Help with annuities!

Jan 2010
17
0
I'm having trouble understanding this question.
any help is much appreciated

Today is your 30th birthday. You intend to retire at age 60 and you want to be able to receive a 20 year, $100,000 annuity with the first payment to be received on your sixty-first birthday. You would like to save enough money over the next 15 years to achieve your objective; that is, you want to accumulate the necessary funds by your forty-fifth birthday.

(a) If you expect your investments to earn 12 percent per year over the next 15 years and 10 percent per year thereafter, how much must you accumulate by the time you reach age 45?

(b) What equal, annual amount must you save at the end of each of the next 15 years to achieve your objective, assuming that you currently have $10,000 available to meet your goal? Assume the conditions stated in Part a
 
May 2010
1,034
272
I'll assume youve been taught standard actuarial notation (ie \(\displaystyle v=\frac{1}{1+i}\) among other things):


The scenario you are given is:
Step 0: You have $10k
Step 1: you save $P per year for 15 years (interest 12%) in arrear (15 payments total). At the end of this you are 45 years old.
Step 2: At age 45, you stop paying into your savings plan. The existing value grows at 10%
Step 3: At age 60, you use the accumulated value of your plan to buy a $100k annuity, payable in arrear for 20 years.

The easiest way to work through these problems is to work backwards

part (a):
Think of the question as:
"How much money must you have on your 45th birthday, in order to pay for $100k per year for 20 years, with the first payment delayed for 16 years".
First note: The interest rate at all times from age 45 is 10%, so use i=0.1 when working this out


Start by working out the lump sum required at age 60 to pay for the annuity. This is just a $100k annuity for 20 years, payable in arrear:

\(\displaystyle Value~~Z = \frac{1-v^{20}}{i} \times 100000\)


Now, work out what you need at age 45 in order to have a lump sum of S at time 60:
Value of Z in 15 years = : \(\displaystyle v^{15}Z\)

This is the answer to the question.
 
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May 2010
1,034
272
part b

The way to think about this is:
"How much money do i ahve to save each month for it to accumulate to the value i worked out in part a, at age 45".

The question tells you that the relevant interest rate is 12%, and we will be making annual savings in arrear.

The accumulated value of £1 paid in arrear for 15 years is:

\(\displaystyle S_{15} = \frac{(1+i)^{15} -1}{i}\)

So, define:
K = the value of the benefit at age 45 (this is the answer of part a)
P = the amount you save each year

You must solve
\(\displaystyle 10000(1+i)^{15} + P \times S_{15} = K\)
 
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Dec 2007
3,184
558
Ottawa, Canada
The scenario you are given is:
Step 1: you save $P per year for 15 years (interest 12%). At the end of this you are 45 years old.
Step 2: At age 45, you stop paying into your savings plan. The existing value grows at 10%
Step 3: At age 60, you use the accumulated value of your plan to buy a $100k annuity, payable in arrear for 20 years.
Step 1: you deposit $10,000 (interest 12%) at age 30.
Step 2: you save $P per year for 15 years (interest 12%) from age 31 to age 45.
Step 3: At age 45, you stop paying into your savings plan. The existing value grows at 10%
Step 4: At age 60, you use the accumulated value of your plan to buy a $100k annuity, payable from age 61 to age 80, at 10%
 
Dec 2007
3,184
558
Ottawa, Canada
So, working backward:

1: calculate value at age 60 required for annuity.
A = 100000(1 - 1/1.10^20) / .10 = 851,356.37

2: calculate value at age 45 required to reach A.
B = A / 1.10^15 = 203,807.95

3: calculate accumulation to age 45 of initial 10,000.
C = 10000(1.12)^15 = 54,735.66

4: calculate required payment from age 31 to 45.
P = .12(B - C) / (1.12^15 - 1) = 3,998.73
 
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