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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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44 views

What does the semicolon (";") mean in this formula? [closed]

I'm working on some statistics work and have to use the formula for the second uncentered moment of a limited loss variable of an exponential distribution. The formula contains this little bit, $Γ(k+...
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1answer
68 views

Understanding whole life Assurance

I asked suggestion for good book here. After reading the suggested book and others I want to ensure what I have learned is correct and want to clear some of my doubt. Whole life Annuity-due The ...
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10 views

Analyzing Reinsurance Effectiveness

Given data for a product (P&C) and a Reinsurance Treaty, how can I achieve the following: Determine claim distributions for frequency, severity and aggregate claims Perform Goodness-of-fit tests ...
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1answer
47 views

Actuarial and Financial mathematics book suggestion

Recently, I was introduced with one of my course named "Introduction to Actuarial and Financial Mathematics". The book I followed was too dry and it's not contain a detail explanation of ...
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17 views

Evaluating the Present Value of a Perpetuity where each term is square [duplicate]

I'm currently studying for SOA Exam FM using Harold Cherry's study manual (11th edition). In practice exam 5, there is a question (#22) that asks the following: Determine the modified duration of a ...
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1answer
42 views

Compound Interest Differential Equation Question

I'm positive someone else has asked a nearly identical problem in the past but I can't find an example of this specific variation. Let's say initially you invest some principal amount ($I_P$) into a ...
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20 views

Accumulation functions and investments

Suppose we invest $\$50$ at time 0.5 and $\$200$ at time 1.5; the accumulation function that applies on the interval $[0,1]$ is $a(t) = 1+t^2$, and simple interest with $i = 5%$ applies on the ...
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2answers
83 views

Integral of a sum under condition

I'm struggling with the following integral, would anyone have an idea on how to approach it? I solved it for N = 2 but the generalization to any positive integer is more difficult. $$\int_{(\omega_1, \...
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1answer
19 views

Question about the mean excess loss function.

I'm new to this risk thing. I am trying to obtain the mean excess loss function evaluated at a point for the following Pareto distribution: $F(x)=1-(\frac{\theta}{x})^{\alpha}$ The excess loss ...
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1answer
36 views

Find expressions for the single sums $X$ with simple annuities

Find expressions for the single sums $X$ equivalent to the set of seven payments of Fig. 5-21 at times $(a)1,(b)5,(c)12$ and $(d)12$, assuming a rate $i$ per period. Solution: $(a)$ At $1$, $X$ is ...
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28 views

Calculate discounted cash flows for a fixed net present value with some restrictions

I was wondering what my maximal investment in year zero can be, given certain ratio that is pre-specified between cash flows, for a fixed net present value: This solution was offered by Bob: If you ...
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1answer
70 views

expected value of geometric distribution

The following question has me extremely confused: At a certain university, registrations for courses have to be made over the telephone. There are so many calls that $90$% of the time you will get a ...
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1answer
40 views

Negative Binomial and Geometric Distributions

So this is the problem I'm dealing with. An actuary has determined that the number of claims per month can take any number 0, 1, 2, 3,... and follows a negative binomial distribution with mean 3 and ...
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30 views

Redington immunization with a bond

A company pays two liabilities: $100$ in $2$ years and $50$ in $4$ years. The company is attempting to protect itself through immunization by purchasing a $3$-year, par-valued bond with annual coupons,...
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1answer
35 views

Annuity & Perpetuity problem

Suppose Joe has been paying $600$ from his monthly salary at the end of every month for the past $n$ years. After $n$ years of payments, he retires having purchased a perpetuity-due plan that begins ...
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37 views

Accumulated Value of Two investments

Suppose the force of interest is: $$F(t) = \frac{0.02 + 0.01t}{1 + 0.02t + 0.005t^2}$$ , where $t$ is the number of years beginning March 31, 2001. An investment of $100$ is made on March 31, 2002, ...
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1answer
37 views

Approximating effective interest rate on a bond

If you have a bond with the following properties: Issued at price $X$; Redeemed at price $Y$; Has a coupon rate of $p$ ; Has a term of $n$ years; then I am told, in my accounting studies, that the ...
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1answer
19 views

Calculating the present value of bond when face value is not equal to value at maturity

