Thank you for asking. We assume that you have enough savings to provide for the same income from age 62 as you will get whenever you start social security draws.
Example: If your social security quote would equate to $24,000 a year at 66, you would need enough savings to be able to draw $24,000 each year from 62 until 66. (All values in today's dollars, but the equations below, that is, inflation-adjusted returns.)
The amount of savings is somewhat dependent on the real return on investment on such savings. The financial equation for the amount of savings is PV(real return, years from 62 to start social security, annual payment equal to the quoted value of social security (x 12) at the age you want to start). If you assume that your after-tax return equals inflation, the equation becomes simply the number of years till you start social security times the annual quoted payments.
Example: Suppose the social security quote at 66 would equal $24,000 per year. The amount of savings needed to support $24,000 of payments for the four year delay would be 4 x $24,000 = $96,000 if returns equaled inflation.
The breakeven age is dependent on how long the same amount of savings would last if you made draws from age 62 to a death year where the draws were equal to the difference in the payments between starting at 62 and starting later. You can use the financial equation for the number of years N(real return, PV savings, payment equal to the annual difference). If the after-tax return equals inflation, the number of years becomes simply Savings divided by Payment Difference.
Example: Extending the previous example, suppose that the age 62 quote equated to $18,000 per year. Then the number of years to breakeven would be $96,000 / ($24,000 - $18,000) = 16 if returns equal inflation. Thus, the breakeven age would be 62 + 16 = 78.
The breakeven age gets higher as the real return increases. With a real return of 4% (likely to be high for a retiree and much more than you would get from an annuity), the breakeven age would be 84.
You can use the free social security calculator on www.analyzenow.com to determine the likely best result for your own case considering various returns, tax effects, spousal benefits and potential death years. This is much more difficult to analyze with simple equations. In general, the addition of spousal benefits points to the high-income spouse delaying social security until age 70 and a low-income spouse to the low-income spouse's full retirement age (65-67) unless savings are insufficient or short life spans appear inevitable.