What's wrong with my calculation?

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thedude
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What's wrong with my calculation?

Post by thedude » Sun Jan 17, 2010 9:24 am

Hi Bogleheads,

This question is directed to the number tinkerers on the forum. I'm trying to understand a calculation on the Bogleheads Wiki and I'm totally mystified. On the I-bonds page, under "Caveats", there is a sample calculation that goes like this:
here is the after-tax, after-inflation value of $1 invested I bonds compounded for 30 years with 3% inflation and 1% fixed rate and then redeemed in the 35% federal income tax bracket:

(1+(1.04015^30-1)*(1-0.35)) / (1.03^30) = $1.0165153416
I don't get it, can someone explain how this calculation works? What are all the factors and terms? I understand the tax factor, (1-0.35), and I think the factor of 1/(1.03^30) accounts for inflation, but I don't get why that works.

Here's how I would do the calculation, and I assume that I'm the one who's wrong:

P(t) = Po[1+(r-i)/n]^nt * (1-0.35)

P(t) = value of your money at year t
Po = P(0) = your initial investment value
r = composite rate = fixed rate + 2 x semiannual inflation rate + fixed rate x semiannual rate
i = annual inflation rate over period of investment, assumed to be a fixed parameter
n = number of compounding cycles per year = 12 for I-bonds

Also, I assume inflation compounds negatively with the same frequency that interest compounds positively.

According to me, the answer should be P(30) = 2.17318, with Po = 1, r = 0.0703 = .01 + 2*.03 + .01*.03, i = .03. So after 30 years you ought to have $2.17 real. Where have I screwed up?

Many thanks. Also: if you can suggest some books to help me familiarize myself with important personal finance calculations, I'd appreciate it. I am a scientist by training and I won't settle for a book that states formulae without providing derivations or proofs.

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Ariel
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Re: What's wrong with my calculation?

Post by Ariel » Sun Jan 17, 2010 10:07 am

thedude wrote:Hi Bogleheads,

This question is directed to the number tinkerers on the forum. I'm trying to understand a calculation on the Bogleheads Wiki and I'm totally mystified. On the I-bonds page, under "Caveats", there is a sample calculation that goes like this:
here is the after-tax, after-inflation value of $1 invested I bonds compounded for 30 years with 3% inflation and 1% fixed rate and then redeemed in the 35% federal income tax bracket:

(1+(1.04015^30-1)*(1-0.35)) / (1.03^30) = $1.0165153416
I don't get it, can someone explain how this calculation works? What are all the factors and terms? I understand the tax factor, (1-0.35), and I think the factor of 1/(1.03^30) accounts for inflation, but I don't get why that works.

Here's how I would do the calculation, and I assume that I'm the one who's wrong:

P(t) = Po[1+(r-i)/n]^nt * (1-0.35)

P(t) = value of your money at year t
Po = P(0) = your initial investment value
r = composite rate = fixed rate + 2 x semiannual inflation rate + fixed rate x semiannual rate
i = annual inflation rate over period of investment, assumed to be a fixed parameter
n = number of compounding cycles per year = 12 for I-bonds

Also, I assume inflation compounds negatively with the same frequency that interest compounds positively.

According to me, the answer should be P(30) = 2.17318, with Po = 1, r = 0.0703 = .01 + 2*.03 + .01*.03, i = .03. So after 30 years you ought to have $2.17 real. Where have I screwed up?

Many thanks. Also: if you can suggest some books to help me familiarize myself with important personal finance calculations, I'd appreciate it. I am a scientist by training and I won't settle for a book that states formulae without providing derivations or proofs.
It looks like there are 3 differences between the two formulae, and I'm quite sure you're wrong in at least two respects - not sure about the third (tax issue).

1) You're growing the whole quantity without any taxes along the way, then applying a tax at the end to everything, including the original amount. However, the original investment should not be not taxed. Big mistake!

2) You're computing a final amount, but then not adjusting it for inflation, which is what the denominator is doing in the Wiki equation. (In other words, the Wiki equation is speaking in terms of real dollars, not nominal dollars.) Big mistake!