I need to calculate PV of the following bond: Face value is $1000$. It pays $50$ every $6$ months. Annual rate is $8\%$. Value at maturity is $1050$. The bond lasts $20$ years. I know how to ...
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1answer
49 views

Difference between yield rate and interest rate (solution verification)

An insurer enters into a four-year contract today. The contract requires the insured to deposit $500$ into a fund that earns an annual effective rate of $5.0\%$, and from which all claims will be paid....
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1answer
74 views

Bonds, Exact Matching Payments, Bank Payments

A bank accepts a 20,000 deposit from a customer on which it guarantees to pay an annual effective interest rate of 10% for two years. The customer needs to withdraw half of the accumulated value at ...
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1answer
75 views

Finding the present value of a continuously varying perpertuity

A perpetuity provides for continuous payments. The annual rate of payment at time $t$ is $1$ if $0\le t<10$ and $(1.03)^{t-10}$ if $t\ge 10$. Using an annual effective interest rate of $6\%$, the ...
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53 views

Modified and Macaulay Duration

Repost but with a proposed solution.* I am practicing for an upcoming examination and came across the following problem: You are given the following information about an asset: (1) The present value ...
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78 views

Swap rate with coupons

Suppose you were given the following information about the prices of zero-bond coupons, all with a maturity of $1$: The price of a $1$ year zero-coupon bond is $0.943$. The price of a $2$ year zero-...
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29 views

Macaulay, Modified Duration

I am practicing for an upcoming examination and came across the following problem: You are given the following information about an asset: (1) The present value of the cash flows at an annual ...
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0answers
40 views

Term structure, spot rates and effective coupon rate

Suppose you were given the following term structure: $r_t = 0.02t + 0.08$ for $t \in \{1, 2, 3, 4\}$. If the annual effective yield rate for a four-year bond with annual coupons is $0.15$, what is the ...
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16 views

Rate of investment return with weighted yield rate

Suppose John has an investment account and at the beginning of the year, this account is worth $300000$. After $m$ months, the value increases to $315000$ and then John withdraws $15000$. Then, $2m$ ...
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1answer
65 views

Joint Life Mortality (Qx) - Actuarial

I've a question regarding the calculation of joint life mortality probabilities based on a "last survivor" principle, i.e. both borrowers need to be dead for the event to be triggered. If I ...
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1answer
34 views

Annuities, Perpetuities, Choosing a Comparison Date - Solution Verification

I am working on the following question: A man turns $40$ today and wishes to provide supplemental retirement income of $3000$ at the beginning of each month starting on his $65$-th birthday. Starting ...
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1answer
29 views

Accumulated amount of annuities question - solution verification

(Old SOA sample problem) Kathryn deposits $100$ into an account at the beginning of each $4$-year period for $40$ years. The account credits interest at an annual effective interest rate of $i$. The ...
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1answer
43 views

Complete Market / Replicating a portfolio

Attempt So Far: 1) First Part: I have shown that the market is arbitrage-free since the only possible portfolio for which $V_1^h\geq0 \ $ given that $V_0^h=0 \ $ is $h=(0,0,0)$ and this clearly ...
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25 views

Expected Value of Unreimbursed Accident Loss

The question goes as follows: A three person family purchases an auto insurance policy that reimburse accident losses up to a maximum on 4 accidents per year. The joint probability distribution for ...
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25 views

Corr(two stocks) = Corr(return of the two stocks)?

Question: A portfolio consists of two stocks, $A$ and $B$. $A$ has expected return $.15$, volatility $.2$, proportion of portfolio $.5$. $B$ has expected return $.18$, volatility $.25$, proportion of ...
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1answer
33 views

Covariance of a Portfolio with a Stock in that Portfolio

A portfolio consists of three stocks: $X, Y, Z$. $40\%$ of the portfolio is invested in $X$, $40\%$ in $Y$, and $20\%$ in $Z$. The three stocks are uncorrelated. The volatilities of the three stocks ...
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1answer
29 views

Variance of return of a portfolio of stocks

Here's a derivation my textbook gives for the variance of the return $R_p$ of a portfolio of stocks with arbitrary weights $x_i$: $$ \mathrm{Var}(R_p) = \mathrm{Cov}\left( R_p, \sum_{i=1}^n x_iR_i \...
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1answer
30 views

