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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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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25 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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59 views

Why would I divide these two equations to solve for i?

I have the following two equations representing a longer actuarial practice question. I properly set up the equations, but am stumped on how to solve them. The book says to divide the first by the ...
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37 views

Insurance claims Poisson problem derving expected value and variance

If I have that claims arrive at an insurance company according to a Poisson process $\{N(t) : t \ge 0\}$ at a rate $\lambda > 0$ and $X_i$ denotes the claim size of the $ith$ claim. I assume that ...
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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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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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20 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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36 views

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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load repayment with increasing annuity

Question is : A loan of $10,000 is to be repaid in ten years by payments at the end of each year. The payments grow by 3% per year, so if the first payment is P, then the second payment is 1.03P and ...
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71 views

Recommended Textbook to prepare for Exam P

Could anyone recommend a good textbook to prepare for the actuarial Exam P? I'm looking for a textbook that explains concepts clearly, provides detailed proofs, and gives difficult questions that ...
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40 views

About annuity immediate calculation

Q1: Find $s_{12}$ if the nominal interest rate payable monthly is $5%$ per annum. What I have done is: $$i^{(12)}=0.05$$ $$1+i=(1+i^{(12)}/12)^{12}$$ which leads to $$i=0.0512$$ ...
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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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60 views

Isolating for i

So this might seem a bit fundamental, but in financial math the following equation gives you the price for a bond $$ P = C \frac {1-(1+i)^{-n}} {i} + B(1+i)^{-n} $$ where $P$ is the price of the ...
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2answers
234 views

Claim from an Actuarial Textbook: limits imply the existence of mean and variance

This is from Actuarial Mathematics for Life Contingent Risks, 2nd ed., by Dickson et al. Some definitions (not directly from the book): Definitions/Notation. $T_x$ is defined to be the future ...
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30 views

Exam FM problem using force of interest. Calculate $P-Q$ [closed]

The foce of interest at time $t$ is given by $\delta_t=.01t$. $P$ is the present value of a 12 yr annuity due of $100$ payable annually. $Q$ is the present value of a 12 yr annuity immediate of ...
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2answers
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What is the probability that at least 2 professors (out of 100) pick the same course (out of 200)?

Suppose each of 100 professors in a large mathematics department picks at random one of 200 courses. What is the probability that at least two professors pick the same course? The answer given in 1 - ...
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41 views

What is the probability for a single hand of bridge to have exactly 3 Aces?

Full question from actuarial exam practice problems: The game of bridge is played by four players: north, south, east and west. Each of these players receive 13 cards. ... b) Consider a single hand ...
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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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2answers
47 views

Need help and clarification in Exam FM problem, future value.

The problem that I am working on is the following. Jim began saving money for this retirement by making monthly deposits of 200 into a fund earning 6% interest compounded monthly. The first ...
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1answer
55 views

Exam FM problem. What does this problem mean?

Danny borrows 4,000 from Genevive at an annual effective rate of interest i. He agrees to pay back 4,000 after 14 years and 5,440.32 after another 14 years. Danny repays the ...
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2answers
27 views

Methods of solving this exam FM problem with geometric-investments.

The problem I am working on is as follows. Matthew makes a series of payments at the beginning of each year for $20$ years. The first payment is $100$. Each subsequent payment through the tenth ...
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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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1answer
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Calculating the yield of a bond purchased at a lower price.

I am working on the following problem. A 10 year bond bearing a $7\%$ coupon rate payable semiannually is bought to yield $5\%$ semiannually. The bond is redeemable at par. If the bond is ...
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Algebraic representation of how values are calculated in TI BA II+?

In order to understand how the BA II+ works, I would like to know the algebraic representation of it. For example, for the problem below Present value of an annual coupon bond that pays 80 per ...
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45 views

Exam FM problem. Bonds

the following problem is what I am working on. Suzan can buy a zero coupon bond that will pay $1000$ at the end of $12$ years and is currently selling for $624.60$. Instead she purchases a $6\%$ ...
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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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82 views

Exam FM Portofolio problem: Using Macaulay Duration

The following problem is what I am working on and I cannot solve it. Under the current market conditions Bond 1 has a price (per 100 of face amount) of $P_1=88.35$ and a Macaulay duration of ...
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36 views

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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41 views

Construct a strategy to profit: Problem involving term structure and interest rates.

I am currently studying about term structure and interest rates such as forward rates, swap rates, etc... The following problem seems like an actual actuarial problem that I might see in the future ...
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1answer
40 views

Find the annual yield rate. Exam FM problem.

I'm trying to solve for the following problem and I cannot get the right #. You are given the spot rates at time $t=1,\ 2 \ \text{and} \ 3$ as $s_0(1)=.15,\ s_0(2)=.10,\ \text{and} \ s_0(3)=.05$ ...
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26 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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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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29 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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31 views

Dollar weighted return. Formula or definition?

I was learning dollar-weighted return and I was a bit puzzled by the following and I would like to have some advice. I understand that it's basically the internal rate return, but using simple ...
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69 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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20 views

Finding out the minimum yield of a premium bond with a different redemption fee. ($F=100, r^{(2)}=10\%, i^{(2)}=8\%, C=110$)

I am working on a specific problem regarding price of bonds and it is the following. A 10% bond with face amount $F=100$ is callable on any coupon date from $t=15.5$ years after issue up to the ...
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43 views

Why would an investor want the minimum yield?

I am puzzled by a problem related to bonds. When a bond is callable, the purchase price (present value of the bond) can fluctuate and I also understand the difference when the bond is purchased at a ...
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28 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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2answers
74 views

Zero-coupon vs. $10\%$ coupon problem

I am working on Bonds and I am having trouble solving this problem. A zero-coupon bond pays no coupons and only pays a redemption amount at the time the bond matures. Greta can buy a zero-coupon ...
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1answer
43 views

Bond prices and how to compare

I have a couple of basic questions regarding bonds that I would like to ask and the following problem is what I used. Find the "price" of the following bonds "redeemable at par". Let $F=100$ be ...
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55 views

Ranking $ d, i, d^{(m)}, i^{(m)}, \delta$

Any actuary or anyone studying mathematics of finance out there? Please help me out. How can I prove or show that $ d< d^{(m)}< \delta< i^{(m)}<i,$ for $m > 1$. Thanks a lot !!!
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1answer
54 views

How to calculate the yield to maturity?

I am looking at an example problem in my textbook and its solution. Can someone look at this picture/ problem and its solution and tell me where they got the yield to maturity.
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1answer
37 views

What does it mean to be an equivalent repay scheme? (sinking fund vs. amortization)

I am having trouble solving the following problem which seems simple, but I cannot quite get it right. Smith can repay a loan $L=250,000$ in one of two ways 1), 30 annual payments based on ...
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$\frac{L}{a_{\overline{n}\rceil i}}(n-a_{\overline{n}\rceil i})$ 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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35 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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Loan Repayments and Yield Rates

Jose loans Martin $12000$. Martin repays the loan by paying $5000$ at the end of two years and $10000$ at the end of $4$ years. The money received at time $t=2$ is immediately reinvested at an annual ...
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1answer
58 views

Calculating the yearly payment in a loan. Amortization with unequal payments.

The following is the problem that I am working on. A 30-yr loan of $1,000$ is repaid with payments at the end of each year. Each of the first ten payments equals the amount of interest due. Each ...
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18 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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23 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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32 views

Reinvestment using $(Is)_{\overline{n}\rceil i}$… calculate $i$

I am working on the following problems which should not involve a difficult polynomial, but I tried a couple of times without success. Victor invests $300$ into a bank account at the beginning of ...