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Questions tagged [actuarial-science]

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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Actuarial practice exam: Find number of exponential data values above threshold given MLE.

I was going thorough an actuarial exam and came across a problem that I can't figure out. Here is the problem as stated on the practice exam: You are given: $\bullet$ An insurance product with ...
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Do 50% of people die on either side of an average life expectancy?

I am no statistician so this may come across as very naive, but I've been trying to get my head around how to interpret life expectancy data. Say you have a population of $n$ and a range of possible ...
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22 views

Prove a general random walk is not stationary

Here is my attempt. $X_t$ is a general random walk is when we have a sequence of independent and identically distributed random variables $Y_1,Y_2,...$ such that $X_0=0,$ $X_1=Y_1$, $X_2=Y_1+Y_2$ and ...
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Widow annuity- change of mortality

I have a task: Take widow annuity for her (x), which pay 1000 per year from the death of him (y). Premium will be paying as continuous annuity until the time of first death with intensity $P$. We ...
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27 views

Sum of compound Poisson processes with dependent counting process

I would like to ask about some literature depending compound process. I want to find some limit theorem for process: $$Z(t)=\sum_{i=1}^{N_{1}(t)}X_{i}-\sum_{j=1}^{N_{2}(t)}Y_{j}, $$ where $N_{1}, N_{2}...
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Construct portfolio given certain payoff function

My question is similar to https://quant.stackexchange.com/questions/37419/replicate-a-portfolio-with-given-payoff but I am not quite sure how to apply this to my problem. A portfolio of European ...
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23 views

Distributions without finite moments

A statistician colleague of mine posed a question to me regarding certain distributions used in loss models. Naturally occurring distributions, such as inverse Pareto, do not have finite moments. But ...
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Black Scholes Framework

Assume the Black-Scholes framework. You are given: •A stock S pays no dividend •The continuously compounded risk-free interest rate is 8.5% •A contingent claim Y pays $\frac{S(4)}{S(2)...
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How can the Gamma distribution be derived from the Negative Binomial Distribution?

I have seen the exponential distribution derived from the geometric distribution here: Deriving exponential distribution from geometric . I would like to use a similar approach to derive the gamma ...
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Barrier Option pricing and valuation

Good day, A reverse knock-out barrier call option expires worthless if the asset price ever goes above a given barrier level. Calculate the value of this barrier option struck at $K = 3$ with ...
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78 views

Hedging Strategy for American Option

Good day, I was asked to devise a hedging strategy for an American Option given the following claims. Note, $r=0$ and the underlying stock pays a dividend of $1$ at time $t=1.5$ \begin{...
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Why is nominal interest defined the way it is?

So if nominal interest is 12% compounded monthly, it is actually 1% compounded each month. It is not 12% effective year, though it is close (It is 12.7%) So why don't we/they say 1% compounded monthly?...
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Beta Distribution Question

X is a random variable for losses. X follows a beta distribution with θ = 1000, a = 2, and b = 1. Calculate TVaR0.90(X) Okay, so I know for this problem I need to know the Value at Risk (VaR) at the ...
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33 views

Finding PV with multiple interest rates…

I've looked for a question that hits on something similar without success. The problem is this: at time 10, an asset pays 1000. Then the teacher gave us a bunch of random interest rates for times 0 ...
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1answer
31 views

Question about answer in 13th edition FM manual for example of a continuous annuity with varying force of interest

Am I just not seeing what happened to the negative sign? I've written the question below, the part I was confused about was between the first and second step in the second equation. Example 2 A 10-...
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1answer
28 views

Using fission method with a series like v^5 + v^12 + v^19+…

I am studying for exam FM and I know we can use the "fission" method to represent a summation like $v^5 + v^{10} + ... + v^{100}$ as $\frac{ \require{enclose}a_{\enclose{actuarial}{100}}}{\require{...
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61 views

Deferred mortality probability [closed]

What is the probability that $20$-year-old person survives $20$ years, then dies in the following $10$ years? In other words, what is the probability that ($20$) dies between age $20+20=40$ and age $...
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1answer
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What does it mean for a force of mortality to be continuous and linear

For (x), you are given that $$\mu_x= \begin{cases} 0.01, &\text{if}& x<50\\ 0.02, &\text{if}& x>60\end{cases}$$ and $\mu_x$ is continuous ad linear on [50,60]. Calculate $_{10}...
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1answer
60 views

