# Present Value of descendants' income.

June 11, 2009 11:32 AM Subscribe

Help me estimate the present value of my descendants' income.

I'd like to calculate a dollar amount that represents the present value of all money that will be earned by an average person living in the United States and all his/her descendants.

The general model that I'm using involves a few assumptions.

1. The ratio of members of one generation to the next is constant.

2. The number of years between generations is constant.

(In other words, each individual has X children immediately upon turning Y years old.)

3. Each individual has the same earning history.

(Starting to get technical...but basically, my grandkid works the same jobs as me at the same ages, and gets paid the same inflation-adjusted salaries.)

3a. Because each individual has the same earning history, each individual's lifetime earnings can be reduced to the same single sum (before taking inflation into account) by discounting all earnings to the start of that individual's career. The point of this isn't really to perform the discounting calculations, but to decide that a single dollar amount can be used to represent each individual's lifetime earnings, making the specifics of the "earning history" irrelevant.

4. The interest rate is constant.

So I now have three variables -- Number of Offspring, Years Between Generations, and Interest Rate. I can figure out a reasonable present value of lifetime earnings for a single individual today, and I can use this as a starting figure for the value of each future individual's lifetime earnings. Now on to my questions:

i. Do I need to add another variable for Inflation Rate, or can I simply choose an Interest Rate that is net of inflation? If I do add a separate Inflation Rate variable, are these the correct uses: Discount Rate = Interest Rate minus Inflation Rate ; Generation 1 lump-sum lifetime earnings at year Y = (Generation 0 lump-sum lifetime earnings at year 0) * (1 + Inflation Rate)^Y ?

ii. The model I've chosen assumes the median, in a sense -- it ignores individuals that don't reproduce as well as those with large families, wealthy as well as penniless, work until old age as well as not-a-day-in-one's-life. Does this introduce significant inaccuracy? Are there any other aspects of the model that introduce significant inaccuracy?

iii. What are reasonable values for the indefinite future for the variables of Number of Offspring, Years Between Generations, Interest Rate, and Inflation Rate? The two rates are going to be the biggest factors in the calculation, and I have the least sense on reasonable estimates for these two rates.

I'd like to calculate a dollar amount that represents the present value of all money that will be earned by an average person living in the United States and all his/her descendants.

The general model that I'm using involves a few assumptions.

1. The ratio of members of one generation to the next is constant.

2. The number of years between generations is constant.

(In other words, each individual has X children immediately upon turning Y years old.)

3. Each individual has the same earning history.

(Starting to get technical...but basically, my grandkid works the same jobs as me at the same ages, and gets paid the same inflation-adjusted salaries.)

3a. Because each individual has the same earning history, each individual's lifetime earnings can be reduced to the same single sum (before taking inflation into account) by discounting all earnings to the start of that individual's career. The point of this isn't really to perform the discounting calculations, but to decide that a single dollar amount can be used to represent each individual's lifetime earnings, making the specifics of the "earning history" irrelevant.

4. The interest rate is constant.

So I now have three variables -- Number of Offspring, Years Between Generations, and Interest Rate. I can figure out a reasonable present value of lifetime earnings for a single individual today, and I can use this as a starting figure for the value of each future individual's lifetime earnings. Now on to my questions:

i. Do I need to add another variable for Inflation Rate, or can I simply choose an Interest Rate that is net of inflation? If I do add a separate Inflation Rate variable, are these the correct uses: Discount Rate = Interest Rate minus Inflation Rate ; Generation 1 lump-sum lifetime earnings at year Y = (Generation 0 lump-sum lifetime earnings at year 0) * (1 + Inflation Rate)^Y ?

ii. The model I've chosen assumes the median, in a sense -- it ignores individuals that don't reproduce as well as those with large families, wealthy as well as penniless, work until old age as well as not-a-day-in-one's-life. Does this introduce significant inaccuracy? Are there any other aspects of the model that introduce significant inaccuracy?

iii. What are reasonable values for the indefinite future for the variables of Number of Offspring, Years Between Generations, Interest Rate, and Inflation Rate? The two rates are going to be the biggest factors in the calculation, and I have the least sense on reasonable estimates for these two rates.

If you are using excel, the discounting part is easy (NPV or PV is your friend).

In the US, we are pretty close to 0 organic growth (pairs have two kids to replace themselves). So each person has one child.

Discount could be at 3% or so I think and be reasonable (make it a variable so you can adjust if need be later)

You ask for Interest as well as Inflation rates, are you going to speculate that they save a certain % of their income? If you are going to do that, you probably want to go as far as an income tax as well.

