Valuing a Deferred Retirement Annuity with Net Present Value
Summary
The document evaluates a retirement contract by discounting its contributions and later benefits to the same date. It treats the payments as end-of-year cash flows, calculates the present value of the 40 annual contributions, discounts the retirement payments from age 65 back to today, and compares the two values. Under the stated 5% annual interest assumption, the accepted calculation finds a positive net present value, so the contract appears financially favorable on that basis.
A second proposed calculation uses a different count for the benefit period and reaches the same qualitative conclusion, though its timing differs. The key caveat is that the answer depends on when payments begin and how many occur; the accepted response assumes payments from ages 65 through 100, inclusive. This is a simplified valuation, with no adjustment for mortality, inflation, taxes, insurer credit risk, or uncertainty about future interest rates.
Key ideas
- Compare a contract’s present value of benefits with the present value of its contributions.
- Discount deferred payments back to today using the stated interest rate and payment timing.
- Count payment dates carefully because the benefit period affects the valuation.
- A positive net present value indicates an attractive deal only under the assumptions used.
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Full text
# NPV of two annuities
# NPV of two annuities
For exam preparation we are given some past papers, however there are no solutions and I would like to know if my logic is correct for the following question:
> Assume you are 25 years old. An insurance company is offering you the following retirement contract: Pay in \$2000 per year for the next 40 years. When you reach 65 years of age, you will receive \$20000 per year until you reach 100. Assume the prevailing interest rate is 5% per year, all payments occur at year end, and it is January 1 now. Is this annuity a good deal?
## Answer by Gabor Bakos (score 1, accepted)
https://quant.stackexchange.com/a/26344
Assume that "I" was born on 1st of January. I pay for $40$ years and receive payments for $36$ years ($65$ to $100$). So I pay when I am $$25, 26 \dots 64$$ and receive payments when I am $$65, 66,\dots 100.$$ Hence the PVs of the annuities are:
$\begin{equation} PV_{paid} = \frac{2000}{0.05}\left(1-\left(\frac{1}{1.05}\right)^{40}\right) = 34318.1727 \quad \text{(to 4dp)}\\ PV_{received~when~65} = \frac{20000}{0.05}\left(1-\left(\frac{1}{1.05}\right)^{36}\right) = 330937.0342\quad \text{(to 4dp)}\\ PV_{received} = PV_{received~when~65}\times\left(\frac{1}{1.05}\right)^{40} = 47008.1768\quad \text{(to 4dp)}\\ \end{equation} $
$$\therefore NPV = PV_{received} - PV_{paid} = 12690.0041 \quad \text{(to 4dp)}. $$ Hence the annuity is a good deal (having positive NPV).
## Answer by user20952 (score 0)
https://quant.stackexchange.com/a/26342
I'm in your class and was stuck on this too
Here's how I make it:
PV of contributions = (2000/0.05)*[1-(1/1.05^40)] = 34318.17
PV of amount received = (20000/0.05)*[1-(1/1.05^35)] = 327483.89
Value today of amount received = PV of amount received / (1.05)^40 = 59369.64
Therefore this annuity is a good deal. Still unsure with this, have you come to a conclusion yet? ThanksShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.