3) The earnings (fixed and inflation adjusted) are presumably taxed along the way and not allowed to compound tax-free, as you've allowed them to do. The Wiki equation taxes the earnings - but not the original amount - each year, if you look closely at the parenthetical groupings. But I don't have I bonds, so I can't comment on the tax aspect.

There's also a slight difference in the way the two equations combine the fixed and inflation rates and compount them annually or semi-annually, but that's much smaller than the issues discussed above. Again, I'm not sure which equation is right in that respect, as that depends on how the US Treasury specifies the rates will be combined.
Do what you will, the capital is at hazard ... - Justice Samuel Putnam (1830), as quoted by John Bogle (1994)

thedude
Posts: 422
Joined: Wed Mar 07, 2007 7:44 pm

Post by thedude » Sun Jan 17, 2010 10:20 am

Wow, I feel like an idiot. Thanks Ariel.

Edit: I feel like an idiot because of the dumb way in which I accounted for taxes. I still don't get why my accounting for inflation is wrong. I think, before going any further with this thread, I need to get myself a good book and run through some practice calculations.

Ugh, I teach a course on dynamical systems, I should be able to do this.
Last edited by thedude on Sun Jan 17, 2010 10:33 am, edited 1 time in total.

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LadyGeek
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Post by LadyGeek » Sun Jan 17, 2010 10:21 am

I'm an engineer by training and won't accept formula without rationale either. Linuxizer highly recommended Fixed Income Mathematics by Frank J. Fabozzi. I just ordered an older used version from Amazon (URL appended with Bogleheads reference). At $10, it's a no brainer. (Update: I took the $10 copy, now the cheapest is $11 with shipping :) )

Go to the current edition (4E) and take a look at some sample content. I could understand it, which is a rarity when crossing between engineering and finance.

Also, some really good tutorials:

Bond Basics: Introduction - enable cookies to skip the nag screens and take the quiz
Advanced Bond Concepts: Introduction

Edit: I deal in math-intensive equations as well, and also feel I should also know this. However, it's not the complexity of the equations. It's the mindset.

Engineering and the sciences work in future time (performance degradation, etc.). Finances work in present time. A lot of these equations work in Present Value, as opposed to Future Value. Note that all the finance experts are very clear to state the reference frame, e.g. "real" or "nominal" rates. The reference frame is translated from future back to present time via the equations which account for time-value-of-money as well as inflation rate. Very nasty when dealing with bonds. The book mentioned above is a good reference.

Update: Corrected Bogleheads reference tag in Amazon.com link
Last edited by LadyGeek on Tue Jan 19, 2010 9:01 pm, edited 5 times in total.
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Ariel
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Post by Ariel » Sun Jan 17, 2010 10:45 am

thedude wrote:I still don't get why my accounting for inflation is wrong.
I think that you've also made a big error in r below:
r = 0.0703 = .01 + 2*.03 + .01*.03
It should be something like:

r = 0.0703 = .01 + 2*.03/2 + .01*.03

since each semi-annual inflation adjustment would be for only half a year's inflation.

Clearly the difference between r and i should only 0.01 plus a tad depending on any compounding. Your calculation has the difference being 0.04 plus a tad.
Do what you will, the capital is at hazard ... - Justice Samuel Putnam (1830), as quoted by John Bogle (1994)

thedude
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Joined: Wed Mar 07, 2007 7:44 pm

Post by thedude » Sun Jan 17, 2010 10:52 am

Ariel wrote:Clearly the difference between r and i should only 0.01 plus a tad depending on any compounding. Your calculation has the difference being 0.04 plus a tad.
Yup, that seems reasonable.

Valuethinker
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Re: What's wrong with my calculation?

Post by Valuethinker » Sun Jan 17, 2010 12:51 pm

thedude wrote:Hi Bogleheads,

This question is directed to the number tinkerers on the forum. I'm trying to understand a calculation on the Bogleheads Wiki and I'm totally mystified. On the I-bonds page, under "Caveats", there is a sample calculation that goes like this:
here is the after-tax, after-inflation value of $1 invested I bonds compounded for 30 years with 3% inflation and 1% fixed rate and then redeemed in the 35% federal income tax bracket:

(1+(1.04015^30-1)*(1-0.35)) / (1.03^30) = $1.0165153416
I don't get it, can someone explain how this calculation works? What are all the factors and terms? I understand the tax factor, (1-0.35), and I think the factor of 1/(1.03^30) accounts for inflation, but I don't get why that works.

