Deriving Shiller's formula for Monthly Total Bond Returns

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Deriving Shiller's formula for Monthly Total Bond Returns

Post by Ben Mathew »

Can someone help me derive the formula Shiller uses for calculating monthly total bond returns in his spreadsheet "US Stock Market 1871 - Present and CAPE Ratio" on his website:

Monthly total bond returns for period t = Yt/Yt+1 + Yt + (1+Yt+1)^(-119)*(1-Yt/Yt+1)

where Yt = 10-Year Treasury Constant Maturity Rate (GS10) divided by 12

Thanks!
Last edited by Ben Mathew on Fri May 07, 2021 12:50 pm, edited 2 times in total.
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

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I too would very much appreciate an explanation!
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by #Cruncher »

I believe Ben is referring to column R, "Monthly Total Bond Returns" on the Data sheet of the Excel file, U.S. Stock Markets 1871-Present and CAPE Ratio (clicking this link will download the spreadsheet!), linked near the top of the web page he references. Note that column R shows the return plus 1. E.g., if the return is 1%, column R will show 1.01. (Also by default the column shows only 2 decimal places.)

Shiller is calculating based on column G, "Long Interest Rate GS10". His calculation assumes a bond selling at par with a coupon equal to the rate. The return of such a bond after one period will equal the sum of one interest payment plus the change in value of the bond. This value after one period can be calculated using these two formulas where r is the new rate and n is the number of periods remaining to maturity:
  • PV$1 = 1 / (1 + r) ^ n = Present Value of $1
  • PVA$1 = (1 - PV$1) / r = Present Value of Annuity of $1
I'll illustrate this with January 2001 from Shiller's Data sheet (row 1569):

Code: Select all

1.00896 = Shiller's value in column R
0.0043  = rate Jan 2001  = 5.16 / 1200
0.00425 = rate Feb 2001  = 5.10 / 1200
0.60370 = PV$1           = 1 / 1.00425 ^ 119
0.40096 = coupon * PVA$1 = 0.0043 * (1 - 0.60370) / 0.00425
1.00466 = total present value after one month [*]
0.00430 = one period's coupon
1.00896 = total value including coupon
 0.896% = return
Ben Mathew wrote: Tue Apr 27, 2021 9:36 pmCan someone help me derive the formula Shiller uses ... t = Yt/Yt+1 + Yt + (1-Yt)^(-119)*(1-Yt/Yt+1)
(You made a typo, Ben. "(1-Yt)^(-119)" should be "(1+Yt)^(-119)".) Here is how to get from my formula above to Shiller's. Note that "x^-n" equals "1/x^n". Where
  • c = coupon (same as initial rate) = Yt
  • r = new rate = Yt+1
  • n = number of remaining periods

Code: Select all

--PV$1--           ------- PVA$1--------
(1+r)^-n + c     * (1    - PV$1      / r     + c  Formula explained above
(1+r)^-n + c     * (1    - (1+r)^-n) / r     + c  Substitute formula for PV$1

(1+r)^-n + (c/r) * (1    - (1+r)^-n)         + c  Rearrange
(1+r)^-n + (c/r) - (c/r) * (1+r)^-n          + c  Extend (c/r)
(1+r)^-n +       - (c/r) * (1+r)^-n  + (c/r) + c  Rearrange
(1+r)^-n * (1    - (c/r))            + (c/r) + c  Factor (1+r)^-n

(c/r) + c + (1+r)^-n * (1 - (c/r))                Rearrange into Shiller's formula
* Equals the sum of the present value of principal after 119 months plus present value of the remaining 119 coupons (1.00466 = 0.60370 + 0.40096). Can also be computed in one step with Excel PV function:
1.00466 = -PV(0.00425, 119, 0.0043, 1, 0)
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Ben Mathew »

#Cruncher wrote: Wed Apr 28, 2021 5:10 pm Shiller is calculating based on column G, "Long Interest Rate GS10". His calculation assumes a bond selling at par with a coupon equal to the rate. The return of such a bond after one period will equal the sum of one interest payment plus the change in value of the bond. This value after one period can be calculated using these two formulas where r is the new rate and n is the number of periods remaining to maturity:
Thanks, #Cruncher! This is super helpful. Makes perfect sense now.

With some rearranging, this is the same formula used by Damodaran to calculate bond returns in his data spreadsheet "Discount Rate Estimation Historical Returns on Stocks, Bonds and Bills - United States" from his website.

