Consider the following mean-square differential equation, driven by a WSS random process with psd The differential equation is subject to the initial condition , where the random variable has zero-mean, variance 5, and….

## Why is the mortality model for Kavis Alpha IV unsuitable for planet Earth?

A lady age 20 wishes to accumulate a fund for retirement by depositing an equal amount at the end of each year for 20 years. Starting at age 60 she will withdraw 50,000 at the beginning of each year for 20 years. If the annual effective interest rate is 10%, how large should each deposit be?

- A bank loans you 50,000 at a nominal interest rate of 4.8% per year, compounded monthly. You agree to repay the loan with level payments at the end of the month for the next 15 years. How much is your monthly payment?

- A family wishes to accumulate 100,000 in a college education fund by the end of 20 years. The fund credits a nominal interest rate of 6% per year, compounded semiannually. They plan to deposit 1000 into the fund at the beginning of every 6 months for the first 10 years and 1000 + X at the beginning of every 6 months for the second 10 years. What is the value of X?

- Fund Alpha earns annual effective 15% interest for the first 5 years and annual effective 6% interest thereafter. Svetlana makes end-of-year deposits of 100 into Fund Alpha for 13 years.

Fund Beta earns annual effective i.Alice makes end-of-year deposits of 100 into Fund Beta for 13 years.

At the end of 13 years the two funds have the same balance.What is the value of i?

- A loan is being repaid by 300 at the end of each year for 25 years. With the 10
^{th}payment, the borrower pays an extra 1000, and then repays the balance over 10 years with a revised annual payment. The annual effective rate of interest is 8%. What is the amount of the revised annual payment?

*For problems 6-9, answer the questions using standard annuity symbols. Do not perform any calculations.*

- You plan to deposit money into a retirement account over the next 10 years. The account credits interest at a nominal interest rate of 6% per year, compounded semiannually. If you deposit 2000 into the account at the end of each month during the first 5 years and 3000 at the end of each month during the second 5 years, what will your balance be at the end of 10 years?

- A family advertises its house for rent for exactly one year at 1000/month, payable at the beginning of each month. A prospective renter prefers to pay the entire rent in a single payment to be made in the middle of the year. If money is worth 18% per year, compounded monthly, what is the amount of the single payment?

- Stacey plans to open a savings account on January 1, 2021 with a deposit of 30,000 and immediately begin withdrawing money from it continuously at the rate of 3,000 per year. The account will credit interest at 4% per year, compounded semiannually. Stacey’s father plans to deposit 2,000 to her account on January 1 of each of the next 5 years (2021-2025) . How much money will Stacey have in her account 5 years after she opens it?

- On July 1, Dimas wins a lottery. His prize is a 20-year annuity-due with payments of 1000 each July 1st and payments of 2000 each January 1. If the payments are all left to accumulate in a new account earning interest at an annual effective interest rate of 5%, what is the accumulated value exactly 20 years after Dimas won the lottery?

- On January 1, an investment account is worth 100,000. On May 1, the account value has increased to 112,000, and 30,000 of new principal is deposited. On November 1, the account value has decreased to 125,000, and 42,000 is withdrawn. On January 1 of the following year, the account value is 100,000. (a) Calculate the investor’s time-weighted yield rate for the year. (b) Write, but do not solve, an equation for finding the investor’s dollar-weighted yield rate. Would you expect the dollar-weighted yield rate to be higher than the time-weighed yield rate? Give a reason for your answer.

- A 30-year loan is to be repaid with equal payments at the end of each year. The interest rate on the loan is annual effective 7%. The lender deposits the loan payments into an account that credits annual effective 6% interest. The balance of the account immediately after the last deposit is 48,420. Find the amount of the loan.

- A loan of 3,835 is scheduled to be repaid by equal payments of 300 at the end of each year for 25 years. The interest rate on the loan is annual effective 6%. Calculate the amount of interest in the second payment.

- You are given that

Calculate (a) p_{x+3},(b) p_{x+2},(c) _{1|2}q_{x}.

- Given that
_{t}p_{99}= (1/2)^{t}, t = 0, calculate each of the following:

- The curtate expectation of life at age 99.
- The approximation to the complete expectation of life at age 99 that is most frequently applied in practice.
- The exact complete expectation of life at age 99.

- For the populations of white mice and gray mice in Hanes Hall, you are given:

(i) Mortality for white mice follows l_{x} = (6-x)^{2}, 0 = x = 6.

(ii) At all ages, the force of mortality for gray mice is twice the force of mortality for white mice.

- In a
**deterministi**c survival model, what do we assume about the number of 36 newly born WHITE mice who will survive to age 3? - In a
**stochastic**survival model, what is the probability distribution of the number of 36 newly born WHITE mice who will survive to age 3? Give the name of the distribution and its parameters, assuming independent future lifetimes. - Calculate the probability that a newly born GRAY mouse will survive to her first birthday.

- For star travelers Tuvok and Jarok:

Age Resident of Mortality model

Tuvok 79 Kavis Alpha IV DeMoivre’s Law with ? = 99

Jarok 79 Cheron m(x) = (1+x)^{-1}, x>0

(a) Calculate Tuvok’s complete expectation of life.

(b) What is the probability that Jarok will survive to at least age 89?

(c) Why is the mortality model for Kavis Alpha IV unsuitable for planet Earth? Is the model for Cheron suitable or unsuitable for Earth and why?

- 1000 new members of the XC Club, all age 90, want to set up a fund that will be used to pay 100 to each one of them who is still alive 2 years later. If the fund earns annual effective 25% interest, calculate how much premium each new club member should pay into the fund at age 90 so that, if the number of actual deaths is equal to the number of expected deaths, the fund will be
**exactly**depleted by the payments to the survivors at age 92. You have decided that p_{90}= 0.5, p_{91}= 0.4, p_{92}= 0.3, and p_{93}= 0 are reasonable survival probabilities for this group.

- Jack and Jill are independent lives, both age 32.

Jack was selected at age 27 and Jill was selected at age 32.

Use the following select-and-ultimate mortality table to find the probability that Jack and Jill will both survive to age 35.

[x] | (1) q _{[x]} |
(2) q _{[x]+1} |
(3) q _{x+2} |
(4) x + 2 |

30 | 0.11 | 0.13 | 0.15 | 32 |

31 | 0.12 | 0.14 | 0.16 | 33 |

32 | 0.13 | 0.15 | 0.17 | 34 |

33 | 0.14 | 0.16 | 0.18 | 35 |

34 | 0.15 | 0.17 | 0.19 | 36 |