How to understand discounting future cash flow to present cash flow?

Question

I was given this question in class.

You have the opportunity to purchase an investment that will pay $2000 at the beginning of each year for three years. You can earn 5% per year on a comparable investment

• What is the current value of this investment?

• How much will you have at the end of the third year if all cash flows are reinvested at 5%?

The calculations go something like this:

I have trouble understanding discounting future cash flows back to current cash flow. Can someone please explain why the $2000? (ie why did the summation ends at 2 and not 3?)

And also how does the timeline affect the calculations? For example what difference do "beginning" of each year and how much at the "end of the third year" make? What if it is beginning and beginning or end and end?

Answer 1

Can someone please explain why the $2000? (ie why did the summation ends at 2 and not 3?)

The summation started at 0, because the initial $2000 is paid "at the beginning of each year". Hence, the beginning of the first year is the zeroth year:

2000 / 1.05^0 + 2000 / 1.05^1 + 2000 / 1.05^2

• Since X to the power of 0 is 1, 2000 / 1.05^0 = 2000 / 1 = 2000.
• Since X to the power of 1 is X, 2000 / 1.05^1 = 2000 / 1.05.

Answer 2

Discounted cashflow has to do with what’s known as the time value of money.

Compare $100 of today’s money with $100 from 100 years ago. Would you prefer to have received $100 a century ago or $100 today? The straightforward answer is that the $100 given a century ago is more valuable today because you could have put it into the bank and earned interest, so today, a century later, that $100 is worth a lot more.

Discounted cashflow turns it the other way around and (in that example) asks what you would have needed to invest a century ago to get today’s $100. Or more typically, what a future sum is worth in today’s dollars.

Now, since you get interest, the timing of the cashflow is important.

If you get some money at the start of the year and invest it for simple interest that is paid at the end of the year, the $100 you get at the start of the year is trivially worth $100 at the start of the year. But if you got $100 at the end of the year, it would normally be worth less at the start of the year because you can invest the smaller sum to get a total of $100 at the end of the year, including interest.

The timing is significant, hence the deliberate choice of beginning/end of year determinations in the calculations.

Basically, you are asking the question, “How much would I need to invest in a something like a term-deposit today to get the equivalent of the cashflow I expect from the investment over time?”

Disclaimer: This is not financial advice. Please consult an appropriate professional before undertaking any financial action.