Link to download Damodaran's spreadsheet: http://www.stern.nyu.edu/~adamodar/pc/d ... tretSP.xls

In the sheet "T.Bond yield and return", he uses a formula for "Return on bond" (column C) that rearranges to Shiller's formula. Except Damodaran's formula uses ^10 instead of the ^9 that I would expect based on your derivation of Shiller's formula above. Any thoughts on why Damodaran is using ^10 instead of ^9?
Last edited by Ben Mathew on Thu Apr 29, 2021 10:40 am, edited 1 time in total.
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

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#Cruncher wrote: Wed Apr 28, 2021 5:10 pm (You made a typo, Ben. "(1-Yt)^(-119)" should be "(1+Yt)^(-119)"
Fixed. Thanks!
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by #Cruncher »

Ben Mathew wrote: Thu Apr 29, 2021 9:45 amLink to download Damodaran's spreadsheet: http://www.stern.nyu.edu/~adamodar/pc/d ... tretSP.xls
... Any thoughts on why Damodaran is using ^10 instead of ^9?
From cell B3 of Damodaran's "T. Bond yield & return" sheet:
To compute the return on a constant maturity bond, I add two components - the promised coupon at the start of the year and the price change due to interest rate changes.
He's valuing each bond at the end of the year assuming it will be priced based on next year's 10-year yield. But in fact it will be priced based on next year's 9-year yield. Since he doesn't have that yield, he's stuck using the 10-year yield. And since his yield is wrong, his pricing will also be wrong whether he assumes the remaining life is 9 years or 10 years. By assuming ten, he magnifies the effect of rate changes: using 10 years makes the loss greater if rates go up, and the gain greater if rates go down.

I computed the effect of using 9 or 10 years for the 51 years 1969 to 2020. It made little difference. Using ten years the Compound Annual Growth Rate (CAGR) is 7.10%. If he'd used nine years, it would have been 7.02%. For the same period I estimated the rate for a bond with nine years to maturity by interpolating between the 7-year and 10-year FRED rates. Using this the CAGR would have been 7.45% over the 51 years, noticeably better than using the ten-year rate. [1] The following table shows the results for three cases of pricing the bonds after one year:
  1. Using the nine-year yield for a bond maturing in nine years. CAGR = 7.45%
  2. Using the ten-year yield for a bond maturing in nine years. CAGR = 7.02%
  3. Using the ten-year yield for a bond maturing in ten years (Damodaran's method). CAGR = 7.10%

Code: Select all

Row       Col A   Col B   Col C    Col D      Col E    Col F    Col G
  4          Growth factor 51 years  -->      39.04    31.87    33.05 [2]
  5      Compound annual growth rate -->      7.45%    7.02%    7.10% [2]
                  -FRED Rates--  Est 9yr    - Value after One Year -
  6                DGS7   DGS10  Mature     Case A   Case B   Case C

Code: Select all

  7  12/31/1969           7.88%
  8  12/31/1970   6.25%   6.50%   6.417%    117.66%  117.07%  117.80% [3]
  9  12/31/1971   5.85%   5.89%   5.877%    110.76%  110.67%  111.01%
 10  12/31/1972   6.29%   6.41%   6.370%    102.68%  102.42%  102.14%
 11  12/31/1973   6.76%   6.90%   6.853%    103.50%  103.20%  102.95%
 12  12/31/1974   7.34%   7.40%   7.380%    103.82%  103.70%  103.45%
 13  12/31/1975   7.68%   7.76%   7.733%    105.29%  105.13%  104.96%
 14  12/31/1976   6.42%   6.81%   6.680%    114.89%  114.00%  114.49%
 15  12/31/1977   7.66%   7.78%   7.740%    100.94%  100.69%  100.24%
 16  12/31/1978   9.23%   9.15%   9.177%     99.47%   99.62%   99.05%
 17  12/31/1979  10.36%  10.33%  10.340%    102.39%  102.44%  102.00%
 18  12/31/1980  12.49%  12.43%  12.450%     99.22%   99.32%   98.67%
 
