Time value of money question. Confused about t

Two years from now, a client will receive the first of three annual payments of \$20,000 from a small business project. If she can earn 9 percent annually on her investments and plans to retire in six years, how much will the three business project payments be worth at the time of her retirement?

Fairly simple question but the wording on these gets me confused. The time never seemed to match up with what the question says.

I solved this using FV of N=3 to get 65,562. I then compounded another year to get 71,462.58. The book gives an answer of compounding the amount for another 2 years to get 77,894.21.

I just can’t understand with the wording of the question how that’s correct.

Starting at t=0, in 2 years the client will receive 20k. Meaning at the beginning of t=2, 20k will start compounding as an annuity. So from 2 to 3, 3 to 4, 4 to 5. We use those 3 years to compound the 20k deposits. At beginning of t=5, we should be left with 65,562. In 6 years from t = 0 is the time she would like to retire. Therefore, from t= 5 to t=6 we are looking to compound the money 1 more year, since In 6 years from t = 0 she is retiring.

Can anybody explain this to me like I’m 5 or something because it really does not make sense to me. For the most part I’m grasping everything but this does not make sense. Thanks

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Draw a timeline.

She gets a payment at time t = 2, at time t = 3, and at time t = 4.

She retires at time t = 6.

Simplify the complicated side; don't complify the simplicated side.

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So her payment at time 4 is compounded until beginning of time 5 i thought? Giving it 1 more year till time 6. 2 to 3, 3 to 4, 4 to 5 is 3 years. 5 to 6 is 1 year. The question is worded that 2 years from today, she receives the 20k. in 6 years she retires. That gives a total of 4 years investing, hence the first 3 then the additional 1.

Not sure if you understand what I’m trying to say but it takes a full year for everything to compound. So if she receives the payment at t = 2, it isn’t until t = 3 that it’s compounded. Same as payments at 3, and 4.

Maybe I’m thinking about it wrong, but hopefully someone can explain it, I’ve done the timeline.

Logically though if you receive 20k to invest in 2 years from today, and in 6 years from today you want to retire, you have 4 years to compound that money. Must just be the wording of the question confusing me.

Einzakin wrote:

Maybe I’m thinking about it wrong, but hopefully someone can explain it, I’ve done the timeline.

Logically though if you receive 20k to invest in 2 years from today, and in 6 years from today you want to retire, you have 4 years to compound that money. Must just be the wording of the question confusing me.

You are on the right track.  The payment received at time 2 has 4 years to compound at 9% until retirement at time 6 (accumulated value is \$28,231.64).  If you follow a similar process for the payments received at times 3 and 4 and compound each to time 6, the accumulated values are \$25,900.48 and \$23,762, respectively.  Added all up, the total account value is \$77,894.20.

Like s2000magician suggested, a timeline helps you visualize the payment stream and by how many years to compound each payment.

“Mmmmmm, something…” - H. Simpson

You could solve the first part as a 3-period ordinary annuity (i.e. in END mode).  For an ordinary annuity, the first payment occurs one period before the PV (if you were calculating it, and the FV is “as of” the same time as the last payment).  Of you timeline these, you’ll see that the last payment occurs at t=4.  Therefore, the FV of the annuity is as of time 4. So compound it for two additional years.

Alternately, you could calculate the PV of the annuity. If you do it as an ordinary annuity, the PV would be as of time 1 (i.e. one period BEFORE the first payment. So you’d compound that initial PV forward for 4 years.

You keep using that word.  I do not think it means what you think it means.