Assessing Recovery Time for Bond Funds

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LMK5
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Assessing Recovery Time for Bond Funds

Post by LMK5 »

I'm hoping some of the math/spreadsheet gurus can help me evaluate the following scenario. I'm trying to figure out how long it will take to recover losses in a bond fund that has experienced rising rates over a prolonged period. Here's the scenario:

Let's say investor #1 buys a 10yr individual bond with a 2% coupon and decides to hold it to maturity. Let's say investor #2 buys a bond fund on the same date, with a fixed duration of 10 years and a current yield of 2%. Over the next 5 years, market rates increase 1% per year for each of the first 5 years, then remain steady for the next 5 years.

Questions:
1) What will be the cumulative return for each investor after the 10-year period?
2) If interest rates remain steady after the 10-year period, how many additional years will it take for investor #2's bond fund to achieve the same cumulative return that investor #1 achieved over the first 10 years?
ruud
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Re: Assessing Recovery Time for Bond Funds

Post by ruud »

I’m assuming investor #2 reinvests the distributions into the fund.
What does investor #1 do with the coupon payments?
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ryman554
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Re: Assessing Recovery Time for Bond Funds

Post by ryman554 »

ruud wrote: Fri May 13, 2022 8:03 pm I’m assuming investor #2 reinvests the distributions into the fund.
What does investor #1 do with the coupon payments?
Perhaps the better analog would be "buy a 10 year CD at 2% with dividends reinvested .. at some relative rate"?

Wouldn't that be similar in intent to what the OP was trying to compare to buying a bond fund, but with no interest rate risk?
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LMK5
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Re: Assessing Recovery Time for Bond Funds

Post by LMK5 »

ryman554 wrote: Fri May 13, 2022 9:00 pm
ruud wrote: Fri May 13, 2022 8:03 pm I’m assuming investor #2 reinvests the distributions into the fund.
What does investor #1 do with the coupon payments?
Perhaps the better analog would be "buy a 10 year CD at 2% with dividends reinvested .. at some relative rate"?

Wouldn't that be similar in intent to what the OP was trying to compare to buying a bond fund, but with no interest rate risk?
Sure, I’m just trying to create a scenario to see how far behind investor #2 gets and how long it takes for him to get to investor #1’s cumulative return after those 5 yearly rate hikes.
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LTCM
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Re: Assessing Recovery Time for Bond Funds

Post by LTCM »

LMK5 wrote: Fri May 13, 2022 9:42 pm
ryman554 wrote: Fri May 13, 2022 9:00 pm
ruud wrote: Fri May 13, 2022 8:03 pm I’m assuming investor #2 reinvests the distributions into the fund.
What does investor #1 do with the coupon payments?
Perhaps the better analog would be "buy a 10 year CD at 2% with dividends reinvested .. at some relative rate"?

Wouldn't that be similar in intent to what the OP was trying to compare to buying a bond fund, but with no interest rate risk?
Sure, I’m just trying to create a scenario to see how far behind investor #2 gets and how long it takes for him to get to investor #1’s cumulative return after those 5 yearly rate hikes.
I think it's the difference in the duration. If the 10 year bond has a duration of 8 and the fund has a duration of 10 then it takes the fund owner 2 more years to break even.

The bond owner is level on the change in rates after 8 years. The fund is level after 10 years.

That might only work exactly for a single rate change rather than the multiple changes you have in your example.
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NoRegret
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Re: Assessing Recovery Time for Bond Funds

Post by NoRegret »

:moneybag
LMK5 wrote: Fri May 13, 2022 7:09 pm I'm hoping some of the math/spreadsheet gurus can help me evaluate the following scenario. I'm trying to figure out how long it will take to recover losses in a bond fund that has experienced rising rates over a prolonged period. Here's the scenario:

Let's say investor #1 buys a 10yr individual bond with a 2% coupon and decides to hold it to maturity. Let's say investor #2 buys a bond fund on the same date, with a fixed duration of 10 years and a current yield of 2%. Over the next 5 years, market rates increase 1% per year for each of the first 5 years, then remain steady for the next 5 years.

