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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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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449 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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What is the intuition behind the probability of the first even toss of a die tossed repeatedly and independently?

This question comes from the Acetex P/1 Actuarial Exam Study Guide: "An ordinary single die is tossed repeatedly until the first even number turns up. The random variable X is defined to be the ...
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349 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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63 views

Sum of Two Continuous Random Variables

Consider two independent random variables $X$ and $Y$. Let $$f_X(x) = \begin{cases} 1 − x/2, & \text{if $0\le x\le 2$} \\ 0, & \text{otherwise} \end{cases}$$.Let $$f_Y(y) = \begin{cases} ...
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What is the best way to study Probability? [closed]

Nowadays, I am studying probability. I want to be an actuary and the first exam that I have to pass is P exam. I just want to know what is the best way and if you can recommend any books please let me ...
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16 views

Functional invariance of exponential stochastic order

I am studying for an exam on stochastic order. I am struggling with a question on functional invariance of exponential order ($\leq_{\mathrm{e}}$), where for r.v.s $X$ and $Y$, $$X \leq_{\mathrm{e}} Y ...
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25 views

Increasing and then level perpetuity

this is my first actuarial question so correct any mistakes I make in formatting! We have a perpetuity with annual payments. The first payment is $ \$500$ and then payments increase by $ \$25$ each ...
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28 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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1answer
230 views

Where does this characterization of an annuity immediate come from?

I'm looking through my notes, and I don't see anywhere that an annuity immediate can be defined as $a_n = \frac{1}{a(1)} + \frac{1}{a(2)} + \cdots + \frac{1}{a(n)}$. I've always seen it as $a_n = v ...
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50 views

How should I start learning Actuarial Science? [closed]

I recently completed my under-graduate studies in pure mathematics, and have been accepted for Masters at one of the top 10 math schools. I have great interest in research and would like to continue ...
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1answer
24 views

Understanding the solution to a basic annuity problem involving an unknown interest rate

The following is the problem and the solution: Before looking at the solution, here is how I approached the problem: Let $X$ be the amount that each child receives. (i) and (ii) imply that ...
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19 views

Stochastic ordering functionally invariant

I am studying for an exam in actuarial science, where I have the following exercise: Prove that the stochastic order relation $\leq_{\mathrm{st}}$ is functionally invariant; i.e. show that $$X ...
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24 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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6answers
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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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1answer
24 views

Is it generally true that $T_x - n \mid T_x \gt n$ has the same distribution of $T_{(x+n)}$

So if $T_x$ is the random variable for future lifetime of age $x$ how can I show that "The distribution of the future lifetime, of a life aged $x$, less $n$ years given the future life time is greater ...
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1answer
33 views

Using $(Ia)_{\overline {n} \rceil i}$ from Exam FM.

I am trying to solve the following problem which I am having a bit of trouble with. Olga buys a 5-yr increasing annuity for $X$. Olga will receive $2$ at the end of the first month, $4$ at the end ...
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19 views

$\frac{L}{a_{\overline{n}\rceil i}}(n-a_{\overline{n}\rceil i})$ vs. $Li \frac{n-1}{2}$ which is larger?

I am having trouble deciding which of the expression is larger. The following is the original problem and I may not have the expression entirely correct, but I am pretty confident. A loan of $L$ ...
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2answers
144 views

Recommended Textbook to prepare for Exam P

Could anyone recommend a good textbook to prepare for the actuarial Exam P? I'm looking for a textbook that explains concepts clearly, provides detailed proofs, and gives difficult questions that ...
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83 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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29 views

Cramér Lundberg Risk Model - exponential distribution of claim sizes

I am studying the classical ruin model, which express the insurance company free surplus at time $t$ as $C_t=u+ct-\sum_{i=1}^{N(t)}Y_i $ where: $ct$ is the premium income up to time t $u$ is the ...
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4answers
67 views

Why would I divide these two equations to solve for i?

I have the following two equations representing a longer actuarial practice question. I properly set up the equations, but am stumped on how to solve them. The book says to divide the first by the ...
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2answers
53 views

Insurance claims Poisson problem derving expected value and variance

If I have that claims arrive at an insurance company according to a Poisson process $\{N(t) : t \ge 0\}$ at a rate $\lambda > 0$ and $X_i$ denotes the claim size of the $ith$ claim. I assume that ...
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24 views

Find the highest price which an investor can pay and still be certain of a yield of:

I'm having trouble understanding this example in Kellison's Theory of interest: Consider a 100 par value 4% bond with semiannual coupons callable at 109 on any coupon date starting 5 years after the ...
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Mean residual lifetime divided by odds of survival?

