Present Value Timing for a Deferred Annuity
Summary
The document works through the present value of thirteen annual payments of $365, with the first payment due four years from now and a 3% interest rate. Its two-stage calculation first values the thirteen-payment annuity at the date three years from today, then discounts that amount back three years to the present. This timing produces a present value close to the stated correct choice of $3,552.
The central point is how to count periods for a deferred annuity. The ordinary annuity valuation treats its first payment as occurring one period after the valuation date, so a stream whose first payment arrives at year four is anchored at year three. Discounting from year three uses three periods, not four. The answer also suggests discounting each payment directly to today as a way to avoid off-by-one timing mistakes. The example illustrates period alignment for this specific annual cash flow and rate; it does not discuss changing rates, irregular payment intervals, or taxes and other cash flow complications.
Key ideas
- An ordinary annuity's first payment occurs one period after its valuation date.
- A first payment at year four places the annuity valuation date at year three.
- The annuity value is discounted three periods to reach today's present value.
- Discounting each payment directly can help avoid period-counting errors.
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# Time-Value of money exercise problem. Any advice on how to solve? # Time-Value of money exercise problem. Any advice on how to solve? ## Problem An investor will receive $365 at the end of each year for thirteen years. The first payment will be received four years from now. Given that the interest rate is 3%, the present value of this cash flow stream is closest to: - $3,552 (Correct solution) - $3,882 - $3,449 (I answered this) ## Solution - N = 13 - PMT = 365 - I/Y = 3 Compute PV: PV = $3881.76 This is the value of the cash flow stream three years from now. We must discount this value back to t = 0 to compute its present value. - N = 3 - I/Y = 3 - FV = -3,881.76. Compute PV: PV = 3552.36 ## Question Why is N = 3 in the bottom half of the solution? I used N = 4 because the initial payment is 4 years from now, resulting in answer C instead. Any clarification on this? Thank you very much! ## Answer by Phil H (score 0, accepted) https://quant.stackexchange.com/a/26167 Summing the PVs of the cash flows, I get 3552. I'm going to guess that your equation assumes the first payment comes at the end of the first period, not at the start of it. For this setup, that would put the "start" date of the instrument at t=3. Personally, I would just discount all the cash flows back to today, rather than once to a forward date and then again to today, to eliminate these kinds of issues. Edit to show how to count: ``` t y 0 spot 1 2 3 0 start for purposes of eqn 4 1 first payment 5 2 ... 16 13 last payment ``` In computer science there are 2 hard problems: cache invalidation, naming things and off by one errors. (Forgotten the source for that)
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