 19  12/31/1981  13.97%  13.98%  13.977%    104.77%  104.76%  104.34%
 20  12/31/1982  10.32%  10.36%  10.347%    134.62%  134.53%  135.88%
 21  12/31/1983  11.77%  11.82%  11.803%    102.61%  102.53%  102.05%
 22  12/31/1984  11.52%  11.55%  11.540%    113.34%  113.28%  113.37%
 23  12/31/1985   8.87%   9.00%   8.957%    127.13%  126.84%  127.92%
 24  12/31/1986   7.09%   7.23%   7.183%    120.74%  120.42%  121.30%
 25  12/31/1987   8.67%   8.83%   8.777%     97.87%   97.57%   96.88%
 26  12/31/1988   9.18%   9.14%   9.153%    106.90%  106.98%  106.85%
 27  12/31/1989   7.97%   7.93%   7.943%    116.63%  116.72%  117.28%
 28  12/31/1990   8.00%   8.08%   8.053%    107.16%  107.00%  106.93%
 
 29  12/31/1991   6.38%   6.71%   6.600%    117.89%  117.12%  117.83%
 30  12/31/1992   6.43%   6.70%   6.610%    107.37%  106.78%  106.78%
 31  12/31/1993   5.53%   5.83%   5.730%    113.38%  112.66%  113.16%
 32  12/31/1994   7.84%   7.84%   7.840%     93.19%   93.19%   92.24%
 33  12/31/1995   5.49%   5.58%   5.550%    123.73%  123.50%  124.81%
 34  12/31/1996   6.34%   6.43%   6.400%    100.10%   99.91%   99.45%
 35  12/31/1997   5.77%   5.75%   5.757%    111.06%  111.11%  111.49%
 36  12/31/1998   4.73%   4.65%   4.677%    113.49%  113.69%  114.39%
 37  12/31/1999   6.55%   6.45%   6.483%     92.44%   92.64%   91.68%
 38  12/31/2000   5.16%   5.12%   5.133%    115.75%  115.85%  116.66%
 
 39  12/31/2001   4.84%   5.07%   4.993%    106.02%  105.47%  105.50%
 40  12/31/2002   3.36%   3.83%   3.673%    115.61%  114.36%  115.21%
 41  12/31/2003   3.77%   4.27%   4.103%    101.81%  100.60%  100.31%
 42  12/31/2004   3.94%   4.24%   4.140%    105.23%  104.49%  104.51%
 43  12/31/2005   4.36%   4.39%   4.380%    103.22%  103.14%  103.05%
 44  12/31/2006   4.70%   4.71%   4.707%    102.11%  102.09%  101.88%
 45  12/31/2007   3.70%   4.04%   3.927%    110.55%  109.68%  110.13%
 46  12/31/2008   1.87%   2.25%   2.123%    119.59%  118.48%  119.91%
 47  12/31/2009   3.39%   3.85%   3.697%     91.34%   90.27%   89.18%
 48  12/31/2010   2.71%   3.30%   3.103%    109.64%  108.07%  108.47%
 
 49  12/31/2011   1.35%   1.89%   1.710%    116.46%  114.87%  116.04%
 50  12/31/2012   1.18%   1.78%   1.580%    104.47%  102.80%  102.89%
 51  12/31/2013   2.45%   3.04%   2.843%     93.44%   91.99%   91.05%
 52  12/31/2014   1.97%   2.17%   2.103%    110.65%  110.08%  110.79%
 53  12/31/2015   2.09%   2.27%   2.210%    101.85%  101.36%  101.28%
 54  12/31/2016   2.25%   2.45%   2.383%    101.36%  100.83%  100.69%
 55  12/31/2017   2.33%   2.40%   2.377%    103.04%  102.85%  102.89%
 56  12/31/2018   2.59%   2.69%   2.657%    100.37%  100.11%   99.89%
 57  12/31/2019   1.83%   1.92%   1.890%    109.25%  109.00%  109.64%
 58  12/31/2020   0.65%   0.93%   0.837%    111.27%  110.43%  111.33%
  1. This is because for most of the years 1970 through 2020, the yield curve rises, meaning the yield of a nine-year maturity was less than that of a ten-year maturity. In such a case one makes extra money by repeatedly selling a 10-year after a year and buying a new 10-year.
  2. Sample formulas for cells E4 & E5:
    E4: Case A: 39.04 = PRODUCT(E8:E58)
    E5: Case A: 7.45% = 39.04 ^ (1 / 51) - 1
  3. Sample formulas for cells E8:G8:
    E8: Case A: 117.66% = -PV(6.417%, 9, 7.88%, 1, 0) + 7.88%
    F8: Case B: 117.07% = -PV(6.500%, 9, 7.88%, 1, 0) + 7.88%
    G8: Case C: 117.80% = -PV(6.500%, 10, 7.88%, 1, 0) + 7.88%
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Ben Mathew »

#Cruncher wrote: Mon May 03, 2021 2:59 pm He's valuing each bond at the end of the year assuming it will be priced based on next year's 10-year yield. But in fact it will be priced based on next year's 9-year yield. Since he doesn't have that yield, he's stuck using the 10-year yield. And since his yield is wrong, his pricing will also be wrong whether he assumes the remaining life is 9 years or 10 years.
Am I understanding correctly that this is a problem with Shiller's calculation as well? i.e. He should be using the 119 month yield to calculate the selling price, but is stuck using the 120 month yield because that's all he has?

