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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Given CDF, F(X), compute E(F(X)) [closed]

I am given Pr(x): 0.1 , x = −1.9 0.1 , x = −0.1 0.3 , x = 2 0.1 , x = 3 0.4 , x = 4 0 , otherwise and F(x): 0 , x < −1.9 0.1 , −1.9 ≤ x < −0.1 0.2 , −0.1 ≤ x < 2 0.5 , 2 ≤ x < 3 0.6 ...
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26 views

Why these two distributions are different when I calculated them?

Two insurers provide bids on an insurance policy to a large company. The bids must be between 2000 and 2200 . The company decides to accept the lower bid if the two bids differ by 20 or more. ...
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37 views

What is the meaning of this notation? $Z: (0,1)$

I understand that it is a mapping onto all real numbers and the piecewise part but not the $Z: (0,1)$ part. Image of Problem Here
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835 views

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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481 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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54 views

Bayes Theorem problem, from Finan #9.4: $P(A\mid B ∩ C)$

The Problem: You are given $\Pr(A) = 2/5, \Pr(A ∪ B) = 3/5, \Pr(B\mid A) = 1/4, \Pr(C\mid B) = 1/3,$ and $\Pr(C\mid A ∩ B) = 1/2$. Find $\Pr(A\mid B ∩ C)$. My work: I know that $\Pr(A\mid B) \Pr(B)...
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51 views

November 2001 P/1 Problem No 35

Problem: Auto Claim amounts, in thousands, are modeled by a random variable with density function $f_x(x)=xe^{-x}$ for $x\ge 0$. The company expects to pay 100 claims if there is no deductible. How ...
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61 views

De Moivre's population recursive formula

For a de Moivre's (i mean, that follows de Moivre's law) population with unknown parameter $\omega$, knowing $\dot{e}_{x+1}$ is given. From the recursive formula find $\dot{e}_x$ I am not certain ...
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42 views

Financial Mathematics Question: Payable

I am trying to solve the following problem and I am not quite sure what it means. A loan of 100,000 is payable over five years with monthly payments of 60,000 commencing one month after the ...
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39 views

Find the price of the bond using its book value

A n year 1000 par-value bond with 8% annual coupons has an annual effective yieled of i, 1+i >0 . The book value of the bond at the end of the third year is 990.92 and the book value of the bond at ...
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52 views

Actuarial Mathematics proof error?

Is there something wrong with this proof or is it just me? Did they forget the -ve sign or does it cancel somehow? (From the second to last to last line)
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78 views

Will conquering the two basic actuary exams, p and fm, be too difficult to do alone? [closed]

I'm currently a PhD student in mathematics at a decent sized graduate school, but I've been questioning my desire to continue on and finish my doctorate after I achieve my master's, which will occur ...
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1answer
46 views

Probability Question [Hypergeometric Distribution]

I was solving the below problem, and I had a few questions: An urn contains five red marbles and three blue marbles. Four marbles are chosen without replacement from the urn and their colors are ...
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2answers
63 views

Calculating accumulated values for annuities

Having difficulty with this problem Find the accumulated value at end of four years of an investment fund in which 100 is deposited at the beginning of each quarter for the first two years and ...
3
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1answer
135 views

Macaulay duration for a coupon bond. Proof

I am working on showing the following. There is a coupon bond redeemable at par with annual coupon rate $r$ per year. The yield to maturity is $i$. The total number of coupons is $n$. Show ...
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72 views

Exact match of assets and liabilities.

Liabilities of $1$ each are due at then ends of periods $1$ and $2$. There are three securities available to produce asset income to cover these liabilities, as follows: (i) A bond due at the end of ...
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79 views

Probability of a car accident [closed]

Recently I was visited by a car insurance policy expert to renew a contract and I was told that I had a (fatal/non-fatal) car accident risk of %35 so my policy would differ. He also provided me with ...
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2answers
84 views

How many interest periods if you always pay double the interest [closed]

A loan of $12,500$ is made at an effective interest rate of $8.5\%$. Payments are made at the end of each interest period. Each payment equals twice the interest due until the borrower pays off the ...
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1answer
76 views

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 use ...
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1answer
37 views

Differential of Fisher-Weil Bond Price using discrete-time compunding

I would like to ask you if my calculations are correct regarding the differential of the price of a bond using a sequence of spot rates as discount factors. \begin{equation} P(i^S ) = \sum_{t=1}^{T} ...
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1answer
60 views

A level continuously paying annuity pay 1500 each month. Find the present value

Here is the full question: A level continuously paying annuty pays $\$ 1500$ each month for eight year. The force of interest is $\delta(t)=\frac{2t}{t^2+5}$ where time is measured in years. Find the ...
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1answer
45 views

Samantha age 40, buys a 20-year certain-and-life annuity due with a single premium. Find the reserve 5 years after issue.

Samantha age 40, buys a 20-year certain-and-life annuity due with a single premium. The annuity makes annual payments of $2,000$. If $i = 6\%$ and you are given the following table, find the ...
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1answer
85 views

Annuity formula proof $\frac{a_{\overline{n}|}}{a_{\overline{k}|}}$

I have the actuarial exam FM in 2 days and there is one more thing that I would like to understand. I cam across a problem having to do with identities and this is the following. A perpetuity ...
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1answer
36 views

Calculating the loan from amortization.

I was not quite sure what the question was asking and I would like to have some input. a), Is it asking us to actually calculate knowing what each principal paid? or b), Working from almost complete ...
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2answers
65 views

Simply this formula

Is there a way to write $\sum_{n=a}^b (Q+P(n-1)) v^n$ in terms of $\sum_{n=a}^b v^n$? So far I've gotten: (Q-P)$\sum_{n=a}^b v^n$ + P$\sum_{n=a}^b nv^n$. *I know the last term can be rewritten as $...
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42 views

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, $1000(1-0.05)^{-10} + 2000(1-0.05)^{-5}= 1000(1-0.1)^{-(10-n)}+2000(1-0.1)^{-(10-2n)}$ $\Longrightarrow\;\; 2584.71=1000(0....
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1answer
29 views

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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0answers
30 views

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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43 views

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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40 views

Life contingency: Amy Santiago buys 1,500,000 dollars of 5-year-term continuous life insurance at age 27. [closed]

I am working on a question in Life Contingencies, and for the life of me, I cannot get the appropriate formula. I am tossing and turning in between two. Any help would be great. The question is: ...
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33 views

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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1answer
84 views

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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29 views

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 $\delta(t)=0....
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1answer
55 views

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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2answers
97 views

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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1answer
30 views

given a discount rate find annual effective rate of interest?

I am reviewing my first midterm and my professor did not provide the solutions to the problems on the midterm when she returned the exam. So I need help with one of the problems I got wrong and am ...
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51 views

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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2answers
41 views

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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1answer
32 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: $min(x_1,...
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1answer
852 views

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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2answers
37 views

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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1answer
37 views

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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67 views

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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1answer
103 views

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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30 views

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= (P_x)(P_{x+1})(P_{x+2})(...
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2answers
32 views

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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1answer
69 views

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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1answer
63 views

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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332 views

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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56 views

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 ...