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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Finding percentile given distance between two percentiles.

The sales for a company are normally distributed with mean $\mu$ and variance $\sigma^2$. The difference between the $90$th and $40$th percentile is $500$. The $70$th percentile is $1700$. What is the ...
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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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23 views

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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Difference between conditional and intersection in probability.

I am having hard time figuring out if it is a conditional probability or an "and" probability under the following types of problems. When a student is absent, the probability of the student being ...
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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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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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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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Redington vs full immunization?

I understand that the present values and duration of liabilities and assets are required to be equal to each other under both cases, and furthermore for Redington immunization the convexity must also ...
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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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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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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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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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439 views

Two questions on nominal rates of interest

I'm reading Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 and have difficulty with two of his questions. Problem 9.6 ...
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Claim from an Actuarial Textbook: limits imply the existence of mean and variance

This is from Actuarial Mathematics for Life Contingent Risks, 2nd ed., by Dickson et al. Some definitions (not directly from the book): Definitions/Notation. $T_x$ is defined to be the future ...
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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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What is the probability that at least 2 professors (out of 100) pick the same course (out of 200)?

Suppose each of 100 professors in a large mathematics department picks at random one of 200 courses. What is the probability that at least two professors pick the same course? The answer given in 1 - ...
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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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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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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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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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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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How to find limits of integration on a convolution of CRVs

In finding the convolution of two independent and continuous random variables, I am struggling with limits of integration. I cannot seem to figure out over what intervals the probability density ...
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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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Question about the mathematics in actuarial studies [closed]

I tried Google but there isn't much information on this and I would really like some insights into actuarial studies, the mathematics involved and how it compars to the mathematics in a bachelor of ...
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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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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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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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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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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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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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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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SOA/CAS Exam P Question (from previous exam (Nov. '09)): Finding percentiles

So this question comes from SOA/CAS Exam P of November 2009, I'm not sure why the solution is the way it is. The question says this: An insurance company sells an auto insurance policy that ...
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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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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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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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Force of interest problem from Exam FM

Tawny makes a deposit into a bank account which credits interest at a nominal interest rate of $10\%$ per annum, convertible semiannually. At the same time, Fabio deposits $1000$ into a different ...