Question - Annuity

Q: Tom will retire 20 years from today and has $200,000 in his retirement account. He believes he will need $40,000 at the beginning of each year for 20 years of retirement, with the first withdrawal on the day he retires. Tom assumes his investment account will return 7%. the amount he needs to deposit at the beginning of this year and each of the next 19 years is closest to: a. $6,500 b. $7,300 c. $7,800 Answer: B how can i calculate that? Plz, help me!

Draw a timeline.

You need to compute the amount that Tom needs in his account 20 years from today to finance the annuity; that’s the present value of the annuity (due). Then you need to compute that payment (also due) that will get him from $200,000 to that PV(annuity) in 20 years.

Good contribution.

I wish to pick up.

If I leave my Calculator in END-Mode, the PV of the retirement annuity is at t = 19, correct? To discount it back to t = 0 with END-Mode I put N = 18 and PV = -$200.000 resulting in a wrong answer for the payments during the depositing period.

How do I apply the END-Mode Alternative correctly? My guess is, I messed up with N?!

Can you advise.

Thanks again.



First you need to calculate the amount needed at retirement t=20 in BGN mode, since its annuity due.

N=20; FV=0; I/Y=7; PMT=40,000; CPT PV=? solve that with calculator

Then you need to calculate the required deposit at t=0 to t=19, Stay in BGN mode

PV=-200,000; N=20; I/Y=7; FV= PV you calculated in step one; CPT PMT this should be your answer.

I would also like to note if you don’t want to use BGN mode and use END mode you need to use t=19 and multiply your answer by (1+I/Y) that’s how annuity due and ordinary annutity are connected. Annuity due= Ordinary annutiy*(1+I/Y)

And a reminder if you use BGN mode don’t forget to change it to END mode ever time you use it.

If you have further questions feel free to ask.

Have a nice a day!

I did the following:

End of Year 20 to End of Year 39 (which are beginning of Y21 and Y40 respectively), the man needs 40,000 dollars each year. The calculation:

With BGN mode: N=20; FV=0; I/Y=7; PMT=40,000; CPT PV=?

With END mode: N=19; FV=0; I/Y=7; PMT=40,000; CPT PV=? + 40,000 (the first 40k which is in beginning of year 21 already and must not be discounted, it is like the “today” money, it is in the date we want). Note: Remember that the PV calculation is with negative sign, so you will need to deduct the (-40,000).

Your PV at the end of year 20 (beginning of year 21) is (-)453,423.8 dollars.

This amount now is the FV of your deposit timeline which has 20 years. But be careful because this is an annuity due again, the first payment (deposit) is done at year 0 and the last deposit is done at the end of year 19 (or beginning of year 20). This tally with withdrawal schedule, it starts at end of year 20 (or beginning of year 21).

The calculation: You need to set your calculator with BGN mode.

With BGN mode: N=20 ; FV= (-)453,423.8 ; I/Y=7 ; PV= 200,000; PMT= ?

PMT = (-)7,306.8

If you do not set the BGN mode, the calculation is done with END mode and N=20, so you will get 7,818.3 which would be answer c). To avoid this, you can use END mode but set N=19 at the previous calculation. The answer will not be the same, but will be approximated already with a value of PMT = (-)7,220.1 and you can choose b) I think.

This time the calculation of BGN / END changing the value of N will not be the same because the initial capital of 200,000 changes everything, so be careful with this kind of problems.



This guy already has $200,000 in the retirement account which @t=19 becomes

200,000 x (1.07)^19 = $723,305.51

The PV of all withdrawals @ t=19 => $423,760.57

The initial balance in the retirement account alone can more than fund all withdrawals

Why would he ever need to deposit money at the beginning of this year and each of the next 19 years ? In fact, he can actually draw a certain fixed amount every year, rather than deposit, such that he just meets the withdrawals

This question doesn’t make much sense and non of the answers do either (they don’t have a negative sign to indicate he can draw the money instead of deposit) Pretty sure it’s not from CFA Institute. Could you provide the source ?

Perhaps I’ve missed something. Can someone clarify where I’ve made an error in the above reasoning ?