Here's how I would do the calculation, and I assume that I'm the one who's wrong:

P(t) = Po[1+(r-i)/n]^nt * (1-0.35)

P(t) = value of your money at year t
Po = P(0) = your initial investment value
r = composite rate = fixed rate + 2 x semiannual inflation rate + fixed rate x semiannual rate
i = annual inflation rate over period of investment, assumed to be a fixed parameter
n = number of compounding cycles per year = 12 for I-bonds

Also, I assume inflation compounds negatively with the same frequency that interest compounds positively.

According to me, the answer should be P(30) = 2.17318, with Po = 1, r = 0.0703 = .01 + 2*.03 + .01*.03, i = .03. So after 30 years you ought to have $2.17 real. Where have I screwed up?

Many thanks. Also: if you can suggest some books to help me familiarize myself with important personal finance calculations, I'd appreciate it. I am a scientist by training and I won't settle for a book that states formulae without providing derivations or proofs.
Just to be clear, the relationship (called the Fisher relation) between inflation and nominal and real rates is

1 + real = (1 + nominal)/ (1+ inflation)

or real rate = ((1 + nominal)/ (1+inflation)) - 1

You cannot subtract inflation from a nominal rate to get a real rate

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baw703916
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Post by baw703916 » Sun Jan 17, 2010 2:37 pm

(mathematical detail) Saying that the real return is 1+ nominal return - inflation rate works approximately if the inflation rate is small. One can write the fraction 1/(1+IR), where IR is the inflation rate, as a series expansion:

1/(1+IR) = 1 - IR + IR^2 - IR^3 + IR^4
If IR << 1, then the higher terms quickly become insignificant.

I do have a more substantive question:

In the formula, I'm not sure where the 1.04015 comes from. Presumably compounding, but is the 1% real return the annualized rate, or is the APY higher because of quarterly compounding? I assume that the 3% inflation rate is annualized?

Brad
Most of my posts assume no behavioral errors.

thedude
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Post by thedude » Sun Jan 17, 2010 2:58 pm

baw703916 wrote:In the formula, I'm not sure where the 1.04015 comes from.
Though I haven't yet revisited the calculation in detail, it seems to me that 1.04015 = principal after 1 year of return since the composite annual rate r is

r = annual fixed rate + 2*semiannual inflation rate + annual fixed rate*semiannual inflation rate = .01 + 2*.015 + .01*.015 = .04015

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baw703916
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Post by baw703916 » Sun Jan 17, 2010 4:12 pm

thedude wrote:
baw703916 wrote:In the formula, I'm not sure where the 1.04015 comes from.
Though I haven't yet revisited the calculation in detail, it seems to me that 1.04015 = principal after 1 year of return since the composite annual rate r is

r = annual fixed rate + 2*semiannual inflation rate + annual fixed rate*semiannual inflation rate = .01 + 2*.015 + .01*.015 = .04015
OK, I'm able to reproduce that number now. I'd forgotten that the inflation adjustment is made semiannually. :P

Brad
Most of my posts assume no behavioral errors.

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LadyGeek
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Post by LadyGeek » Sun Jan 17, 2010 6:17 pm

I updated the wiki to show the derivation of the 1.04015.
To illustrate the point, here is the after-tax, after-inflation value of $1 invested I bonds compounded for 30 years with 3% inflation and 1% fixed rate and then redeemed in the 35% federal income tax bracket:

(1 + (1.04015^30 - 1)*(1 - 0.35)) / (1.03^30) = $1.0165153416
where 1.04015 is the principal return + composite rate (.01 + 2*.015 + .01*.015 = .04015) after 1 year
Please see I Savings Bonds on the Bogleheads Wiki.

If it's not worded correctly, please let me know (wiki editors can updated directly).
Wiki To some, the glass is half full. To others, the glass is half empty. To an engineer, it's twice the size it needs to be.

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