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.

Filter by
Sorted by
Tagged with
2
votes
1answer
24 views

Proof of continuous force of interest with infinitely compounded interest rate. For actuaries, delta and i upper infinity

Specifically, we know the following: $$(1+\frac{i^{(2)}}{2})^2 =(1+\frac{i^{(4)}}{4})^4=(1+\frac{i^{(12)}}{12})^{12}= 1+i, $$ Where $i^{(2)}$, $i^{(4)}$, and $i^{(12)}$ are the interest rates ...
0
votes
0answers
27 views

Certainty equivalent related to the net present value when there is no interest

Throughout we work with the assumption that there is no force of interest (i.e. the interest rate is $i=1$). Consider a lottery $X$ that pays $1000$ with probability $e^{-1}$ and $0$ otherwise. Then ...
1
vote
1answer
31 views

What is the actuarial present value of a life insurance given in this question?

I need help on approaching a particular question regarding actuarial science. I just need a rough concept on how to do it, so I won't provide any extra data to keep this simple. For these questions, ...
1
vote
1answer
63 views

Present value of varying annuities

I need help with the following questions. In order to answer these questions, I am only allowed to use two of the tables of values for $\ddot a_n , \bar a_n$ (continuous annuity) and $(I \ddot a)_n$ ...
1
vote
1answer
57 views

Calculate the probability that a randomly chosen claim on this policy is processed in four hours or more

I'm not sure how to approach this question, can someone please help me? An insurance policy is written to cover a loss X where X has density function f(x) = (2/9)x for 0 ≤ x ≤ 3 and 0 ...
1
vote
1answer
114 views

Recurrence relations for annuities

I need help figuring out recurrence relations for various annuities. I've attacked the questions below and my responses. I'm not too sure what the recurrence relation is for d) however. I have ...
0
votes
1answer
27 views

accumulated value in geometric progression

In a general form, i know how to compute the PV of an annuity that follows a geometric progression. But how do i compute it when it's the accumulated value? let's say a payment of 100/ year with ...
0
votes
0answers
20 views

Survival Analysis - finding the survival function from hazard function

If the hazard function (i.e the instantaneous rate of death/failure) was given as, for a random survival time: $h(t) = λ(2 + 2 \exp(−λt))$ for all $t > 0$, how could find the survival function $S(...
0
votes
1answer
39 views

accumulated value on the last payment of an annuity due

A $20$-year annuity-due makes payments of $100$ each year for the first $10$ years and then each subsequent payment decreases by $5$ for the next $10$ years. The effective annual interest rate is $9\%...
0
votes
1answer
42 views

Maximum likelihood estimators of the parameter of an aggregate loss (Poisson frequency, exponential loss)

Question: $$N\sim Poisson(\theta), X\sim exp(\theta)$$ $$S = X_1 + X_2 + ... + X_N$$ With $4$ observed aggregate loss $s_1, s_2, s_3, s_4$. What's the maximum likelihood estimator of $\theta$? My ...
1
vote
1answer
54 views

Variation of $_n d_x = l_x - l_x+n$ (Actuarial notation used)

I am having trouble deriving this equation I understand that $l_x=l_0 s(x)$ $s(x) = e^{-\int_0^x \mu (u)du}$ and from there I derived $l_{x+n} = l_xe^{-\int_x^{x+n}\mu(u)du}$. I tried playing ...
0
votes
1answer
47 views

Solving for the amount of level payments proportional to interest due from sample SOA exam

this is a problem from practice exam FM that I am studying for: A 20-year loan of 1000 is repaid with payments at the end of each year. Each of the first ten payments equals 150% of the amount of ...
0
votes
1answer
33 views

Find the jump of force of mortality under Balducci assumption

I need to find the jump of $\mu_x$ (force of mortality) under Balducci assumption at the point n $\in \Bbb N$. Under the assumption of Baoducci force of mortality function has the form: $$\mu_{x}= \...
0
votes
1answer
34 views

Please help me with this probability question [closed]

enter image description here this is the problem. I already integrated the given integral for part (a) and I believe its finite for all values of p, but I'm confused about how I'd go about the ...
0
votes
1answer
29 views

Find yield-to-maturity of a bond

Given: A one-year zero-coupon bond has an annual yield of $6.25\%$. A two-year zero-coupon bond has an annual yield of $7.00\%$. A three-year zero-coupon bond has an annual yield of $7.50\%$. A three- ...
2
votes
1answer
33 views

What is the present value of an immediate annuity over 12 years with 4 yearly payments and an interest of i = 2%?