Saying that all folks have kids at 25 or so would be a decent start (it really varies on a lot of things, but I would start with that, and make it a variable so you can tweak with it later).

This sounds like an interesting little experiment, let us know how it works out.

posted by milqman at 11:50 AM on June 11, 2009

In the US, we are pretty close to 0 organic growth (pairs have two kids to replace themselves). So each person has one child.

Discount could be at 3% or so I think and be reasonable (make it a variable so you can adjust if need be later)

You ask for Interest as well as Inflation rates, are you going to speculate that they save a certain % of their income? If you are going to do that, you probably want to go as far as an income tax as well.

Saying that all folks have kids at 25 or so would be a decent start (it really varies on a lot of things, but I would start with that, and make it a variable so you can tweak with it later).

This sounds like an interesting little experiment, let us know how it works out.

posted by milqman at 11:50 AM on June 11, 2009

What is the time frame of interest? is there an infinite chain of descendents all the way down? I would think, then, that the present value of the money would be infinite, given a net positive lifetime earnings per generation, or zero/negative given a net negative. Any PV earnings of way-way-future descendants would be miniscule, but they would add up to infinity.

Over a shorter (but still very long) scale, the present value of distant-future dollars is still very very low for any reasonable discount rate, so the graph of delta PV flattens out and approaches 0 at some distant number of generations, based on the discount rate, reproduction rate and generation spacing you choose.

So I'm guessing you want to model out to a point on that graph where delta PV is close to 0 but not to an infinite number of generations? That sort of captures the spirit of your experiment, but that number also changes based on your inputs. I'd be interested to hear how you are defining the boundaries of the model.

i. That said, I think you should choose a generic discount rate net of inflation, for simplicity's sake. You've got plenty of variables already. What hypothesis are you testing? Does adding another variable get you any more accuracy in answering the question?

ii. Sure it'll be inaccurate and totally speculative. It's a model. The fellas on wall street had models too. Are you accounting for wars? Shifting morals regarding abortion? Advances in health care? The future is vague.

iii. seems to me that the more developed a country becomes, the lower its birth rate. We've hit a plateau for now, but population may even shrink some. Hard to say. Years between generations has been highly variable even in the last century. If lifespans increase and the period of youth/education lengthens, will people give birth even later? As for discount rates, I think those numbers are total black magic, they incorporate so many variables. It's the aggregate pricing of all risk and opportunity costs of your input, right? So maybe start by looking at the risk and opportunity cost of having a child vs. not. (that may be the wrong way to go about it, but I would give it a shot, anyway).

posted by Chris4d at 5:50 PM on June 11, 2009

Over a shorter (but still very long) scale, the present value of distant-future dollars is still very very low for any reasonable discount rate, so the graph of delta PV flattens out and approaches 0 at some distant number of generations, based on the discount rate, reproduction rate and generation spacing you choose.

So I'm guessing you want to model out to a point on that graph where delta PV is close to 0 but not to an infinite number of generations? That sort of captures the spirit of your experiment, but that number also changes based on your inputs. I'd be interested to hear how you are defining the boundaries of the model.

i. That said, I think you should choose a generic discount rate net of inflation, for simplicity's sake. You've got plenty of variables already. What hypothesis are you testing? Does adding another variable get you any more accuracy in answering the question?

ii. Sure it'll be inaccurate and totally speculative. It's a model. The fellas on wall street had models too. Are you accounting for wars? Shifting morals regarding abortion? Advances in health care? The future is vague.

iii. seems to me that the more developed a country becomes, the lower its birth rate. We've hit a plateau for now, but population may even shrink some. Hard to say. Years between generations has been highly variable even in the last century. If lifespans increase and the period of youth/education lengthens, will people give birth even later? As for discount rates, I think those numbers are total black magic, they incorporate so many variables. It's the aggregate pricing of all risk and opportunity costs of your input, right? So maybe start by looking at the risk and opportunity cost of having a child vs. not. (that may be the wrong way to go about it, but I would give it a shot, anyway).

posted by Chris4d at 5:50 PM on June 11, 2009

Let's just talk in time periods of 20 years. Say you are only one person, in the first generation, and your total lifetime income is $x. If every time period yielded the same $x, and the discount rate for that time period was

But in your model, x is growing. So the income may be x at first, but if each of your children marry and have two children, then the next generation will have total income of 2x, and the next 4x, and so on. That's a 100% increase each period, but let's generalize and say it grows at rate p. Then the NPV = x + x*p/d + x*p^2/d^2 + x*p^3/d^3 + .... So clearly, if the descendant growth rate is greater than the discount rate, then p/d > 1 and the NPV doesn't converge so the present value of all your descendants is infinite. However, if p/d <>D, and it does converge, back to the same formula of x*D/(D-1).

