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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annuities - equations of value

Chuck needs to purchase an item in $10$ years. The item costs $ \$200$ today, but its price inflates at $4 \%$ per year. To finance the purchase, Chuck deposits $ \$20$ into an account at the ...
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Perpetuity Immediate Present Value Question

A perpetuity-immediate pays $X per year. Brian receives the first n payments, Colleen receives the next n payments, and Jeff receives the remaining payments. Brian's share of the present value of ...
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76 views

simple interest rate to product of interest rates

Which simple interest rate over six years is closest to being equivalent to the following: an effective rate of discount of 3% for the first year, an effective rate of discount of 6% for the second ...
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97 views

Present Value Cash Flow Questions - Discounting

At an annual effective interest rate of i, i > 0 all the following are equal: i. the present value of 10,000 at the end of 6 years ii. the sum of the present values of 6000 at the end ...
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108 views

What is some math subject area that could widely apply to acturial science?

What is some math subject area that could widely apply to actuarial science? I know that an actuary mainly deals with stochastic processes (stochastic calculus) / probabilty, statistics, calculus.
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58 views

Trying to understand an integral algebraically and conceptually $f'(t) = \frac{Ae^t}{(0.02A+e^t)^2}$

It is defined that $f'(t) = \frac{Ae^t}{(0.02A+e^t)^2}$ is the rate of population. It is also given that the population at $t=0$ is given as 6. Our goal is to find the time $t$ where the population ...
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111 views

I am having trouble proving this ,$\int^{\infty}_0 x^n e^{-cx} dx = \frac{n!}{c^{n+1}}$.

To be more precise, $n$ is a nonnegative integer and c is a positive constant. $\int^{\infty}_0 x^n e^{-cx} dx = \frac{n!}{c^{n+1}}$ supposedly, but I cannot prove it. I've tried integration by ...
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138 views

Solving i for annuities equation without financial calculator

I would like to know if there was a way to approximate i here without a financial calculator, in the following equation: $\displaystyle -50000 + \frac{12992}{1+i} + \frac{12992}{(1+i)^2} + ⋯ + ...
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207 views

Question about the mathematics in actuarial studies

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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90 views

Variable substitution in probability

In modeling the number of claims filed by an individual under an automobile policy during a three-year period, an actuary makes the simplifying assumption that for all integers $n \ge 0$, ...
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136 views

Memorylessness and Expectation

I have a specific problem I'm working on. Let $X$ be an exponential random variable, and let $Y$ be a random variable defined by: $$ Y = \begin{cases} 0 & \text{ if } X < d \\ ...
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118 views

How to derive the variance premium formula

How to derive formula (5.5) with Taylor's expansion in the following link? The difficult part of my question is that i had never seen Var(S) appear in Taylor's expansion ...
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49 views

Calculating $(IA)_{50}$

Mortality of (50) follows De Moivre's law with w=100 and i=0.06. I find it hard to evaluate the value of $(IA)_{50}$. Any tip will be much appreciated
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1answer
20 views

Writing and expression to find the probability

Suppose (x) and (y) are independent lives. $T(x)$ and $T(y)$ denotes the future lifetimes. How do I write an expression to find out the probability that exactly one of the dies within the next 10 ...
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157 views

SOA Exam P Question: Exponential Distribution

Here is an Exam P problem as I have it. That is, it was passed down to me from someone else and I am unsure if the wording is exactly as it was originally posted. I've tried searching for this ...
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SOA Exam P Question: $P$ is a random point on the Cartesian Coordinate Plane. Find the variance of the area of a circle formed by $P$.

Caution: This problem was "passed down" to me and I think the wording was altered or lost along the way. I will post the problem as I have it and then make suggestions on what I think it should be. ...
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69 views

What is a two point support in this lemma?

What is the terminology of two point support in this lemma?
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71 views

How to calculate $\Pr[\max(X,Y)<4]$?

Suppose the joint PDF of X,Y is $f(x,y)=1/40$ and $0 < x < 5$ and $0 < y < 8$. How to calculate $\Pr[\max(X,Y)<4]$?
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97 views

What is some application of variance in actuarial science/insurance risk?

What is some application of variance in actuarial science/insurance risk? I learn that a lot of applied math book of actuarial science have variance of probability distribution frequently. Don't ...
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265 views

Closed form for Exponential Conditional Expected Value & Variance

I am wondering if there is a closed form for finding the expected value or variance for a conditional exponential distribution. For example: $$ E(X|x > a) $$ where X is exponential with mean ...
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51 views

Probability of $X^{2} > Y^{3}$ over distribution other than uniform

I was working on a problem in which $X$ and $Y$ are continuous random variables which both have the uniform distribution $U[0, 1]$. The question was: what is the probability that $X^2 > Y^3$. ...
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190 views

Bonds and Force of Interest

Studying for FM/2 and ran into this problem dealing with bonds; A 1,000 par value 3 year bond with annual coupons of 50 for the first year, 70 for the second year, and 90 for the third year is bought ...
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424 views

Finding moment generating function from piecewise CDF

I am stuck on this problem: I am given the CDF, and unless there is a shortcut I am not remembering, I need to find the PDF before I can get the moment generating function and solve the problem. ...
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216 views

Probability of being within one standard deviation of mean given only a moment generating function

I am trying to figure out how to approach this: For starters, I calculated M'(0) and M"(0), which allowed me to find the mean and variance. I got 2 for the mean and 1.6 for the variance. But now, ...
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How often does it happen that the oldest person alive dies?