Questions:
1) What will be the cumulative return for each investor after the 10-year period?
2) If interest rates remain steady after the 10-year period, how many additional years will it take for investor #2's bond fund to achieve the same cumulative return that investor #1 achieved over the first 10 years?
I looked at this problem a while ago, albeit not deeply. My conclusion was 2.5 years + duration to get to “even” for the bond fund investor. The result can be generalized to X years of linear rate increase and taking X/2 + duration years to get even up to X= 2 times duration.

EVEN is defined as total principal and dividends = original investment + original yield * years of holding.

Somebody should check my math.
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jeffyscott
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Re: Assessing Recovery Time for Bond Funds

Post by jeffyscott »

I think you would need to know more than just the initial duration of the fund.
https://www.bogleheads.org/wiki/Bonds:_ ... #Convexity

Another comparison would be something like:
Investor #1 buys a single 10 year bond.
Investor #2 buys a rolling ladder: 10 year, 9 year, 8 year, 7 year,..., 1 year, when each matures, buys a new 10 year. After the first year, when the 1 year bond matures, the new 10 year rate is 3%, etc.

That seems to me to be more comparable choices by the two investors, since they will each hold the same average maturity over the 10 year period. The 10 year bond will have a gradually declining remaining maturity, which will average 5 years. The rolling ladder will have a (nearly) constant average remaining maturity of 5 years.
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Re: Assessing Recovery Time for Bond Funds

Post by #Cruncher »

LMK5 wrote: Fri May 13, 2022 7:09 pmLet's say investor #1 buys a 10yr individual bond with a 2% coupon and decides to hold it to maturity. Let's say investor #2 buys a bond fund on the same date, with a fixed duration of 10 years and a current yield of 2%. Over the next 5 years, market rates increase 1% per year for each of the first 5 years, then remain steady for the next 5 years.
For simplicity I'll assume zero-coupon bonds [ a ] initially with a yield-to-maturity (YTM) of 2%. Again for simplicity I'll assume the "fund" holds a single 10-year bond replacing it every year with a new 10-year bond. [ b ] I'll also assume a flat yield curve; e.g., the same yields for 1-year as for 10-year bonds. Here are the values each year assuming we start with a $1,000 face value zero-coupon bond.

Code: Select all

Year    YTM    --- Single 10-year Bond ---   -- 10-year Bond Rolled Every Year --
   0     2%     820.35 = 1000.00 / 1.02^10    820.35 = 1000.00 / 1.02^10
   1     3%     766.42 = 1000.00 / 1.03^9     766.42 =  820.35 * 1.02^10 / 1.03^9
   2     4%     730.69 = 1000.00 / 1.04^8     723.66 =  766.42 * 1.03^10 / 1.04^9
   3     5%     710.68 = 1000.00 / 1.05^7     690.51 =  723.66 * 1.04^10 / 1.05^9
   4     6%     704.96 = 1000.00 / 1.06^6     665.74 =  690.51 * 1.05^10 / 1.06^9
   5     7%     712.99 = 1000.00 / 1.07^5     648.50 =  665.74 * 1.06^10 / 1.07^9
   6     7%     762.90 = 1000.00 / 1.07^4     693.90 =  648.50 * 1.07^10 / 1.07^9
   7     7%     816.30 = 1000.00 / 1.07^3     742.47 =  693.90 * 1.07^10 / 1.07^9
   8     7%     873.44 = 1000.00 / 1.07^2     794.44 =  742.47 * 1.07^10 / 1.07^9
   9     7%     934.58 = 1000.00 / 1.07^1     850.05 =  794.44 * 1.07^10 / 1.07^9
  10     7%   1,000.00 = 1000.00 / 1.07^0     909.56 =  850.05 * 1.07^10 / 1.07^9
  11     7%                                   973.23 =  909.56 * 1.07^10 / 1.07^9
  12     7%                                 1,041.35 =  973.23 * 1.07^10 / 1.07^9
LMK5, in same post, wrote:
  1. What will be the cumulative return for each investor after the 10-year period?
  2. If interest rates remain steady after the 10-year period, how many additional years will it take for investor #2's bond fund to achieve the same cumulative return that investor #1 achieved over the first 10 years?
  1. After ten years the return for the single bond is naturally 2%. For the "fund" it's 1.0%.
    2.0% = (1000.00 / 820.35) ^ (1 / 10) - 1
    1.0% = ( 909.56 / 820.35) ^ (1 / 10) - 1
  2. The "fund" will have a 2% return after another 1.98 years. [ c ]
    $1,040 = 820.35 * 1.02 ^ 11.98
    $1,040 = 909.56 * 1.07 ^ 1.98
  1. The math is simpler with zero-coupon bonds and we don't have to worry about reinvesting coupons.
  2. The duration of the single bond will start out at 10 years and gradually decline to 0 at maturity averaging 5. The duration of the "fund" will start each year at 10 and decline to 9 at year's end, averaging 9.5.
  3. Calculated as follows using Excel LN function:
    1.98 = (LN(909.56) - LN(820.35) - 10 * LN(1.02)) / (LN(1.02) - LN(1.07))
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Re: Assessing Recovery Time for Bond Funds

