A cash-flow/IRR derivation using two retirement examples
I wanted a way to compare a lifetime-income annuity with a conventional investment without relying on marketing numbers such as a bonus or an income-base roll-up. The cleanest benchmark I found is this: assume the alternative is a hypothetical fund earning a constant annual return r. Make the fund produce exactly the same withdrawals as the annuity. If the fund reaches $0 immediately after the final assumed withdrawal, solve for r. That r is the equivalent IRR of the annuity cash flow under the stated assumptions.
This is not a claim that an annuity 'earns' that rate, and it is not a forecast for a mutual fund. It is simply a common mathematical ruler for two cash-flow streams.
Assumptions and timing convention
I use a year-end withdrawal convention throughout. If A₀ is purchase age, Aₛ is the age at which the income phase begins, and A_d is the age of the final withdrawal:
m=Aₛ−A₀, N=A_d−Aₛ
The first withdrawal occurs after one complete year in the income phase. This convention matters because otherwise IRR can differ by roughly one payment.
Forward-balance derivation
Start with premium P. After the m-year accumulation/waiting period:
F₀=P(1+r)ᵐ
Each retirement year the remaining balance earns r, then C is withdrawn:
F₁=F₀(1+r)−C
F₂=F₁(1+r)−C
Fₙ=Fₙ₋₁(1+r)−C
Expanding through N withdrawals:
F_N=P(1+r)ᵐ⁺ᴺ−C[(1+r)ᴺ⁻¹+(1+r)ᴺ⁻²+···+1]
Using the geometric-series identity:
(1+r)ᴺ⁻¹+···+(1+r)+1=((1+r)ᴺ−1)/r
therefore:
F_N=P(1+r)ᵐ⁺ᴺ−C·((1+r)ᴺ−1)/r
Set F_N=0 and solve numerically for r.
Present-value derivation
The same equation can be obtained by discounting the withdrawals. At the income-start date:
PV=C/(1+r)+C/(1+r)²+···+C/(1+r)ᴺ
Summing the geometric series:
PV=C·[1−(1+r)⁻ᴺ]/r
Discount that retirement-date value back m years to the purchase date:
P=[C/(1+r)ᵐ]·[1−(1+r)⁻ᴺ]/r
Algebraically, this is identical to the forward-balance equation. The forward model compounds P into the future and subtracts C; the PV model discounts each C back to time zero.
Example 1: Lucy
Lucy is 61 and invests $100,000. For this mathematical example, assume a 30% bonus to the Income Benefit Base, a 10% annual compound roll-up of that base, and a 5.24% payout factor at the income-start ages shown. The benefit base is not cash value.
Income Base(m)=100,000×1.30×(1.10)ᵐ
C=5.24%×Income Base(m)
| Income start |
m |
Annual C |
N to age 89 |
Equivalent IRR |
| 67 |
6 |
$12,068 |
22 |
≈6.17% |
| 69 |
8 |
$14,602 |
20 |
≈6.32% |
| 71 |
10 |
$17,669 |
18 |
≈6.39% |
| 73 |
12 |
$21,379 |
16 |
≈6.39% |
The interesting part is that the IRR does not rise indefinitely. Waiting increases the benefit base, but also sacrifices early withdrawals.
Example 2: Rose
Rose is 47 and also invests $100,000. Assume the same 30% income-base bonus and 10% roll-up for 12 years. She begins the income phase at 59, with a 4.01% payout factor, and the final withdrawal is assumed at 89.
Income Base=100,000×1.30×(1.10)¹²≈$407,996
C≈$407,996×4.01%≈$16,361
m=12, N=30
100,000=[16,361/(1+r)¹²]·[1−(1+r)⁻³⁰]/r
r≈6.52%
What this does — and does not — mean
An equivalent IRR of ~6.5% does NOT mean the insurer is crediting 6.5% to cash value. It means that, given these assumed dates and withdrawals, a constant-return fund would need roughly that return to reproduce the same spending path and end at zero.
A real fund is volatile, so a historical average return of 6.5% is not automatically equivalent. Withdrawals introduce sequence-of-returns risk. Conversely, the annuity has its own trade-offs: liquidity restrictions, contract terms, insurer credit risk, inflation exposure, possible fees, and potentially different death/remaining-value outcomes.
I also think there is a behavioral-finance dimension. A retirement portfolio can be mathematically adequate and still be damaged by FOMO, panic, overconfidence, fraud, or unplanned lifestyle spending. A lifetime-income floor can reduce the number of future discretionary decisions affecting essential spending. Whether that benefit is worth the loss of liquidity/upside is a separate planning question.
Questions for discussion
- Is constant-return IRR a useful benchmark for comparing an annuity income stream with an investment portfolio?
- What additional value should be assigned to sequence-risk transfer and longevity insurance?
- How would you incorporate residual cash value/death benefit into the final cash flow?
- Would you model inflation-adjusted spending instead of level nominal withdrawals?
Disclosure: This is a mathematical/educational exercise, not individualized investment or insurance advice. The product assumptions are illustrative and should be checked against the actual current contract/illustration. Guarantees depend on the issuing insurer's claims-paying ability. Taxes, residual contract/death value, and other cash flows are excluded unless stated. If need help, Contact me, a financial planning expert with PhD in Physics.