So examples: Every generation grows by 20%, p=1.2, and the discount rate for those 20 years is 15%, d=1.15, and the present value is infinite. However, if the discount rate were 30%, D = 1.3/1.15 = 1.13, so the net present value of all descendants converges to x*(1.13/.13) = 8.7x. So all your descendants will net you close to 9 times what you yourself have made.

I'm not really sure what realistic numbers for p and d are, though.

Oh, and I could be wildly off base here.

posted by losvedir at 6:49 PM on June 11, 2009

*d*, then the net present value NPV = x + x/d + x/d^2 + x/d^3 + ... = x*d/(d-1).But in your model, x is growing. So the income may be x at first, but if each of your children marry and have two children, then the next generation will have total income of 2x, and the next 4x, and so on. That's a 100% increase each period, but let's generalize and say it grows at rate p. Then the NPV = x + x*p/d + x*p^2/d^2 + x*p^3/d^3 + .... So clearly, if the descendant growth rate is greater than the discount rate, then p/d > 1 and the NPV doesn't converge so the present value of all your descendants is infinite. However, if p/d <>D, and it does converge, back to the same formula of x*D/(D-1).

So examples: Every generation grows by 20%, p=1.2, and the discount rate for those 20 years is 15%, d=1.15, and the present value is infinite. However, if the discount rate were 30%, D = 1.3/1.15 = 1.13, so the net present value of all descendants converges to x*(1.13/.13) = 8.7x. So all your descendants will net you close to 9 times what you yourself have made.

I'm not really sure what realistic numbers for p and d are, though.

Oh, and I could be wildly off base here.

posted by losvedir at 6:49 PM on June 11, 2009

Thanks for the comments so far...

@madcap, I hadn't thought of considering whether the planet could contain or sustain an infinite number of generations...so let's just say that whatever I calculate is possibly an overestimation.

@milqman, if a pair has two kids, I'm considering that two children, because I'm only looking at it from the perspective of one of those parents.

@Chris4d, the sum of an infinite sequence can converge (finite) or diverge (infinite), it just depends on the sequence. 1/1 + 1/2 + 1/3 + ... diverges; it has no upper bound. 1/1 + 1/2 + 1/4 + 1/8 + ... is bounded (it approaches 2), and even 99/100 + (99/100)^2 + (99/100)^3 + ... is bounded (it approaches 99). This model is more like the latter, a geometric series:

1 + [X / (1+r)^Y] + [X / (1+r)^Y]^2 + ...

The thing to remember is that X only happens every Y years. So if the discount rate is 3%, in Y = 25 years that amounts to about 2.09, and if X is anything less than that, such as 2, then the sequence is geometrically decreasing and the series converges.

posted by RobinFiveWords at 6:59 PM on June 11, 2009

@madcap, I hadn't thought of considering whether the planet could contain or sustain an infinite number of generations...so let's just say that whatever I calculate is possibly an overestimation.

@milqman, if a pair has two kids, I'm considering that two children, because I'm only looking at it from the perspective of one of those parents.

@Chris4d, the sum of an infinite sequence can converge (finite) or diverge (infinite), it just depends on the sequence. 1/1 + 1/2 + 1/3 + ... diverges; it has no upper bound. 1/1 + 1/2 + 1/4 + 1/8 + ... is bounded (it approaches 2), and even 99/100 + (99/100)^2 + (99/100)^3 + ... is bounded (it approaches 99). This model is more like the latter, a geometric series:

1 + [X / (1+r)^Y] + [X / (1+r)^Y]^2 + ...

The thing to remember is that X only happens every Y years. So if the discount rate is 3%, in Y = 25 years that amounts to about 2.09, and if X is anything less than that, such as 2, then the sequence is geometrically decreasing and the series converges.

posted by RobinFiveWords at 6:59 PM on June 11, 2009

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(I'm being deliberately vague because I don't

wantto make a precise statement about how often a given future person would be descended from a given present person via multiple pathways; that would require making precise assumptions and doing lots of calculations, although perhaps somebody's already done them.)So if someone is your descendant via two different pathways, do they get counted twice? Or just once?

Sorry I made your problem more complicated.

posted by madcaptenor at 11:47 AM on June 11, 2009