$ES_X(p)$ of a Lognormal

I have been trying to find the expected shortfall for $X \sim Lognormal(\mu, \sigma^2)$. I have calculated the following; $$ ES_X(p)=\frac{1}{p}\exp\left(\mu+\frac{\sigma^2}{2}\right)\left[1-\Phi\...
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1answer
23 views

What is the distribution of $P_M(M_B(t))$

$M_X(t)= P_M(M_B(t))$ $P_M(s)= (1-q+qs)^2 $ $M_B(t)= \frac{\beta}{\beta-t}$ Where P(x) is the probability generating function and M(x) is a moment generating function I identified M as $M \sim ...
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44 views

Finding a future value given force of interest

One of my actuarial exam problems asks to find $\ddot{s}_{\bar{2}|}$, given $\delta_t=\frac{2t}{10+t^2}$ $(t\geq 0)$. Here's what I've done: From the definition of $\delta_t=\frac{A'(t)}{A(t)}$, $A(t)...
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114 views

Digital Option risk neutral probability

In the $T$-period binomial model, if the asset price is $S$ at any time, the next period's price will be either $SU$ or $SD$. The interest rate per period is $r>0$ and $D^* < 1 < U^*$, ...
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1answer
154 views

Determine fair price of a digital option

A digital option pays one dollar at time $t = T$ if the asset price is above a fixed level (strike) $K$ and is worthless otherwise. Consider the following model, with $r = 0$: \begin{...
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Deferred Term-Limited Annuity

I'm looking for a formula to calculate the EPV of an annuity (in terms of $a_x $ , $ a_{xy} $ etc) that is payable during life y, for a maximum of n years, but only beginning on the death of life x. ...
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102 views

Risk-neutral probability measures

Consider a market with $\Omega= (\omega_1, \omega_2, \omega_3)$, $r = 0$ and one asset $S$. Suppose that $S(0) = 2$ and $S$ has claim $\bar S = (1, 3, 3)$ at time 1. Find all the risk-...
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1answer
95 views

Sure Thing Arbitrage

Consider the following model with assets $S_1, S_2$ and three states, and suppose that $r = 10\%$ \begin{array}{|c|c|c|c|} \hline n&S_n(0)& S_n(1,\omega_1) & S_n(1,\omega_2) & S_n(1,\...
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1answer
125 views

Is there arbitrage?

An economist writes a 1-period expectation model for valuing options. The model assumes that the stock starts at S and moves to $2S$ or $\frac{1}{2}S$ in 1 year's time with equal probability. ...
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1answer
27 views

Compound interest with discounted rate in the time period

A note for \$250 dated August 1st 2010, is due with compound interest at $i=9\%$, 4 years after date. On November 1st 2011, the holder of the note has it discounted by a lender who charges $i=7.5\%$. ...
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1answer
24 views

What would the interest rate have to be to make the two shares to be equally valuable?

Marco dies and leaves behind a perpetuity paying $1000/month. His wife is entitled to the payments for the first 20 years, and then payments go to Marco's favourite charity. Interest rates are 6% ...
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132 views

Market quotes and arbitrage argument

DUPLICATE ON HOLD Use an arbitrage argument to construct a formula relating the price of the European to the price of an American. Good day, I wanted to ask for help with a question from one of my ...
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102 views

Calculate 3 month future of Oil

This question is from an exercise sheet for financial mathematics. Calculate the $3$-month future on Oil given the following information: Spot Oil is trading at $90$ USD per barrel. USD ...
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1answer
107 views

Fair Strike of a 1 Year Forward

Good day, I have been tasked with this question and I am unsure if my understanding and work so far is correct. On January 1st 2018 AAA shares were trading at $30USD$ per share. The shares ...
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1answer
71 views

Why is the calculator giving me different answers???

