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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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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Drop Payment -Actuarial Science [on hold]

Vanessa 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, ...
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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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9 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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7 views

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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26 views

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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17 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 ...
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32 views

Hard to understand the strategy of the following investment.

Upon attempting to solve a problem regarding annuities, I am a little puzzled regarding the following strategy of investment. $10,000$ can be invested so that one can purchase an annuity-immediate ...
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29 views

Annuity problem from Exam FM

I am working on the following problem and I was wondering if someone could help me solve it. Chuck needs to purchase an item in 10 yrs. The item costs $\$200$ today, but its price inflates $4\%$ ...
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16 views
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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 $i$. ...
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43 views
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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. ...
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32 views

Choosing final year project [closed]

I'm studying pure math now yet I would like to develop a career in actuarial science. For my final year project, I can choose to either the topic of probability analysis or statistic. While I know ...
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32 views

Operations Research: Problems of choice with delayed effects

A company buys a manufacturing plant at a cost of $ 500,000. The manufacturing plant has a term of 5 years, with a value of elimination equal to $ 200,000 The manufacturing plant only requires a ...
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1answer
38 views

Question regarding interest method during a year. $i-\delta$

B and P deposits the amount $100$ in separate bank accounts. B's account have a nominal rate convertible semiannually. P's account has a force of interest $\delta$. After $7.25$ years, each ...
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1answer
22 views

Force of interest problem from Exam FM

Tawny makes a deposit into a bank account which credits interest at a nominal interest rate of $10\%$ per annum, convertible semiannually. At the same time, Fabio deposits $1000$ into a different ...
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15 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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20 views

Force of interest :$e^{\int_0^t{\delta_t}ds}$ vs $(1+\frac{i^{(2)}}{2})^2$

I was wondering if someone could help me confirm the answer for the following problem regarding force of interest. X deposits $1$ at time $t=0$ with force of interest $\delta_t=\frac{t^2}{k}$. Y ...
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38 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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28 views

Discount rates vs. Interest rate problem.

I am working on a problem as follows. A discounted value $X$ that is due when $t=0.5$ has a present value \$4992. Calculate the value of $X$ when a), the effective annual compound interest rate ...
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22 views

Finance problem - Canadian bond with simple interest

I am having trouble understanding a lot of things from the following problem. It would be very helpful if I could get some explanations. Smith purchases a Canadian bond for 1000 with an issue ...
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14 views

Financial math vocab; “convertible”, “roll over”?

I am having trouble understanding what is going on regarding the following problem. Smith receives income from his investments in yen. He finds a bank that will issue a term deposit that allows ...
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27 views

Am I plugging this into the calculator correctly?

I want to make sure I am plugging these into my BA II plus correctly in order to help me when I'm taking an exam. I keep looking up how to work the calculator and I think I have gotten the steps down ...
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19 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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2answers
34 views

Proof $(1+i)^t < 1+it$ if $0 < t < 1$

I am trying to prove that compound interests grow slower than simple interest in the first year. I attempted to do this with the following approach. Let $$f(t) = (1+i)^t$$ then $$f'(t) = ...
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1answer
18 views

Find expression for $\frac{d}{dn}\overline{s}_{\overline{n|}}$

I got a question from book Mathematics of Investment and Credit question 2.3.25 b) Find expression for: $$\frac{d}{dn}\overline{s}_{\overline{n|}}$$ I got the solution, but there is one part that I ...
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21 views

How to interpret this annuity-immediate equation?

I am trying to verbally interpret this annuity equation of annuity immediate, payable m-thly $$a^{(m)}_{\overline{n|}i}=\frac{i}{i^{(m)}}a_{\overline{n|}}$$ I drew a time line and try to make ...
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38 views

Expressing a summation using matrix algebra

Consider the $n \times r$ 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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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 ...
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2answers
44 views

What is a discount?

I am learning some financial terms and am having trouble understanding what a discount $d$ is. Numerically, I understand that it is defined as $\frac{i}{1+i}$ but I do not intuitively understand what ...
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45 views

Master in financial mathematics or actuarial science?

I study mathematics and want to study a master degree in Europe that allows me to develop in a different teaching (at least for now) workforce. I am interested in financial mathematics but I've also ...
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15 views

Different types of interest

I'm currently taking actuarial mathematics one. I have a quiz and exam coming up, but I'm very confused between all these interest rates and what they represent. There is annual effective interest ...
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65 views

conditional expected value - Poisson process plus random variable

I've struggled with this actuary excercise for a while and I don't know how to do it: Each claim can be characterized by two random variables $(T,D)$, where $T$ is the moment of reporting the claim ...
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Saving account & annual effective rate

A bank pays 6% per annum on its savings accounts. At the end of every 3 years a 2% bonus is paid on the balance. Determine the annual effective rate of interest, earned by an investor if the deposit ...
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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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70 views

Pure Mathematics to Actuarial Science.

I did my undergraduate in pure mathematics and recently after a three year break I decided to get a master's degree in Actuarial Science. I am in the second week of classes and I feel completely lost. ...
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77 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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21 views

Force of mortality calculation

The force of mortality of a non smoker and smoker aged x is μx and c.μx respectively (c>1). What is the probability that a non smoker live longer than a smoker? I guess I haven't really understand ...
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18 views

Let $f(x,y)=2(x+y)$ for $-<x<y<1$ Find $E[Y]$

This may be a dumb question, but I thought that to get the expected value, we were supposed to calculate $$A = \int_0^1 \int_x^1 y 2(x+y) dydx$$ which gave me a different answer to $$B = \int_0^1 ...
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170 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 ...
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1answer
56 views

Unfamiliar notation. (actuarial science)

I am a math instructor self studying for the actuarial exam and I am trying to understand the following notation that I have encountered today. $$E[X \land d]$$ The explanation in the book told me ...
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47 views

Finding the variance problem

I am working on the following problem and the explanation was not clear to me, so I am seeking for help. The following is the problem. A fire occurs with a probability of 0.01. The damage Y ...
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1answer
25 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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2answers
65 views

Maximum of Three Uniform Random Variables

Here is the question, I am studying for exam P, and am using a study guide with a solution guide. I am stumped on this problem, and the solution in the back was very confusing. Any clarity I can get ...
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21 views

Deffered annuity with perpetuity

An annuity immediate has $40$ initial quarterly payments of $20$ followed by perpetuity of quarterly payments of $25$ starting in the eleventh year. Find the present value at $4\% $ convertible ...
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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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60 views

When to use alternate parametrization of Gamma distribution?

In Loss Models, 4th ed., by Klugman et al., the following parametrization is given for the Gamma distribution: $$f(x) = \dfrac{(x/\theta)^{\alpha}e^{-x/\theta}}{x\Gamma(\alpha)}\text{.} $$ When ...
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48 views

Proof of $\text{TVaR}_p(X)$ and $\text{VaR}_{u}(X)$ relationship

From Loss Models, 4th ed., by Klugman et al.: Definition. Let $X$ denote a loss random variable. The Value-at-Risk of $X$ at the $100p$% level, denoted $\text{VaR}_{p}(X)$ or $\pi_p$, is the ...
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43 views

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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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 ...