If so, I can see that Shiller's monthly return calculation may be more accurate than Damodaran's annual return calculation (using ^9 to keep the formula the same) because the difference in yield between 119 months vs 120 months is less than the difference in yield between 9 year and 10 year bonds. But then, maybe the fact that the smaller monthly error is being made 12 times per year wipes out that advantage?

To check, I added Shiller's estimates to your comparison. Shiller's CAGR was 7.45%, exactly the same as your calculation for the nine-year yield for a bond maturing in nine years. But though the average was correct, it had deviated in the middle, so it's not clear that it's better than Damodaran's estimate. So maybe a small monthly error * 12 is not better than a bigger annual error?

(And, as you said, Damodaran^9 and Damodaran^10 are practically the same. But it's helpful to know that I wasn't missing some reasoning behind his choice to use ^10.)

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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Ben Mathew »

I thought it would be interesting to compare the above estimates against a hypothetical that does not include price changes (i.e. return for the year is assumed to be the starting yield for the year). The results are plotted in the graph below as the "coupon only" series.

Image

There was a significant difference. "Coupon only" averaged a CAGR of 6.16%, quite a bit lower than the CAGR of 7.02% to 7.45% when you include price changes. It pulled ahead briefly in the 70s as interest rates rose, but lagged since then as interest rates fell.

Something to keep in mind when simulating bond returns is that the historical return included a bonus from interest rate declines that we cannot count on repeating going forward. In other words, the historical return from holding 10 year bonds was greater than the expected return because the price changes have been favorable in recent decades. Something similar would be true for stocks as well--historical stock returns have been aided by the tailwind of valuation increase, and our forward looking expectations should correct for that.
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Aristarchus »

Hello from Athens Greece.

I had a similar question recently that eventually led me here.
First of all, thank you Ben and Cruncher, you enlightened me on the issue.

If we were to adjust the formula in order to extract the returns on a daily basis (utilizing daily yields), would a modification of 1200 to 25200 and -199 to -2519, be sufficient?

Thank you in advance.
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Ben Mathew »

Aristarchus wrote: Wed Feb 09, 2022 12:02 pm If we were to adjust the formula in order to extract the returns on a daily basis (utilizing daily yields), would a modification of 1200 to 25200 and -199 to -2519, be sufficient?
I assume -199 above is a typo and should be -119.

Looks like you are assuming 252 days in a year. If so:

- for coupon calculation divide by 252 days per year *100 = 25200 (instead of divide by 12 months per year * 100 = 1200)

- In the return calculation, use 2520 days in 10 years - 1 = 2519 days (instead of 120 months in 10 years - 1 = 119 months)

So, yes, that looks right.

In case it's helpful, here are the notes that I had made at the time to help me understand the derivation based on #Cruncher's response:

Derivation of Shiller and Damodaran's return formula
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Aristarchus »

Ben Mathew wrote: Thu Feb 10, 2022 11:08 am
Aristarchus wrote: Wed Feb 09, 2022 12:02 pm If we were to adjust the formula in order to extract the returns on a daily basis (utilizing daily yields), would a modification of 1200 to 25200 and -199 to -2519, be sufficient?
I assume -199 above is a typo and should be -119.

Looks like you are assuming 252 days in a year. If so:

- for coupon calculation divide by 252 days per year *100 = 25200 (instead of divide by 12 months per year * 100 = 1200)

- In the return calculation, use 2520 days in 10 years - 1 = 2519 days (instead of 120 months in 10 years - 1 = 119 months)

So, yes, that looks right.

In case it's helpful, here are the notes that I had made at the time to help me understand the derivation based on #Cruncher's response:

Derivation of Shiller and Damodaran's return formula
Hello Ben,

Yeah, brain messed up which number should be typed twice. :happy
Thank you for your comment and your kindness to share your notes. Very much appreciated.

