As a result of the helpful links provided by Cludgie in http://boards.fool.co.uk/Message.asp?mid=12011549 and Arb in http://boards.fool.co.uk/Message.asp?mid=12014965 (many thanks to both of them!), I have been able to do some calculations of the chances of outliving your capital that take the variability of both how long you will live and investment returns into account. (As a bit of shorthand, I'll use "failure" in the rest of this post to refer to the event of outliving your capital.)

As a preliminary warning, the calculations are still very crude, both because of limitations of the data sources I have available (more about some of those limitations towards the end of this post) and because of limits on the amount of work I'm willing to put into this investigation at present. Also, if anyone wants similar calculations done on different assumptions, expect to do them yourself! They involve feeding the FIRECalc retirement calculator http://firecalc.com/ in Arb's link with lots of different scenarios, which is a decidedly tedious job that I feel I've done my fair share of now - I'll help with how to do such calculations if people don't see it from the description below, but not with actually doing them.

The basic technique I used was:

1) Use the UK life expectancy data in Cludgie's link to split up the range of possible remaining lifespans into short subranges, using both the 'age you live to with 99%/95%/90%/75%/50%/25%/10%/5%/1%' and the 'chances of living to age 60/70/80/90/100' data points to get as many subranges as reasonably possible (except that I merged some silly small subranges into one of their adjacent subranges - e.g. the first example below would have had an 'age 99.9 to 100' subrange if I hadn't done that!).

2) For each subrange, I decided on a "years to go" figure by rounding the midpoint of the range to the nearest year and subtracting the starting age assumed in the example. (Special case: for the 100+ subrange, I chose 103 as its "midpoint" on the basis of visual inspection of the top 'tail' of the distribution.)

3) Then use the FIRECalc calculator on each of those numbers of years to get its estimate of the chance of failure if you happen to live for that number of years.

4) Finally, take a weighted average of the FIRECalc results for each subrange, weighted by the chance implied by the UK life expectancy data of your remaining lifespan lying in that subrange.

All the examples I tried used the following assumptions:

* The investor initially has an all-equity portfolio of value £250k and needs (not just "wants") to take £15k per year out of it - they'll take it from income if possible, but if there is an income shortfall, they'll take it from capital. That produces three changes from the FIRECalc defaults that I used in all calculations: on the "Start Here" tab, 'Spending' is set at 15000 and 'Portfolio' at 250000, and on the "Your Portfolio" tab, the percentage box in the 'Total market' item is set at 100%.

* In the calculations in which the start of retirement is delayed and the investor works on, they don't draw the £15k per year from the portfolio and instead save £10k per year into it until they retire. (This is easily done using FIRECalc's "Not Retired?" tab.)

* Everything is inflation-adjusted, both because I think that's the right thing to do for decades-long planning and because it's what FIRECalc does anyway.

The rationale behind the £250k capital and £15k income drawn assumptions was basically that they are realistic levels for a HYPer who is getting close to retirement, mainly or solely funded by their HYP, but at a sufficiently high drawdown rate (£15k/£250k = 6%) to probably require at least a bit of capital drawdown. That's basically a level I regarded as decidedly risky, but likely to be fixed by a few more years of saving - the point of the exercise being to confirm whether / when that is the case.

As regards the 'all-equity' part, by the way, that's because I'm looking at it from the point of view of the HYP, not necessarily the investor's entire financial situation. For example, I don't really care whether the HYP is the only income source they've got, or whether they've also got a state pension of £5k per year and need £20k per year in total, or any of the many other scenarios that lead to needing £15k per year from the HYP - just that there is that need. And if anyone wants to do more complex calculations involving strategies that do things like rebalancing between equities and bonds, feel free to use the FIRECalc facilities or other methods to try out such things!

