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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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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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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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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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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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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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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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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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 ...
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124 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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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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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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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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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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34 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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50 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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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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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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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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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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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\;\; ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...