See the question above, the result should be 10.689. I tried using the temporary annuity-due formula (see below): $$ \ddot{\mathbf{a}}_{n}^{[m]}=\frac{1-v^{n}}{d^{[m]}} $$ where: $$ d^{[m]}=m \cdot\...
0
votes
1answer
23 views

Comparing variances of insurance models $I(x)=kx$ vs $I_d(x) = x-d , \quad x \in (d, 100)$- Need verification

I am trying to show that $$Var[X-I(X)] > Var[X-I_d(X)]$$ and I don't think this is true in general, so I would like to have some confirmation. I am given $X \sim Unif(0,100)$ and $I(x)=kx, \quad k\...
0
votes
2answers
66 views

Expected value of sum of two discrete random variables

Correct Answer: 1.1 My work: I'm not sure, but I tried to find the value for $c$ by summing joint pmf over all possible values of x and y, in order to use Law of Unconscious probabilist to calculate ...
0
votes
1answer
30 views

constructing portfolio at time 0 using Euro put option, shares and cash with same payoff as Euro call

Now I have to construct a portfolio using a European Put option, shares and cash such that the payoff is equal to that of an European call option contract at time 0. I understand how to do this in ...
0
votes
1answer
46 views

Find $P(Z_1>0, Z_2<0)$

Let $Z_t$ be standard Brownian motion. I would do $$P(Z_1>0, Z_2<0) {= P(Z_1>0 | Z_2<0) P(Z_2<0)\\=P(Z_1-Z_0>0 | Z_2-Z_0<0) P(Z_2-Z_0<0) \\=^{(*)} P(Z_1-Z_0>0)P( Z_2-Z_0<...
1
vote
0answers
36 views

Average remaining lifetime for him and her

She (40) is chosen from the population with maximum age 121. Average remaining lifetime at age $x$ is equal $\frac{121-x}{2.4}$ for $x<121$. He (30) is chosen from the population with maximum age ...
0
votes
1answer
41 views

Calculating interest rate for Perpetuities

So I have this homework question for actuarial mathematics: Two perpetuities have the same annual effective interest rate. Perpetuity A pays \$4 at the end of each year for the first 20 years and ...
1
vote
2answers
59 views

Serial bond price

On August 15, 2015 a corporation issues a 10% serial bond with face amount 50,000,000. The redemption is scheduled to take place at 5,000,000 every August 15 from 2025 to 2029 and 25,000,000 on August ...
0
votes
0answers
106 views

degree advice (actuarial science vs honours math)

Currently, I'm a first year student doing my undergrad in actuarial science. I went into ac sci because I really love math, but it's not what I thought it'd be. I am not too fond of the material I'm ...
0
votes
0answers
31 views

No arbitrage pricing examples

Good day, I have a question about FTAP (No arbitrage Theorem). I have the following two problems to solve and I am not quite sure about them. Asset prices and payoffs are given by $(\pi,D)$. ...
0
votes
1answer
43 views

$X=10(Y-1) \forall y \ge 2$. Otherwise $X=0$. $Y$ follows a Poisson distribution with mean $0.3$. What is $Var(X)$?

$X=\begin{cases} 0: &y=0,1 \\10(Y-1): &y=2,3,4,... \end{cases}$ $E[X] = E[10(Y-1)] -(10(0)-1)P(Y=0)-(10(1)-1)P(Y=1)$ $E[X]=10E[Y]-10+P(Y=0)-9P(Y=1) = 10(0.3)-10+e^{-0.3}-9e^{-0.3}(0.3)$ $E[...
1
vote
1answer
32 views

Percentile of positive values of $\max(0,X-500)$ given the CDF of $X$

$X$ follows a distribution with distribution function $$F(x)=\begin{cases}1-\left(\frac{2000}{2000+x}\right): &x\ge0\\ 0: &\text{otherwise}\end{cases}$$ Let $Y=max(0,X-500)$. Calculate the $60^...
0
votes
0answers
28 views

Under constant force of mortality assumption show that tqx <= u*t.

I think I know how to prove a) because I know that the Taylor series expansion for e^x is 1 + x + x^2 /2! + x^3/3! + ..... and the Taylor expansion for 1/(1-x) is 1+x+x^2+x^3+ .... and thus e^x<- 1/...
1
vote
0answers
53 views

Find an equivalent risk-neutral measure (martingale measure)

Consider a market model with risk-free rate $r > -1$ and one risky asset that is such that $\pi^{1} > 0 $. Assume that the distribution of $S^1$ has a strictly positive density function $f:(0,\...
1
vote
1answer
37 views

Linear Combination of Normal Random Variables

An exercise from my textbook: $X$ is normally distributed with mean $100$ and variance $6400$. $Y$ is normally distributed with mean $350$ and variance $10000$. Calculate the probability that $12X - ...
1
vote
1answer
69 views

Are moment generating functions still multiplicative if their random variables are not independent?

Given $X$ is a random variable with moment generating function $M(t)$, I am suppose to determine whether $M(t)M(5t)$ is the MGF of some random variable. The solution in my textbook says it is, if I ...
1
vote
1answer
78 views

Present Value of Continuous Life Annuity - Derivation of Standard Integral Using Integration by Parts

The standard integral expression for the present value of a continuous annuity on a life aged $x$ is: $$\overline{a}_x = \int^\infty_0 v^t {}_t p_x dt $$ Notation: The symbol on the left hand side is ...
0
votes
0answers
22 views

prospective liabilities

I studied a article about "Dynamics of solvency risk in life insurance liabilities" and I saw a formula for "prospective reserve". How is this formula justified, where $T > 0$ is a finite time ...
1
vote
2answers
50 views

Help with understanding bond duration.