Post by jeffyscott »

#Cruncher wrote: Sat May 14, 2022 12:50 pm Again for simplicity I'll assume the "fund" holds a single 10-year bond replacing it every year with a new 10-year bond. [ b ] I'll also assume a flat yield curve; e.g., the same yields for 1-year as for 10-year bonds.
Wouldn't it actually be only the 10 and 9 year yields that need to be the same, since that's all that the fund owns/trades?
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LMK5
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Re: Assessing Recovery Time for Bond Funds

Post by LMK5 »

#Cruncher wrote: Sat May 14, 2022 12:50 pm
LMK5 wrote: Fri May 13, 2022 7:09 pmLet's say investor #1 buys a 10yr individual bond with a 2% coupon and decides to hold it to maturity. Let's say investor #2 buys a bond fund on the same date, with a fixed duration of 10 years and a current yield of 2%. Over the next 5 years, market rates increase 1% per year for each of the first 5 years, then remain steady for the next 5 years.
For simplicity I'll assume zero-coupon bonds [ a ] initially with a yield-to-maturity (YTM) of 2%. Again for simplicity I'll assume the "fund" holds a single 10-year bond replacing it every year with a new 10-year bond. [ b ] I'll also assume a flat yield curve; e.g., the same yields for 1-year as for 10-year bonds. Here are the values each year assuming we start with a $1,000 face value zero-coupon bond.

Code: Select all

Year    YTM    --- Single 10-year Bond ---   -- 10-year Bond Rolled Every Year --
   0     2%     820.35 = 1000.00 / 1.02^10    820.35 = 1000.00 / 1.02^10
   1     3%     766.42 = 1000.00 / 1.03^9     766.42 =  820.35 * 1.02^10 / 1.03^9
   2     4%     730.69 = 1000.00 / 1.04^8     723.66 =  766.42 * 1.03^10 / 1.04^9
   3     5%     710.68 = 1000.00 / 1.05^7     690.51 =  723.66 * 1.04^10 / 1.05^9
   4     6%     704.96 = 1000.00 / 1.06^6     665.74 =  690.51 * 1.05^10 / 1.06^9
   5     7%     712.99 = 1000.00 / 1.07^5     648.50 =  665.74 * 1.06^10 / 1.07^9
   6     7%     762.90 = 1000.00 / 1.07^4     693.90 =  648.50 * 1.07^10 / 1.07^9
   7     7%     816.30 = 1000.00 / 1.07^3     742.47 =  693.90 * 1.07^10 / 1.07^9
   8     7%     873.44 = 1000.00 / 1.07^2     794.44 =  742.47 * 1.07^10 / 1.07^9
   9     7%     934.58 = 1000.00 / 1.07^1     850.05 =  794.44 * 1.07^10 / 1.07^9
  10     7%   1,000.00 = 1000.00 / 1.07^0     909.56 =  850.05 * 1.07^10 / 1.07^9
  11     7%                                   973.23 =  909.56 * 1.07^10 / 1.07^9
  12     7%                                 1,041.35 =  973.23 * 1.07^10 / 1.07^9
LMK5, in same post, wrote:
  1. What will be the cumulative return for each investor after the 10-year period?
  2. If interest rates remain steady after the 10-year period, how many additional years will it take for investor #2's bond fund to achieve the same cumulative return that investor #1 achieved over the first 10 years?
  1. After ten years the return for the single bond is naturally 2%. For the "fund" it's 1.0%.
    2.0% = (1000.00 / 820.35) ^ (1 / 10) - 1
    1.0% = ( 909.56 / 820.35) ^ (1 / 10) - 1
  2. The "fund" will have a 2% return after another 1.98 years. [ c ]
    $1,040 = 820.35 * 1.02 ^ 11.98
    $1,040 = 909.56 * 1.07 ^ 1.98
  1. The math is simpler with zero-coupon bonds and we don't have to worry about reinvesting coupons.
  2. The duration of the single bond will start out at 10 years and gradually decline to 0 at maturity averaging 5. The duration of the "fund" will start each year at 10 and decline to 9 at year's end, averaging 9.5.
  3. Calculated as follows using Excel LN function:
    1.98 = (LN(909.56) - LN(820.35) - 10 * LN(1.02)) / (LN(1.02) - LN(1.07))
Great stuff #cruncher. You stated: "After ten years the return for the single bond is naturally 2%. For the "fund" it's 1.0%.". Wouldn't the 2% and 1% returns be average annual total returns and not cumulative returns?
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Re: Assessing Recovery Time for Bond Funds

