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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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. ...
7
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1answer
984 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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2answers
139 views

Prove that $\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}$ is a strictly decreasing function.

This is part of an actuarial science problem. Unfortunately, the official solution of this problem takes the derivative of $$\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}\text{, } \quad x \geq 0\text{.}$$ and ...
5
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3answers
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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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4answers
28k views

Are the actuarial exams hard? [closed]

I heard that they are difficult. Is this true? Are they like the qualifying exams in grad school? For example, is the probability exam and the financial math exam comparable to qualifying exams (e.g. ...
4
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2answers
63 views

Probability question : meaning of the sentence

The following is the problem I am working on. The probability of a passing car being an import is defined as $p(i)=1/4$ and the probability of it being domestic is $p(d)=3/4$. Find the probability ...
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3answers
154 views

Probability problem - fake and real diamonds -

A box contains 35 gems, of which 10 are real, 25 are fake. Gems are randomly taken out of the box, one at a time without replacement. What is the probability that exactly 2 fakes are selected ...
4
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2answers
216 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 \\ ...
4
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1answer
96 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 ...
4
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1answer
240 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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1answer
82 views

What is the pdf of $Z=X/\max(X,Y)$ with $X,Y$ exponentials of lambda parameter?

Given $X,Y$ 2 independent r.v.'s both distributed as $\exp(λ)$, what is the pdf of $Z=X/\max(X,Y)$?
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3answers
3k views

Masters in Actuarial Science

I am applying to a grad school for the Masters in Actuarial Science. Now i am getting cold feet. I do love math, i was always good in math (not excellent or a genius). Did all adv. calculus classes ...
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2answers
238 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 ...
3
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1answer
65 views

Isolating for i

So this might seem a bit fundamental, but in financial math the following equation gives you the price for a bond $$ P = C \frac {1-(1+i)^{-n}} {i} + B(1+i)^{-n} $$ where $P$ is the price of the ...
3
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1answer
146 views

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. ...
3
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2answers
82 views

Zero-coupon vs. $10\%$ coupon problem

I am working on Bonds and I am having trouble solving this problem. A zero-coupon bond pays no coupons and only pays a redemption amount at the time the bond matures. Greta can buy a zero-coupon ...
3
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2answers
215 views

Expressing a summation using matrix algebra

Consider the $r \times n$ matrix $$\begin{pmatrix} X_{11} & X_{12} & \cdots & X_{1n} \\ X_{21} & X_{22} & \cdots & X_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ ...
3
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1answer
4k views

Confused about Effective Rate of Discount- Theory of Interest

I'm currently reading Kellison's book, The Theory of Interest. I've reached the chapter on Effective Rate of Discount and it's somewhat confusing. The book explains it as a loan where interest is paid ...
3
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4answers
156 views

Beta distribution for exam P?

I had a quick question regarding the beta distribution and exam P for actuaries. From the recommended books that I have seem, beta distribution does not seem like it is likely to show up on the P ...
3
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1answer
232 views

Text on Probability Theory applied to Actuarial Science

I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life ...
3
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1answer
359 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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3answers
187 views

Expected value and life

Let $e_{x} = \int_{0}^{\infty} p_{x}(t) \ dt$ where $p_{x}(t)$ is the probability that a person aged $x$ will survive at least $t$ more years. Why is $e_{x} \leq e_{x+1}+1$? We know that $e_{x} \geq ...
2
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3answers
39 views

Deriving a property regarding variance.

I am studying for the P-exam for actuaries and I've encountered a property that said, $Var(x\pm y)=Var(x)+Var(y)$ I come from a math major and it has been years since I was taught statistics or ...
2
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4answers
294 views

Remembering the mean and variance of Poisson vs Exponential

I am having the P-Exam for actuaries on September and I am trying to work on my details. One of the troubles that I have is to remember the difference between Poisson and Exponential distribution. I ...
2
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1answer
83 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 ...
2
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1answer
72 views

A more theoretical than computational interest theory problem involving amortization

I am working on the following problem: A borrower has a mortgage that calls for level annual payments of 1 at the end of each year for 20 years. At the time of the seventh regular payment an ...
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2answers
32 views

Methods of solving this exam FM problem with geometric-investments.