My initial example was based on a UK male aged 55 considering immediate retirement. Using the data from http://www.wolframalpha.com/input/?i=life+expectancy+uk+male... to obtain the age subranges, the calculation went:

Age range      Chance   Years   Failure
                        to go   chance
---------------------------------------
55 to 56.7      1.0%      1       0.0%
56.7 to 60      2.4%      3       0.0%
60 to 61.8      1.6%      6       0.0%
61.8 to 66      5.0%      9       0.0%
66 to 70        6.5%     13       2.4%
70 to 73.9      8.5%     17      17.1%
73.9 to 80     19.2%     22      28.0%
80 to 81.5      5.8%     26      32.5%
81.5 to 87.9   25.0%     30      40.9%
87.9 to 90      7.0%     34      46.2%
90 to 92.8      8.0%     36      51.0%
92.8 to 95.3    5.0%     39      54.5%
95.3 to 100     4.0%     43      57.7%
100+            1.0%     48      55.4%
---------------------------------------
Overall chance of failure:       32.0%
(= average of "Failure chance",
 weighted by "Chance")

As I expected, pretty risky - nearly a 1 in 3 chance of running out of money before the investor dies. I wouldn't take that risk unless I absolutely had to - which in turn says that at age 55, planning to use even a little bit of capital drawdown is likely to be too risky.

What happens if the investor works on a few more years? Repeating the above calculation with the date on the "Not Retired?" tab set to 2011, 2012, etc, produced:

Overall chance of failure on immediate retirement:    32.0%
Overall chance of failure after working 1 more year:  25.4%
Overall chance of failure after working 2 more years: 18.3%
Overall chance of failure after working 3 more years: 13.4%
Overall chance of failure after working 4 more years:  7.8%
Overall chance of failure after working 5 more years:  3.7%
Overall chance of failure after working 6 more years:  2.4%
Overall chance of failure after working 7 more years:  0.7%
Overall chance of failure after working 8 more years:  0.0%

Until the end, each extra year of working and saving produces a significant improvement in the risk. Assuming that data actually applied to me (i.e. in particular, that I thought it a plausible distribution for my remaining life expectancy), I would probably be thinking in terms of working another 5 years - enough to get the chance of failure pretty low (below 5%) without going over the top about it. But I can certainly imagine some variation on either side of that depending on just how risk-averse the investor is.

I also used FIRECalc to calculate the average wealth at retirement after each of those numbers of years, and how big a drawdown yield £15k is of that amount:

Average wealth on immediate retirement:    £250k (6.0% drawdown yield)
Average wealth after working 1 more year:  £280k (5.4% drawdown yield)
Average wealth after working 2 more years: £313k (4.8% drawdown yield)
Average wealth after working 3 more years: £347k (4.3% drawdown yield)
Average wealth after working 4 more years: £385k (3.9% drawdown yield)
Average wealth after working 5 more years: £425k (3.5% drawdown yield)
Average wealth after working 6 more years: £467k (3.2% drawdown yield)
Average wealth after working 7 more years: £514k (2.9% drawdown yield)
Average wealth after working 8 more years: £565k (2.7% drawdown yield)

The '5 more years' point at which the chance of failure has dropped below 5% therefore corresponds roughly to the point at which the rate of drawdown I require is about 3.5%. That's very roughly the market average dividend yield in the UK; as a HYP's yield will typically be a percentage point or two above it, it says far from wanting to use capital drawdown, the investor wants to draw down only part of the dividend income, leaving a safety margin within the portfolio. (In this example, that safety margin gets reinvested in more shares for the HYP - but that's basically just a consequence of the way the calculations are done: basically, they assume capital drawdown is done if the required income exceeds the available dividend income, which has the natural consequence that they also assume that 'negative capital drawdown' (i.e. reinvestment) happens if the available dividend income exceeds the required income.)

Part of the reason for that is that the 55-year-old male investor I've assumed in that example has a life expectancy of 80.47 years, i.e. about 25 more years, and could easily live quite a lot longer than that - periods of up to 40 years are within the reasonably-likely range, and 50 or even more years are possible. The amount of capital drawdown that is safely possible on such long-term investments is very limited even for entirely predictable investments, and is easily exceeded by the amount of leeway you need to allow for the not-so-predictable returns of shares.

