Pension Problem

[FINHELP]

The question I need to figure out is:

Exactly 15 years from now Mr. J.R. Smith will start receiving a pension of $20,000 per year. The pension payments will continue for 10 years after retirement. How much is the pension worth now if the money is worth 10 percent per year?

Answer: $29,419

I need to figure out how to get from one to the other. I have a lot of formulas listed and examples in my book - but nothing specific to this question so I am not sure which one to use? Can anyone give me a jump start?

 

[tkhunny]

Basic Principles. Forget ALL formulas. Learn where they come from.

i = 0.10 -- This is given
v = 1/(1+i) = 1/1.10 -- Annual Discount Factor

That's all you need. Now write it down.

\(\displaystyle 20000\cdot\left(v^{15} + v^{16} + v^{17} + ... + v^{24}\right)\) -- Done!

If you need a little algebra lesson...

\(\displaystyle 20000\cdot\left(v^{15} + v^{16} + v^{17} + ... + v^{24}\right) = 20000\cdot\left(v^{15})*(1 + v + v^{2} + ... + v^{9}\right) = 20000\cdot\left(v^{15}\right)\cdot\left[\frac{1-v^{10}}{1-v}\right]\)

With a little arithmetic:

\(\displaystyle 1-v = 1 - \frac{1}{1+i} = \frac{1+i-1}{1+i} = \frac{i}{1+i} = i\cdot v = d\)

\(\displaystyle 20000\cdot\left(v^{15}\right)\cdot\left[\frac{1-v^{10}}{d}\right] = 20000\cdot\left(v^{15}\right)\cdot\left[\frac{1-v^{10}}{i\cdot v}\right] = 20000\cdot\left(v^{14}\right)\cdot\left[\frac{1-v^{10}}{i}\right]\)


Which formula? The one you invent that is exactly correct and you NEVER have to winder if you picked the right formula. Get good at it and you will not have a speed problem some may suggest.

By the way, I'm not really seing the $29,419. You'll have to show me what assumptions were made to get that one. Perhaps the pension pays monthly or something like that?

 

[FINHELP]

Thanks

 

[TchrWill]

The question I need to figure out is:

Exactly 15 years from now Mr. J.R. Smith will start receiving a pension of $20,000 per year. The pension payments will continue for 10 years after retirement. How much is the pension worth now if the money is worth 10 percent per year?

Answer: $29,419

I need to figure out how to get from one to the other. I have a lot of formulas listed and examples in my book - but nothing specific to this question so I am not sure which one to use? Can anyone give me a jump start?

The present value of an ordinary annuity is the sum of the present values of the future periodic payments at the point in time one period before the first payment.

.................................P = R[1 - (1 + i)^(-n)]/i for the 10 year period

where P is the present value, R is the periodic payment, i is the periodic interest rate in decimal form = %Int./(100) and n is the number of interest bearing periods.


20,000 = R[1.1^-10]/.1 making R = $122,891 the value at the start of payments

In its most basic use, if P is an amount deposited into an account paying a periodic interest, then Sn is the final compounded amount accumulated where

..........................Sn = P(1+i)^n for the 15 year period

where Sn is the accumulated sum of P dollars deposited, i is the periodic interest rate in decimal form = %Int./(100m) and n is the number of interest bearing periods.

122,891 = P/(1.1)^15 making P = $29,419

Looks like you are right on the money

 

[Denis]

YEAR PAYMENT INTEREST BALANCE
  0                   29,419
  1            2,942  32,361
....
 14           10,156  111,718
 15           11,172  122,890
 16 -20,000   12,289  115,179
....
 24 -20,000    3,471   18,182
 25 -20,000    1,818       00

Get it?

 

[tkhunny]

I just don't like this language at all. Who said it was an "Ordinary Annuity"? I realize this reproduces the suggested result, but it is NOT in the problem statement.

I am certain that I would lose in court if I wrote a contract that stated "Exactly 15 years from now Mr. J.R. Smith will start receiving a pension..." and I later told Mr. Smith that he didn't actually get anything until the END of year 15, viz. Exactly 16 years.

I thoroughly disagree with $29,419 as the problem is stated.

My Views. I welcome others'.

 

[Denis]

Exactly 15 years from now Mr. J.R. Smith will start receiving a pension of $20,000 per year.

I guess better wording would be:
Exactly 15 years from now, Mr. J.R. Smith will have a pension plan that will permit 10 payments
of $20,000 per year, 1st due end of 15th year...

 

[TchrWill]

I just don't like this language at all. Who said it was an "Ordinary Annuity"? I realize this reproduces the suggested result, but it is NOT in the problem statement.
I am certain that I would lose in court if I wrote a contract that stated "Exactly 15 years from now Mr. J.R. Smith will start receiving a pension..." and I later told Mr. Smith that he didn't actually get anything until the END of year 15, viz. Exactly 16 years.
I thoroughly disagree with $29,419 as the problem is stated.
My Views. I welcome others'.

You are right tkhunny, in that the problem statement did not clearly state that an Ordinary Annuity was involved. Clearly, if Mr. Smith put $X into an account for 15 years, paying 10%, compounded annually it would receive 15 interest payments and accumulate to S = P(1 + .1)^15.

The same day, or the next day, of the end of the 15 year period, Mr Smith begins receiving a check for $20,000 and does so for the next 10 years, meaning, or implying,that he receives 11 checks, meaning that the present value of the pension derives from P = R [1 - [1 - (1 + .1)^(-10)]/i.

Therefore, the value of the pension account on the first day of the 10 year period, the day of the first check, followed by 10 more checks to the end of the 10th year, is $122,891 + 20,000 = $142,891.

The present value of this amount 15 years prior to the first check being issued is therefore, P = 142,891/(1.1)^15 = $34,207.

My views.
All others graciously accepted.