The problem I am working on is as follows. Matthew makes a series of payments at the beginning of each year for $20$ years. The first payment is $100$. Each subsequent payment through the tenth ...
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2answers
41 views

Identity of $I_t$ under annuity with principal $1$

I am trying to prove an identity and quite not get there. The following is the premise. One deposits $\$1$ at time $t=1,2, \cdots ,n$. evenly spaced. The effective interest per payment is $i$. ...
2
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1answer
530 views

Darth Vader Rule: what is the reason for its name, and a formal proof?

I often hear the term "Darth Vader Rule" when calculating the expected value using the survival function and taking the integral where it is defined. I am not quite sure why it is called that (is it ...
2
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1answer
300 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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1answer
1k 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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2answers
91 views

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 ...
2
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1answer
716 views

Converting an Annuity due to Annuity immediate

I'm working on the following problem at the moment while preparing for an exam. Find the present value of payments of 200 every six months starting immediately and continuing through four years from ...
2
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1answer
19 views

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 ...
2
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1answer
30 views

Can any function of the second moment of a random variable be recovered from its quantile function?

Summary of question It is known that the expected value of a random variable can be obtained from integrating its survival function. This is easily restated in terms of the quantile function as: $$ ...
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2answers
202 views

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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1answer
71 views

Annuity problem, calculating the accumulated value.

the following is the problem I am trying to work on. Kathryn deposits 100 into an account at the beggining of each 4 year period for 40 years. The account credits interest at an effective annual ...
2
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1answer
70 views

Intuitive understanging of re-investment.

There was an interesting problem that I would like to have some input from people who knows a bit of finance. The following is the situation. Smith loans $\$10,000$ for $i=5\%$ for $10$ years. ...
2
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1answer
54 views

Notation in Actuarial text

I am not very knowledgeable in Actuarial Sciences, but I was looking at a text, and found a notation, which I cannot find any references to. What does: $_{2|2}q_{1} = 0.288$ mean? I don't think it ...
2
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2answers
638 views

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 ...
2
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1answer
249 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 $n \ge 0$, ...
2
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1answer
273 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 ...
2
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1answer
4k 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 ...
2
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0answers
62 views

Simplifying $\sum_{t=1}^{n}t^2v^t$ using actuarial notation.

In financial mathematics involving immunization, I encounter situations where I am trying to calculate $$(A) \quad v+4v^2+9v^3+ \cdots +n^2v^n=\sum_{t=1}^{n}t^2v^t $$ where $v$ is the present value ...
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0answers
46 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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2answers
80 views

Are there other accumulation functions that holds $a(n-t)={a(n) \over a(t)}$?

This might be a beginner's question regarding accumulation methods and their functions, but so far I have learned that compound interest satisfy $$a(n-t)={a(n) \over a(t)}$$ Which allows nice ...
2
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2answers
145 views

How to remember the Jacobian

the following is the problem that I was working on. Let $f(x,y)=8xy$ for $0<x<y<1$. What is the joint density function of $W={X \over Y}$ and $Z=Y$? Since I am self studying this ...
2
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0answers
75 views

“Taxes and Option Prices” (question about Derivatives Markets by McDonald)

Thanks in advance for any help, and please tell me if there's anything I can do to make things clearer. I am having trouble understanding appendix 10.A to Derivatives Markets by Robert L. McDonald. ...
2
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0answers
45 views

Reinvesting the interest (generalized version)

If I deposit \$1 at $t=0$ into an account which credits interest at the end of each year at a force of interest $\delta_t$ (assume it's integrable.) Then, if I reinvest the interest at an annual ...
2
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0answers
184 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 ...