So what happens for older investors? I tried a 65-year-old investor with otherwise the same assumptions, and got the following results:

Overall chance of failure on immediate retirement:    18.0%
Overall chance of failure after working 1 more year:  13.3%
Overall chance of failure after working 2 more years:  8.8%
Overall chance of failure after working 3 more years:  5.6%
Overall chance of failure after working 4 more years:  2.8%
Overall chance of failure after working 5 more years:  1.2%
Overall chance of failure after working 6 more years:  0.5%
Overall chance of failure after working 7 more years:  0.1%
Overall chance of failure after working 8 more years:  0.0%

A similar sort of picture: rather a high risk of failure for immediate retirement and a few more years of working and saving would look advisable to me. But the risk of failure is not much more than half of what it was at age 55, making it rather less bad if you have to retire; the number of extra years of work needed is lower (just over 3 years to get down to a 5% risk of failure rather than just under 5 years); consequently, you can take about 4.2% income from the portfolio rather than about 3.6% if you target a 5% chance of failure (in round-number terms, that means that you can run a HYP with a dividend yield of 5-5.5% at about a 25% income safety margin rather than about 50%). These are all consequences of the fact that the life expectancy of the 65-year-old investor is 82.35 years, i.e. about 17.5 more years compared with the 55-year-old's 25.5 years, but still with considerable variation above that possible - up to 30 years reasonably likely, 40 or even more not beyond the realms of possibility.

And finally, I did a 75-year-old investor on the same assumptions:

Overall chance of failure on immediate retirement:     6.1%
Overall chance of failure after working 1 more year:   3.8%
Overall chance of failure after working 2 more years:  2.0%
Overall chance of failure after working 3 more years:  0.9%
Overall chance of failure after working 4 more years:  0.3%
Overall chance of failure after working 5 more years:  0.1%
Overall chance of failure after working 6 more years:  0.0%

If I were in that unfortunate position of still working at age 75 and only having built up my investments to the point where I need to draw down a 6% yield from them, I would undoubtedly retire immediately. The chance of failure is already down near the 5% mark and I don't have many expected remaining years of life - life expectancy 85.49, so a bit over 10. There's still some risk due to another 10 beyond that being in the reasonably-likely range and 30 or even more conceivable, but it would be relatively unimportant!

Furthermore, there might be a risk reduction method available to me that was not available at earlier ages, because by age 75, the top index-linked annuity rates available for a single male have gone over 6%. Depending on the tax situation, this might allow part of the capital to be used to produce a reliable income at over the required rate, albeit at the cost of losing that part of the capital, while the rest is kept in order to have capital available and with a correspondingly lower income requirement. It's not likely to do the job in any case where the capital and income is likely to be wanted after death, though, and in particular not for married males.

Finally, some (not by any means all!) of the limitations of the above calculations:

* The FIRECalc calculations use a US index tracker as the equity portfolio, not a HYP. I've gone as far as I can towards a HYP by making the portfolio 100% that index tracker rather than the mixture of equities and fixed-interest recommended by the calculator, but it still isn't all that close to a HYP! As a result, I would only really treat the results of the calculations as ballpark numbers, not as anything precise - but equally, I don't think a huge amount of precision is required in such calculations: whether you're prepared to take a 5%ish risk of running out of money is far more important than whether the risk is actually 4%, 5% or 6%. (I should add that the choice of 100% equities is purely an attempt to make the calculations as close as possible to the subject matter of this board, and not a recommendation that 100% HYP is the way to go. I don't have a recommendation about what the overall strategy of an investor should be - and if I did, it would be something for e.g. the Investment Strategies board and not this one.)

* They don't take individual variations in life expectancy into account - as the most notable example, they're all on male investors and not females, and it's well-known that female life expectancies are higher than those of males (something I intend to complain about vigorously when I find whoever's in charge... ;-) ). I do think the calculations give some sort of guidance about at least some other situations - for example, the 'UK male aged 75' and 'UK female aged 78' cases have very similar distributions for their remaining lifespans, so the calculations above for the 75-year-old male apply just about equally well (or badly!) to a 78-year-old female. I do also recognise though that there are situations with very different life expectancy distributions from the three used above - all I can say about such situations is that if you can get hold of reasonable data about the distribution, it's possible to do calculations along the above lines to get results that apply to it.

* The FIRECalc calculations are based on historical US real returns. If you believe that future real returns from equities are likely to be significantly lower, you want to be correspondingly more conservative.

* Life expectancies are increasing at a noticeable rate and it's possible that people's real life expectancies are above the estimates used in the calculations - something that you might want to take into account and err a bit on the conservative side of what the above calculations suggest.

Gengulphus