Wanting to verify my calculations, I compared the results derived from the 10 year yields with IEF (7-10 years ETF) and the 30 year yields with VUSTX (Long-Term Treasuries MF). The results are somewhat off, not in an alarming way, but I guess this can be partly attributed to the handful assumptions (i.e. a....daily coupon) or the not identical maturities/durations of the bonds within the baskets of these funds.

Thanks again!
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Peter23454 »

hi all,
thanks for these calculations.
I have an additional question: I'm trying to understand the meaning of the annualized bonds real return. I understand this is what i would have earned had I invested in 10-year T-bonds. But what I actually want to know is what I would have made had I put my money in a savings account at the bank. Is this annualized bonds real return a good measure of that, because it is calculated from 10-year long-interest rates? Or would I need a different dataset for this (and if so, do you have recommendations for which one)?
thanks for the help!
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Ben Mathew »

Peter23454 wrote: Fri Jun 10, 2022 3:41 am I understand this is what i would have earned had I invested in 10-year T-bonds. But what I actually want to know is what I would have made had I put my money in a savings account at the bank. Is this annualized bonds real return a good measure of that, because it is calculated from 10-year long-interest rates? Or would I need a different dataset for this (and if so, do you have recommendations for which one)?
I'm guessing that 10 year T-bonds will generally have higher interest rates than bank savings accounts. Shorter term bonds might be a better proxy. You may want to check out these sources for alternate data series:

Simba's backtesting spreadsheet [a Bogleheads community project]


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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Peter23454 »

thanks Ben. How would I then calculate the total (or average annual) interest for putting money in a savings account at the bank? Would it be the same formula as that Shiller uses for bond returns?
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Ben Mathew »

Peter23454 wrote: Wed Jun 15, 2022 2:32 am thanks Ben. How would I then calculate the total (or average annual) interest for putting money in a savings account at the bank? Would it be the same formula as that Shiller uses for bond returns?
The formula will work for bonds of any maturity. Just adjust for the number of months. So if you're rolling over 5 year bonds, you would use 60-1=59 months instead of 120-1=119 months for rolling over 10 year bonds.

The formula above assumes monthly coupons. Short term bonds may pay no coupons (T bills don't). Then the math becomes simpler. For a 1 year bond that pays no coupons, the monthly return would be:

Monthly bond return for period t = ((1+Yt)/((1+Yt+1)^(11/12))) - 1

where Yt = bond yield in month t
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Peter23454 »

Hi all,
I have another question related to Shiller's bond data. I'm interested to calculate nominal historical annual bond interests, so uncorrected for inflation. To keep things as simple as possible: do you think I could use Shiller's file for this and just change all the values in the CPI column to the same number, e.g. "1"? That would result in annualized bond and stock interest rates, uncorrected for inflation right? Or would I be making a calculation error then?
thanks for your input,
Peter
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Re: Deriving Shiller's formula for Monthly Total Bond Returns

Post by Telloverture »

Ben Mathew wrote: Fri May 07, 2021 7:26 pm
Something to keep in mind when simulating bond returns is that the historical return included a bonus from interest rate declines that we cannot count on repeating going forward. In other words, the historical return from holding 10 year bonds was greater than the expected return because the price changes have been favorable in recent decades. Something similar would be true for stocks as well--historical stock returns have been aided by the tailwind of valuation increase, and our forward looking expectations should correct for that.
This is a spot-on observation. People who bought intermediate term bond funds 5 years ago when the yield was 2% should not be surprised that the value of their bond fund fell when rates rose this year. Their bonds are performing exactly as you would expect them to and they'll probably end up making right around 2%.

That said, I think even the "coupon only" series may be optimistic in the long run. If I understand the construction, you are "reinvesting" every year at the coupon rate. You lag the total return series because you're not recognizing annual gains when interest rates fall. But you're also not recognizing annual losses when they go up (probably a good part of the reason why the coupon series is ahead in those years).

I've been mulling this issue for some time. The way I tried to get at this was to take the income distribution yield (basically the coupon payments on the bonds) from a Vanguard fund but ignore the cap gains distributions. The distribution data on Vanguard's site only goes back 10 years. For those ten years, compounding the monthly yield gave me a CAGR of 1.7% for the Intermediate Term Treasury Fund and 2.7% for the Long-Term Treasury Fund. The Shiller 10-year interest rate data, which falls in between those funds in terms of maturity, compounded at a CAGR of 2.1% for the same period. So in a low/declining rate environment your approach approximates pretty well.
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