I have a question with regards to calculating annuity due in terms of annuity immediate. I thought that one way to do so is annuity due for n periods = annuity immediate n-1 periods + 1 (where the ...
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1answer
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Determine the AV of 13 annual deposits of $1,429 one year after the last deposit, at 2.10% effective

I got confused when calculating AV in this question. Apparently the correct formula would be $1,429\cdot 1.021 \cdot \frac{1.021^{13} - 1}{1.021 - 1}$. Or at least, this equals \$21.511.03, which is ...
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1answer
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during the first 4 years, interest is credited using a simple

I'm having trouble with the following problem from actuarial exam FM: During the first 4 years, interest is credited using a simple interest rate of $5\%$ a year. After 4 years, interest is credited ...
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How to implement an insurance risk model

So the problem goes as follows: "Suppose that the different policyholders of a casualty insurance company generate claims according to independent Poisson processes with a common rate $λ$, and that ...
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82 views

SOA Practice Exam: How am I to understand P(Z=z)?

Let $X$ denote the loss amount sustained by an insurance company’s policyholder in an auto collision. Let $Z$ denote the portion of $X$ that the insurance company will have to pay. An actuary ...
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Investment Method

can I ask which of the investment method below in accumulating wealth is better? Option 1: deposit P dollar at the end of every month for 25 years at a dividend rate of 5% per annum. Option 2: ...
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Prove that deferred continuous annuity has APV: $E(e^{-\int_w^t \delta(s)ds}\int_0^{T-w}e^{-\int_w^{w+t} \delta(s)ds}dt\mathbb I_{T>w})$

I want to prove that deferred continuous annuity has the APV: $$E(e^{-\int_w^t \delta(s)ds}\int_0^{T-w}e^{-\int_w^{w+t} \delta(s)ds}dt\mathbb I_{T>w})$$ $$=E(\int_0^{T-w}e^{-\int_0^{w+t} \delta(s)...
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1answer
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To prove the function is strictly decreasing (nominal interest rate convertible $p$ times a year)

How would one prove that the function $f(p)=p[(1+i)^p-1]$ is strictly decreasing? Here $i>-1$. I've tried the usual approach, i.e. finding the derivative of $f$ and got a pretty terrible ...
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Multiplicative Perpetuity of Dividend

"A share pays dividends annually and the next dividend payment is due in 3 months time and is expected to be 5 cents per share. It is expected that future dividends will grow at a rate of 4% p.a. ...
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1answer
44 views

Issue with Given Solution for Problem - SOA FM Exam Practice

I've been preparing for the FM exam coming up next month and am having some trouble understanding the solution given to Problem 24.15 in the book "A Basic Course in the Theory of Interest and ...
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43 views

Accumulation Function and Discount Rate (Monthly) (Financial Mathematics)

A deposit of \$300 is made into a fund at time 0. The fund pays interest for the first three years at a nominal monthly rate of discount $d^{(12)}$. For $t=3$ to $t=7$, interest is credited ...
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1answer
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how does this perpetuity relate to the infinite sum $1+x+x^2+\cdots$

You are given that $v=0.9$. You are also given a non-level perpetuity that pays the amount of $\frac{1}{k(k+1)}$ at times $k=1,2,3,\ldots$. Find the present value of this perpetuity. I am given that ...
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1answer
78 views

Conditional probability with uniform distributions: A company will experience a loss X that is uniformly distributed between 0 and 1

I'm trying to solve the problem: "A company will experience a loss X that is uniformly distributed between 0 and 1. The company pays a bonus to its employees that is uniformly distributed on the ...
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1answer
29 views

Actuarial Notation (Future Value Annuity Due)

Does anybody recognize this notation? I know the two dots means annuity due, the s refers to future value, but I'm not sure about the rest. I'm assuming that the latter value (without the (m)) ...
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2answers
62 views

How do I sum 1/6( e^t + e^2t + … + e^6t)? [duplicate]

I have a question about a sum when calculating moment generating function. The question is : "Find the moment generating function for each of these two random variables. (i) $X$ = outcome a die toss, ...
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1answer
55 views

Why is P(maximum of 3 functions >=3 ) = 1- prob. their intersection <= 3

I have this question: In a small metropolitan area, annual losses due to storm, fire, and theft are independently distributed random variables. The pdf's are: Storm $e^{-x}$ Fire $\frac{2e^{-2x/3}}...
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Exchanging Increasing Annuity Immediate for a Perpetuity

for the most part I understand this question, but i'm missing something. Any help would be appreciated. ...
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70 views

Accumulated Value of Investment

So I have been given this questions €500 is invested over a period of 4-years. In year 1 a nominal rate of interest of 6% p.a. convertible quarterly applies. In year 2 a nominal rate of discount of ...