Here's the problem:

• You have annuity with total worth of $A which will pay $P per year until you drop dead, perhaps increasing with inflation.
• You can convert any fraction of $P into a lump sum payment ...

Now suppose you decide to reduce your annuity payments from $5000 to $4000.
The $4K payment is 5% of the total worth of the annuity. That makes that total worth 4000/[0.05] = $80,000.
>So you just divide your annuity payment by the annuity rate, eh?
Yes, and that means the insurance company can give you a lump sum of $20,000 and still provide the reduced $4K annual annuity with the remaining $80K.

In general:

• For an annuity with total worth $A which provides an annual payment of $P, the annuity rate is P/A.
• If you reduce your payments from P to P-1, the total worth is reduced from $A to $B where B = (P-1) / [P/A].
  (That's payment / rate.)
• That gives a total worth of B = A - A/P, a reduction of $A/P.
• You should then receive a lump sum of A/P = 1/rate for each $1.00 reduction in annuity payment.

>And the Answer is?
Okay, suppose:

• You invest this 1/rate lump sum at a return of R (with R = 0.08 meaning an 8% annual return on the investment).
• From this investment you withdraw $1.00 each year
  (to make up the $P annual income you would have received if you hadn't reduced your payments from P to P-1).
• Assume that the annuity payments increase with inflation at the annual rate I (where I = 0.03 means 3% inflation).
• At this return and with withdrawals increasing (from $1.00) at the inflation rate I, then the 1/rate lump sum investment would be worth
  (1/rate)(1+R)ⁿ - (1+I) {(1+R)ⁿ - (1+I)ⁿ}/(R - I) after n years.
• This investment would drop to $0 when n = LOG[1+(I-R)/{(1+I)rate}] / LOG[(1+I)/(1+R)]

>Isn't that like withdrawing from a portfolio and Safe Withdrawal Rates and ...? Exactly. With the above assumptions a portfolio would last 34 years if we withdrew at the (initial) rate of 5% our withdrawals ... increasing with inflation. >So if you drop dead before 34 years, then ... Then reduce your annuity, take a lump sum and invest it ... and you'll have money in your investment to leave to the kiddies when you're six feet under. >And if you can't manage 6.5% return or what if the annuity rate is 6% or what if ...? Here's a picture (where the big dot illustrates the above example) ... and here's a calculator:

>Are you suggesting that, if those Years until Investment is $0 is 50 years and I don't expect to live another 50 years then I should cash in everything and ...
We've assumed a constant inflation and a constant return. If both are variable (and unknown!) then a 50 year investment withdrawal can't be guaranteed whereas the annuity IS guaranteed. That's (dare I say it?) a Safe Withdrawal Rate problem so I guess the thing to do is to jump over to some Monte Carlo spreadsheet, stick in rate as your withdrawal rate (increasing with inflation), then assume some inflation rate, some average investment return, some volatility and determine the Monte Carlo probability that the investment will last until you drop dead. In particular, you compare that rate with what you regard as a Safe rate.

>Yeah, so if I don't cash in all of my annuity, what fraction should I cash in?
Hmmm ... you have a certain Monte Carlo probability that your portfolio will last N years (until you drop dead) and you get a reward when you win (that's the investment amount left when you die) and a penalty if you lose (the loss of investment income if your portfolio doesn't last until you die) and you want to know how much of your annuity to commit to a lump sum payment ... sounds like a Kelly Ratio problem, eh?

Now, if we could identify some ratio of Winning dollars to Losing dollars (that's W / L in the Kelly formula), we might use: Kelly Ratio = fraction of annuity to cash in as a lump sum = p - (1-p) / (W/L) where p is the Monte Carlo probability of surviving until we drop dead. >Mumbo jumbo. Let me think on it. In the meantime take a look at this and this.