I'm currently studying for a pure maths degree, and so have no background knowledge of bonds. I'm reading through some material regarding actuarial work, and came across the following definition of ...
-2
votes
1answer
23 views

Expected value of a Guarantee Actuarial Science [closed]

I have this problem and I've never seen a problem about actuarial present value with pareto distribution A 3-year warranty for a new computer pays 500(4-k) if there is a failure in the k-same year, ...
0
votes
2answers
26 views

Problem with principal payment

Hey I am supposed to determine the principal payment of the following situation: Loan amount: 50 000$ Interest rate: 5% p.a. Number of years: 30 What I did: I calculated the ...
2
votes
0answers
27 views

Reserve for Credit Loss

This is a regulatory way to calculate a reserve for credit loss. I want to interpret this formula. $$Lifetime\ Reserve=\frac{PD\times LGD\times EAD}{(1+i)}\left[\frac{1-(1-PD)^n}{PD}\right]-$$ $$\...
0
votes
1answer
44 views

What did I do wrong in this question? [closed]

I did $P(X >= 2,000,000) = P(z >= (2,000,000 - 2,400,000)/80,000)$, but I got a negative number $P(z >= -5)$. I forgot what I need to do in this case.
0
votes
0answers
14 views

Conversion of whole life insurance actuarial present value from annually to m-thly

Let the fractional independence assumption holds if $_sq_{x+k} = H(s)q_{x+k}$ for all $k =0,1,2,...$and $0\leq s \leq 1$. I want to prove $A_x^{(m)} = \frac{i}{i^{(m)}}A_x$ assuming UDD, I first ...
0
votes
0answers
28 views

Linear change in force of interest

The force of interest $\delta$(t) is 10% p.a. at the start of the year, 7% p.a. at the end of 9th month and 5% p.a. at the end of the year. $\delta$(t) is linear during the two periods before and ...
0
votes
1answer
36 views

Exam P/ Discrete Probability

I'm trying to understand the question from the 'Probability for Risk Management 2e' book and the solution I found online vs. the solution on the answer key does not match. For the insurance policy ...
0
votes
0answers
15 views

Example - Consumption and saving income profiles

Two women $Y$ and $Z$ have the following income profiles: $Y$: $w_1$ at time $1$ and $w_2$ at time $2$. $Z$: $w_1$ at time $1$ and $w_2 + X$ at time $2$, where $X$ is a non-degenerate random ...
0
votes
1answer
63 views

How to express conditional probability?

For planes of a certain type, the actuarial estimate of the probability of a crash during a flight (including takeoff and landing) from New York to Chicago is $P_1$; from Chicago to L.A., it is $P_2$. ...
0
votes
2answers
48 views

How do we derive the formula for continuous compounding by writing an ODE?

Recently, I encountered a question on continuously compounding interest. The solution started with the following differential equation $$\frac{dN}{dt} - rN = 0$$ where $r$ is the interest rate and $...
0
votes
0answers
24 views

Financial Appraisal - Discounting

Suppose we were given a discounted cost of $50,000, which has been generated using a discount factor of 3.5% and a time-horizon of 45 years. Is it possible to obtain what the non-discounted cost is ...
0
votes
0answers
21 views

Variable life insurance payout

How do I calculate the payout required at a given time such that my wife can replace my income with a combination of investment returns and withdrawals such that it has dwindled to zero by the end of ...
1
vote
1answer
48 views

Financial Maths Question: How to calculate the duration of a cash flow

The question reads: A firm has liabilities as follows: £2,910 at time t = 0 and £7,501 at time t = 4 (time is measured in years). On the asset side the firm has two payments, each for £5,000, at time ...
1
vote
2answers
63 views

Pesky Bond Pricing Question involving Redemption Yield

Consider a bond with face value £100 with semi-annual coupons at a rate 3% per annum and redeemable at par in ten years. If the bond is to produce a gross redemption yield of 3.5% per annum, what is ...
1
vote
2answers
54 views

Finding the annual yield of an investment with a changing interest rate

Let's say that there was a variable annual effective interest rate on a capital invested for two years that followed these changes: i(t) = 2%, t ∈ [0, 0.5), 3%, t ∈ [0.5, 1.5), 0.5%, t ∈ [1.5, 2] ...
1
vote
1answer
91 views

Annuities over Non-Integer Periods/ Loan Repayment Question

Let's say a loan of £1000 was to be repaid by payments of £100 at the end of each quarter, and a smaller final payment made one quarter after the last regular payment. If the interest rate is 10% p.a, ...

1
2 3 4 5
10