Today, we are brought the sad news that Europe's oldest woman died. A little over a week ago the oldest person in the U.S. unfortunately died. Yesterday, the Netherlands' oldest man died peacefully. ...
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Increasing and decreasing annuity

Going in circles..... Find an expression for the present value of an annuity-immediate where payments start at 1, increase by 1 each period up to a payment of $n$, and then decrease by 1 each period ...
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141 views

Continuous annuity calculation

This is a problem from Marcel Finan's Exam FM/2 course. It is not homework but I am studying for the FM exam and trying to get through this. You are given $\frac{d}{dt}\bar{s}_t$ = $(1.02)^{2t}$. ...
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546 views

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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138 views

$e^{F(x,y)}$ Type Multi-variable Exponential Integrals

I am sure all you integration buffs can do this faster than I can type it. Your help with a quick explanation and solution is appreciated. $$F _{XY} = \int_0^\infty\int_0^\infty xye^{-\frac{x^2 + ...
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223 views

Stuck on solving for x in exponential to find variance

The problem seems simple: Let X be an exponential random variable such that $P(X \le 2) = 2P(X > 4)$. Find the variance of X. Easy, right? $ P(x \le 2) = 1 - e^{-2\lambda} $ and $ P(x > 4) = ...
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531 views

Procedure for Max Function with identical random variables

I am studying for the P/1 actuarial exam, and I keep encountering one type of problem which I can't seem to solve and can't seem to find any generalized explanation for how to do it in my textbooks. ...
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2k views

How to convert interest rate to discount factor

I'm studying on Kellison's Theory of Interest and I'm stuck on the exercise 20/a of the 1st chapter. If the $i=0.1$ then $d = 0.0901$ $d_5=\frac{A_5-A_4}{A_5}$ when I insert $d$ into this ...
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Where is the logic with this trinomial probability?

I am studying for Exam P to take in a few hours so I'd appreciate very much if someone got to me befoe I left. The problem says that we have three categories of risk, high, medium and low risk. The ...
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89 views

Monthly rental fee to achieve given profit on average, given probabilities of numbers of rentals

I have this problem here and I'm very unsure of how to start this. I have an idea but I'm not sure where to go from a certain point. The problem says: A video rental store is analyzing a flat fee ...
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128 views

Are waiting times always going to be exponentially distributed?

I'm studying for CAS/SOA Exam P/1 and a question I have here is: We have a portfolio of $20$ insurance policies. The number of claims per policy in a $3$-month period has a Poisson distribution ...
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569 views

Calculating a continuously varying, continuously paid annuity

I'm unsure how to calculate a continuously varying, continuously paid annuity. I'll write up my solution (which I suspect is wrong) to one, sample question, and I would greatly appreciate any ...
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94 views

Question about polynomials in actuarial science

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 resolved my difficulty with one of his questions ...
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672 views

What does it mean that the probability density function is proportional to a function?

I'm studying for SOA/CAS Exam P and I have a problem that says that $X$ is a continuous and positive random variable whose probability density function is proportional to: $$\frac{1}{(1+x)^5}$$ Where ...
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254 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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101 views

Finding principal on two loans of equal term given different rates and amounts

I'm reading through Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 on my own and am unsure how to proceed with a question ...
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120 views

Finding ratio of interest rates

I'm reading through Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 on my own and am unsure how to proceed with a question ...
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143 views

Two questions about rate of discount

I'm reading through Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 on my own and I want to make sure I understand ...
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156 views

A question in Finan's FM/2 book

Problem 6 on page 47 of Marcel B. Finan's A Basic Course in the Theory of Interest and Derivatives Markets: A Preparation for the Actuarial Exam FM/2 is: Fund A is invested at an effective annual ...
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Simple insurance firm model.

This question is a part of an larger question in actuarial mathematics. It is a model of an insurance firm with periodic stochastic outflows of $X_i$, an initial wealth of u and some income paid ...
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359 views

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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Actuarial / Modelling question - difference between RUN date and VAL date

My understanding is that RUN date is the date you do the RUN (eg in prophet / moses / igloo / whatever, the date on which you performed the calculations) Whereas VALuation date is the initial 'as at' ...
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How would I determine whether these events are independent?

I'm studying for CAS/SOA Exam 1/P and I'm stumped on this question. It says: From the set of families with two children a family is selected at random. Let $X_1=1$ if the first child of the family ...
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Which methods are used by actuaries in practice?

Recently I read a comment from an actuary that a lot of the math they studied as part of the program they never actually used. I'm not interested in becoming an actuary, but I'm interested in ...
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80 views

How is this the right answer?

In a survey of customer satisfaction, participants are asked to give a score of 1,2,3 or 4 to each of the 6 questions. If participants are instructed not to give the same numerical score to more than ...
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203 views

Derivation of formula for continues annuities

I'm currently working through a book on the Theory of Interest. Currently, I'm looking at continuous annuities, or annuities that are paid continuously. The book gives the following formula, the ...