Approximation of portfolio distribution using Gamma distribution. [closed]

Let $X\sim Poi(5)$ and $N\sim Bin(60,1/2)$. We have portfolio given by: $$S_N=X_1+\dots+X_N$$ I already did some calculation in Maple and ended up with this density plot (on the graph the pink line is ...
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1answer
24 views

Finding and examinating the distribution of portfolio [mostly Maple question] [closed]

So that's more Maple related question but I'll give it a try. Assume we have portfolio $$S_N=X_1+\dots+X_N$$ Where $X\sim Poi(5)$ and $N\sim Bin(60,1/2)$. We want to study the distribution of $S_N$. ...
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1answer
48 views

(a,b,0) class of distributions moments [closed]

(a,b,0) class of distributions is such as $p_{n} = (a+b/n)*p_{n-1}$ n=1,2,3,... where $p_n=pr(N=n)$ How can we prove that if N is a random variable from the (a,b,0) family then $$E[X]=(a+b)/ (1-a)$$ $$...
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113 views

Finding transition probabilities from an intensity/transition rate matrix for Markov Chains.

I've just started studying Markov chains this year using my brother's old notes from college and I've been finding them quite useful so far. However, there is one thing that I'm not that sure of. If I'...
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31 views

Formula for convexity of bonds

I have to write a code in R that calculates the convexity of bond when parameters $T,y$ are given and coupons are paid once every year. Recall that convexity of bond is given by: $$C=\frac{1}{v}\cdot\...
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15 views

calculate the OB using prospective method

I'm learning about the prospective for loan repayment but I'm having trouble creating the equation. Here was my problem (loan repayment- finding the loan if end payment increases by certain amount) I'...
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1answer
51 views

loan repayment- finding the loan if end payment increases by certain amount

What happens if a loan has an effective annual interest rate $i = 10$% is repaid with 10 yearly payments starting one year after the loan. The amount of the first payment is $500$ but each subsequent ...
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1answer
56 views

loan repayment- find the loan and interest paid

I have a loan at a nominal annual interest rate compounded monthly $i^{(12)} = 12$% and is repaid with $120$ monthly payments starting one month after the loan. The monthly payments are $600$. I am ...
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2answers
48 views

Find the m.g.f., the mean and the variance of the aggregate claims?

Assume that the number of losses $N$ has a geometric distribution with parameter $p$, while the claim amounts,${\{X_i}\}_{i≥1}$, is a sequence of i.i.d. Gamma distributed r.v. with parameters $a$ and $...
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2answers
30 views

Present value of perpetuity with one extra payment

Consider the perpetuity that pays $3$ by the end of the 2nd year and then every $4$ years. I need to calculate the present value of it when $i=0.05$. So the second payment will be at 6th year. I ...
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2answers
34 views

Varying payment- with both increasing and decreasing annuity

I am trying to find the present value of an annuity-immediate such that the first payment is $1000$, and each subsequent payment increases by $100$ until the payments reach $2000$, but then decreases ...
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1answer
24 views

Present values of two perpetuities

I have two perpetuities: Pays $1$ at the beginning of every year Pays $1.8$ at the beginning of every odd year I have to show which one is more profitable if $i=0.05$. I calculated present value of ...
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0answers
12 views

Interpreting the deferred probabilities of a life aged x will survive for n years but die during the subsequent m years

I am a little bit confused when trying to interpret $_{n|m}q_x$, which I understand is the probability that a life aged x will survive for n years but die during the subsequent m years. However when I ...
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1answer
56 views

annuity- find the present value of varying payments

I am trying to find the present value of an annuity, where we have a $20$-year annuity immediate where the first payment is $1000$, and each subsequent payment is increased by either $50$ for the ...
1
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1answer
39 views

Why is probability divided by time = annual rate? (Force of Mortality)

In force of mortality, one divides the probability of failure in a time interval $(x, x+\Delta x)$ by $\Delta x$. Apparently, this gives an annualised instantaneous rate of failure. In this article, a ...
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1answer
41 views

calculation expect pension payout given survival probabilities

Just wondering. How is the expected pension payout calculated, given an agreed fixed annual payment of a made up figure of 1000, whilst linking it to survival probabilities coming from life tables (I ...

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