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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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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347 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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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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84 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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27 views

Forward rate example, switching the investment.

I need explanation regarding forward rates for the following specific example. A zero coupon with spot rate $s_0(1)=.08$ and $s_0(2)=.09$ are available. a), Smith borrows $1$ and is obliged ...
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99 views

Fail time with joint density $f_{X,Y}(x,y)=6e^{-x}e^{-2y}, \quad 0<x<y< \infty$

The following is the problem I am solving for. Let $X$ and $Y$ be the fail time for a machine with two components. The component $Y$ will start working if and only if component $X$ fails. The ...
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77 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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46 views

Dollar weighted method vs. Time weighted method Problem. Exam FM

The following is the problem that I am working on and I am having trouble. On Jan 1 2005, an investment account is worth 100. On Apr 1 2005, the value has increased to 103 and 8 was withdrawn. ...
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83 views

Exam FM problem: Financial calculator necessary for finding $i$ from $a_{\overline{n}\rceil i}$? Edited

I am currently studying for the Exam FM for actuaries, and the calculator that I have is a TI 30X IIS, which was very helpful for me during the Exam P. I cam as far as studying bonds, and the ...
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29 views

Are yield rates different from rate of return? (Bonds)

There is a puzzling thing that is bothering me regarding bonds and I would like to have some help. The following is the situation I am dealing with. A 20-yr 8% bond has semi=annual coupons and a ...
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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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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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28 views

An actuary question but more of an algebraic manipulation

I'm stuck in this actuary question. If $a_{\overline{n|}}=x$ and $a_{\overline{2n|}}=y$, express $d$ as a function of $x$ and $y$ Hints: $a_{\overline{n|}}=\frac{1-v^n}{i} $, $v=\frac{1}{1+i}$, ...
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88 views

182-Day T-Bill vs. 91-Day T-Bill

I am trying to understand how T-Bills work and it would be great if someone could explain me using the following question At $t=0$ Smith buys a 182-Day T-Bill with a simple annual discount rate of ...
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23 views

Problem regarding Investments and selling the right.

I have trouble solving the following problem algebraically. Smith lends $\$1000$ to Jones at time $t=0$. Jones is supposed to repay Smith by paying $\$100$ at time $t=1$ and $2$, and $\$1000$ at ...
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27 views

128 factories close down first

I found this problem on http://poissonlabs.com/blog/how-insurance-works. I'm not quite sure how to solve it. Suppose that $1000$ factories belong to an industry, and that each factory faces an ...
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147 views

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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55 views

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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37 views

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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445 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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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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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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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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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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“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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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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181 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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15 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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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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145 views

Calculating the interest rate for an annuity (Exam FM)

I have been searching for a way to solve for the interest rate given the monthly payments of a loan. I would like to set up a problem as the following. $X$=monthly payment , $i$=effective ...
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Finding drop payment with varying interest rates..

Person A is accumulating a 10,000 fund by depositing 100 at the end of each month starting September 1,2002. If the nominal interest rate on the fund is 12% convertible monthly until May 1,2005, and ...
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119 views

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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142 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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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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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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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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21 views

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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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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Purchasing a unit on fund $X$ calculating the dollar weighted and the time weighted rate of return.

I am currently working on the following problem trying to figure out the rate of return. Fund $X$ has unit values which are $1.0$ on Jan 1 05, $0.8$ on Jul 1 05 and $1.0$ on Jan 1 06. A fund ...
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38 views

Refinancing an amortized loan - find the maximum value of $k\%$

The following problem is quite complex for me to see what I am solving for and I would appreciate to receive a bit of help. Jenny borrows $L$ at $i^{(12)}=12\%$ from Bank A, requiring level ...
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24 views

Amortization problem with yeild rates from Exam FM.

I am tackling the following problem. I have a tendency to not do well with yield rates, and I am thinking that that's the part that I am not getting. Betty borrows $19,800$ from Bank X. She ...
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26 views

Calculating the loan from amortization.

I was not quite sure what the question was asking and I would like to have some input. a), Is it asking us to actually calculate knowing what each principal paid? or b), Working from almost complete ...
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29 views

Increasing annuity problem

I learned increasing/decreasing annuity and am tackling the following problem for hours now without success. Amy deposits $Z$ into a bank account that has effective annual interest rate of $5\%$ ...
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Pension calculation with annuity and a geometrically progressing deposits.

I am a bit confused regarding how to solve the following problem. An employee serves $37$ years before retiring on a pension. His initial salary was $18,000$ per year and increased by $4\%$ each ...
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Reinvestment problem involving finding loan payments given the yeid rate (From Exam FM)

I am having trouble computing the interest rate for the following problem. Sally lends $10,000$ to Tim. Tim agrees to pay back the loan over 5 years with monthly payments at the end of each ...
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Expected payment problem. $d=4, u=5, f(x)=.125e^{-x/8}$

I am working on the following problem. Let $X$ be a random variable for a loss that is exponential with mean 8. Let $Y$ be the payment by the insurance company given a positive loss, with ...
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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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408 views

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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50 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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65 views

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 ...