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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Interest Theory- Annuity Withdrawals/Deposits

"Consider an investment of $5,000 at 6% convertible semiannually. How much can be withdrawn each half−year to use up the fund exactly at the end of 20 years?" To solve this problem, an equation of ...
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Bayesian Statistics

Question: Given $N$, $X$ is distributed as $\mathrm{B}(N,\theta)$. Derive the unconditional distribution of X assuming N is distributed as $P(\lambda)$. This is what I have tried so far: $$x|N \sim ...
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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 ...
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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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Continuous pyments with continuous compounding

At time $t=0$ saving account balance is $0$. Then we start continuous payments with intensivity $C_t$. Continuous compounding intensivity is $\delta_t=\frac{1}{1+t}$. Accumulated value of funds at ...
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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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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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Actuarial Science - Amortization

Kevin takes out a $10$-year loan of $L$, which he pays by the amortization method at an annual effective interest rate of $i$. Kevin makes payments of $1000$ at the end of each year. The total amount ...
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Discounting Perpetuity Question

"A project pays a dividend of $0.75 next year and then grows at 12% for 3 more years, and then grows at 8% indefinitely thereafter, find PV" Okay so first step is to find the initial value of ...
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Question on duration matching and reddinggton's immunisation

An insurance company has liabilities of 6 million due in 8 years’ time and 11 million due in 15 years’ time. The assets consist of two zero-coupon bonds, one paying X in 5 years’ time and the other ...
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financial mathematics question

An investor is interested in purchasing shares of ABC company. The company pays annual dividends, and a dividend payment of 1.2 per share has just been made. Future dividends are expected to grow at ...
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Poisson Process Basic Question

The sum of independent interarrival times for the poisson process is a gamma random variable. in general does the sum of exponentials have to be independent to sum to gamma? also this would produce ...
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poisson and discrete distribution

Business failures are due to three mutually exclusive risks: market risk, credit risk, and operation risk, which account for 20%, 30%, and 50%, respectively, of all business failures. Suppose the ...
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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 ...
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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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“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. ...
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Probability of remaining lifetime using force of mortality

I've been stuck on this question for the past half hour and still have no idea how to solve it... I don't think it's supposed to be very difficult but I'm struggling: There are two independent live ...
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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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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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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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Expected value problem with given condition.

the following is the problem I am working on. A fair die is rolled repeatedly. X is the necessary number of rolls until 5 shows up, and Y is the necessary number of rolls until 6 shows up. Find ...
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Derive the $m$thly death benefit in terms of the annual benefit

How do I approximate the $m$thly death benefit $A^{(m)}_x$ in terms of the annual benefit $A_x$ using the uniform distribution of deaths assumption (UDD1), i.e., ${}_{s}q_x=sq_x$ to show that ...
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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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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 ...