Is there a name for the mean residual lifetime divided by odds of survival? Does it have an intuitive meaning or interpretation? Example: $P(X=\{0,1,2,3\}) = (0.40, 0.30, 0.2, 0.1)$ ...
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Having trouble solving this Exam FM problem with zero coupon bonds.

You have two 4-year annual-coupon bonds, each one of them has a face value of 8000 and a redemption value of 8000. The coupon rate of first bond is 7% and its price is 7908.57, while the second has ...
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2answers
125 views

Geometric progression in annuity

I am working on the following problem that involves annuity which deposits form a geometric progression. Stan elects to receive his retirement benefit over $20$ years at the rate of $2,000$ per ...
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39 views

Force of mortality calculation

The force of mortality of a non smoker and smoker aged $x$ is $μx$ and $c\cdot μx$ respectively $(c>1)$. What is the probability that a non smoker lives longer than a smoker? I guess I haven't ...
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20 views

load repayment with increasing annuity

Question is : A loan of $10,000 is to be repaid in ten years by payments at the end of each year. The payments grow by 3% per year, so if the first payment is P, then the second payment is 1.03P and ...
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47 views

About annuity immediate calculation

Q1: Find $s_{12}$ if the nominal interest rate payable monthly is $5%$ per annum. What I have done is: $$i^{(12)}=0.05$$ $$1+i=(1+i^{(12)}/12)^{12}$$ which leads to $$i=0.0512$$ ...
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80 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 ...
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1answer
85 views

Exam FM question. Bonds with loss at the last moment.

I was working on the following problem and the answer that was given to me looks a little shady and I wanted someone to confirm my thoughts. As of 12/31/2005, an insurance company has a known ...
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64 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 ...
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236 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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1answer
33 views

Exam FM problem using force of interest. Calculate $P-Q$ [closed]

The foce of interest at time $t$ is given by $\delta_t=.01t$. $P$ is the present value of a 12 yr annuity due of $100$ payable annually. $Q$ is the present value of a 12 yr annuity immediate of ...
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159 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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2answers
50 views

What is the probability for a single hand of bridge to have exactly 3 Aces?

Full question from actuarial exam practice problems: The game of bridge is played by four players: north, south, east and west. Each of these players receive 13 cards. ... b) Consider a single hand ...
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198 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 \\ ...
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Need help and clarification in Exam FM problem, future value.

The problem that I am working on is the following. Jim began saving money for this retirement by making monthly deposits of 200 into a fund earning 6% interest compounded monthly. The first ...
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74 views

Quadratic Utility Function

Before this homework, "Calculate the corresponding premium for a quadratic utility function", we got to solve this example: Suppose the insurer has an exponential utility function with parameter ...
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1answer
57 views

Exam FM problem. What does this problem mean?

Danny borrows 4,000 from Genevive at an annual effective rate of interest i. He agrees to pay back 4,000 after 14 years and 5,440.32 after another 14 years. Danny repays the ...
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31 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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1answer
18 views

Algebraic representation of how values are calculated in TI BA II+?

In order to understand how the BA II+ works, I would like to know the algebraic representation of it. For example, for the problem below Present value of an annual coupon bond that pays 80 per ...
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1answer
51 views

variance unchanged under subtracting mean - application in portfolio theory

I’m working through the following passage: We will be interested in the variance of the portfolio returns given the variances of individual asset’s returns. If we have assets with returns ...
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1answer
23 views

Calculating the yield of a bond purchased at a lower price.

I am working on the following problem. A 10 year bond bearing a $7\%$ coupon rate payable semiannually is bought to yield $5\%$ semiannually. The bond is redeemable at par. If the bond is ...
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1answer
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Exam FM problem with loans. $(1.0075)^2$ or $(1.0075)^3$?

I am a bit confused about the following problem and I would like to have clarification. A loan of $12,000$ was made with annual rate of $12\%$ convertible quarterly. Smith plans to make a ...
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1answer
55 views

Exam FM problem. Bonds

the following problem is what I am working on. Suzan can buy a zero coupon bond that will pay $1000$ at the end of $12$ years and is currently selling for $624.60$. Instead she purchases a $6\%$ ...
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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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1answer
98 views

Exam FM Portofolio problem: Using Macaulay Duration

The following problem is what I am working on and I cannot solve it. Under the current market conditions Bond 1 has a price (per 100 of face amount) of $P_1=88.35$ and a Macaulay duration of ...
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43 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 ...