here is a sample question: X and Y have the following joint pdf: $f_{X,Y}(x,y) = 4xy, 0 ...
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Explicit formula for recursive sequence.

Consider the series defined by $$P_n=(P_{n-1}-a)b\ .$$ $$P_0=c$$ Basically the number sequence $P_n$ represents the current Principal balance of a debt and constants $a$ and $b$ are constants or ...
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Clarification on a collared stock being equivalent to a bull spread?

The following is a question on financial math from the financial math actuarial exam: Earlier in the manual, the author stated that a collared stock is equivalent to a bull spread. Therefore, in ...
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Order Statistics for IIND Variables

Given 3 exponential random variables with different means (for example 1, 2, 3), how can one calculate E(X) for MIN(X1,X2,X3)?
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Conditional expectation of insurance payment

I'm trying to solve the following problem: An insurance policy is written to cover a loss, X, where X has a uniform distribution on (0, 1000). At what level must a deductible be set in order for the ...
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Univariate transformation/change of variable with integration

I'm trying to solve the following problem: My thought process is to do a univariate change of variable, so that $Y = \frac{V}{100,000}$. Then we have $\frac{\delta}{\delta v}[\frac{V}{100,000}] = ...
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What is the intuition behind the probability of the first even toss of a die tossed repeatedly and independently?

This question comes from the Acetex P/1 Actuarial Exam Study Guide: "An ordinary single die is tossed repeatedly until the first even number turns up. The random variable X is defined to be the ...