Sam and Sally
motivated by questions raised by KenM
 Sam has a \$1M portfolio. He's read all the financial literature and did a jillion Monte Carlo simulations and concludes that, at a 4% withdrawal rate, his portfolio has a 99% probability of surviving the next 40 years. Every year Sam says to himself: "There's a 99% chance of my portfolio surviving to the end of 40 years." Ten years later, Sam's portfolio is worth \$A. Sally notices that her portfolio also \$A. She hears Sam reciting: "There's a 99% chance ..." Sally starts in with: "There's a 99% chance of my portfolio surviving to the end of 30 years." Is she right?

Overheard at the local coffee shop:

Sam: I'll live for 40 years and need \$40K per year and a 4% withdrawal from my \$1M portfolio is safe so I'm happy.

Sally: I'll also need \$40K per year from my \$1M portfolio. But I'll live for just 30 years, so I'll take out 10 years worth of \$40K - that's \$400K - leaving me with \$600K. I'll spend the \$400K on fun-and-games and withdraw \$40K per year from what's left - that's the \$600K.

Sam: But you're withdrawing \$40K from a \$600K portfolio, not a \$1M portfolio!
That's a withdrawal rate of, let's see: 40/6 = 6.7%. You think this is safe? You're dreaming!

Sally: Well, suppose I take out 5 years, that's 5 x \$40K for fun & games. That leaves me with \$800K. How's that?

Sam: You'll be withdrawing \$40K from an \$800K portfolio, that's ...

Sally: I can do the math! That's 40/8 or 5%. Is that safe, for just 30 years?

Sam: I doubt it ... even for 30 years.

Sally: How many years can I take out ... to play with?

Sam: If you want \$40K per year and you take out N year's worth, that's N x \$40K, then, let's see ...

Sally: I can do the math! What's left in my portfolio is \$1M - \$40K*N and I'm withdrawing \$40K per year from this so ...

Sam: That's a withdrawal rate of 40K/(1M - 40K*N) or 40/(1000-40N) or, as a percentage, that's 4000/(1000-40N)%.

Sally: I don't know if I agree with your formula. Let me check:
For N = 10 years that's 4000/(600) = 6.7% and for N = 5 years it's 4000/(800) = 5%.
Yes, that's what I got above, so I agree.

Sam: Suppose you think x% is safe, then you'll want 4000/(1000-40N) = x so N = 25 - 1000/x meaning that ...

Sally: I can do the math! I think maybe 4.5% is safe, for my 30 years.
So I should be able to take out N = 25 - 100/4.5 = 2.8 years worth.
What! That's all? I drop dead 10 years earlier yet I only get to play with 2.8 years worth of withdrawals?
 Sam: Just think: In order that your portfolio lasts 30 years instead of 40, you can certainly expect to withdraw at a larger rate. But if you take out N years worth of withdrawals, your effective withdrawal rate, as percentage of this reduced portfolio ... this effective rate increases with N. If N is too large then your withdrawal rate won't be safe, eh? Sort of like Sally: So? Sam: So, I suggest you live for 40 years.

submitted by KenM

Overheard at the local pub:

Sam: 10 years ago I decided I'll live for 40 years and my friend Mr Monte Carlo told me that withdrawing \$40,000 per year adjusted for inflation from my \$1M portfolio would be 99% safe.

Sally: That sounds a great idea. Do you think I could do that starting now?

Sam: Well, after 10 years of withdrawals, my next one should be \$55,000 but the market isn't too good right now and my portfolio's only worth \$800,000. However I trust Mr MC and as he told me at the beginning of the 40 years that I would be 99% safe, I intend to take the full \$55K.

Sally: That sounds even better, I really like your friend Mr MC . I only expect to live for another 30 years and my portfolio's coincidentally worth the same as your current \$800,000. So I'll start withdrawing \$55,000 now and still have the same 99% safety as you for the next 30 years.

Sam: But that doesn't seem fair. My initial withdrawal rate was 4%. Yours will be 6.9%. I'd better get Mr MC to buy me a free lunch.

Note: If Sam's portfolio is one of those 99% that survived, then Sally can assume Sam's portfolio and withdrawal amount (at the 10-year point) and her portfolio is guaranteed to survive another 30 years ... and theres a 99% probability that this is the case. Getting a higher initial withdrawal rate shouldn't be surprising. Of a jillion \$1M starting portfolios, after 10 years, some may be at \$500K and some may be at \$5M and they all have identical withdrawals ... yet 99% of them survive another 30 years. However, the withdrawal rate (at the 10-year mark) can vary widely. For example a \$50K withdrawal (at the 10-year mark) might give \$50K/\$500K =10% or \$50K/\$5M = 1%.

See this this online spreadsheet to see how 30- and 40-year withdrawal rates compare.

Check out the online spreadsheet where inflation is incorporarted
and random returns are generated each time you press F9
and it's assumed that Sally takes out N years worth of Sam's withdrawals to start
... well, just try it and see.

Questions questions ...

Sam has a \$1M portfolio and withdraws at the "Monte Carlo 40-year Safe Rate" (say 4%).
Suppose there's a 99% chance of Sam's \$1M portfolio lasting 40 years.
That means that, of 1000* 40-year MC simulations, 990 will survive.
* or a jillion

If Sally jumps in at the 10-year mark, then her portfolio is identical to his (for the last 30 years)
... hence there's a 99% probability that hers is one of those 990 surviving portfolios, right?

Of the 990 surviving portfolios, there will be plenty whose withdrawal rate, at the 10-year mark (as a percentage of the 10-year portfolio value), will NOT be the "MC safe" rate for 30 years.

That poses interesting questions:

• Consider a jillion \$1M portfolios over 40-years with a 4% withdrawal rate (where 99% of them survive).
• Now look at these jillion portfolios at the 10 year mark.
• (Some are \$5M, some are \$500K and, sadly, some are \$0.)
• Look at the withdrawal amount \$A, at the 10-year mark.
• (This is the same for all jillion portfolios if we assume a fixed inflation rate.)
• For each of the jillion portfolios, we calculate \$A/\$Portfolio, the withdrawal rate at the 10-year mark.
• Look at the Distribution of these withdrawal rates (at this 10-year mark).
• Some will be large, some small ... and some will be Sally's 6.9%!
Question #1: What fraction of the jillion 10-year portfolios will survive another 30-years?

If one does MC simulations, then this Question might have an answer like 95%.
But we've assumed that 99% of the jillion portflios survive 40 years!
So how come only 95% survive the last 30 years?

Now forget all about the jillion portfolios and calculate the 30-year "MC safe" rate (say 4.5%).

Question #2: How does the 4.5% "MC safe" rate for 30 years compare to the Distribution (at the 10-year mark)?

For a fuller discussion of this subject, see this excerpt from the NoFeeBoard.