Post by #Cruncher »

jeffyscott wrote: Sat May 14, 2022 6:18 amAnother comparison would be something like:
Investor #1 buys a single 10 year bond.
Investor #2 buys a rolling ladder: 10 year, 9 year, 8 year, 7 year,..., 1 year, when each matures, buys a new 10 year. After the first year, when the 1 year bond matures, the new 10 year rate is 3%, etc.
That would be complicated. But here's a simpler scenario that I suspect would produce a similar comparison:
  • Investor # 1 buys a single 11-year zero-coupon bond yielding 2% and holds it to maturity. The bond has a face value of $1,000 and costs $804.26 (1000 / 1.02 ^ 11). Average duration is (11 + 0) / 2 or 5.5 years.
  • Investor # 2 invests the same $804.26 in a "fund" that initially holds a 6-year zero coupon bond which it rolls over every year to a new 6-year bond. Average duration is (6 + 5) / 2 or 5.5 years.
Here are the results with the same progression of interest rates. The "fund" does better.

Code: Select all

Year  Yield       Bond    "Fund"    "Fund" Value Formula
   0     2%     804.26    804.26  = 1000.00 / 1.02^11
   1     3%     744.09    781.29  =  804.26 * 1.02^6 / 1.03^5
   2     4%     702.59    766.78  =  781.29 * 1.03^6 / 1.04^5
   3     5%     676.84    760.19  =  766.78 * 1.04^6 / 1.05^5
   4     6%     665.06    761.25  =  760.19 * 1.05^6 / 1.06^5
   5     7%     666.34    769.92  =  761.25 * 1.06^6 / 1.07^5
   6     7%     712.99    823.82  =  769.92 * 1.07^6 / 1.07^5
   7     7%     762.90    881.48  =  823.82 * 1.07^6 / 1.07^5
   8     7%     816.30    943.19  =  881.48 * 1.07^6 / 1.07^5
   9     7%     873.44  1,009.21  =  943.19 * 1.07^6 / 1.07^5
  10     7%     934.58  1,079.85  = 1009.21 * 1.07^6 / 1.07^5
  11     7%   1,000.00  1,155.44  = 1079.85 * 1.07^6 / 1.07^5
However, if interest rates fell in a symmetrical fashion, the "fund" would do worse -- even though its value never goes down.

Code: Select all

Year  Yield       Bond    "Fund"    "Fund" Value Formula
   0     7%     475.09    475.09  = 1000.00 / 1.07^11
   1     6%     558.39    532.78  =  475.09 * 1.07^6 / 1.06^5
   2     5%     644.61    592.16  =  532.78 * 1.06^6 / 1.05^5
   3     4%     730.69    652.24  =  592.16 * 1.05^6 / 1.04^5
   4     3%     813.09    711.91  =  652.24 * 1.04^6 / 1.03^5
   5     2%     887.97    769.92  =  711.91 * 1.03^6 / 1.02^5
   6     2%     905.73    785.32  =  769.92 * 1.02^6 / 1.02^5
   7     2%     923.85    801.03  =  785.32 * 1.02^6 / 1.02^5
   8     2%     942.32    817.05  =  801.03 * 1.02^6 / 1.02^5
   9     2%     961.17    833.39  =  817.05 * 1.02^6 / 1.02^5
  10     2%     980.39    850.05  =  833.39 * 1.02^6 / 1.02^5
  11     2%   1,000.00    867.06  =  850.05 * 1.02^6 / 1.02^5
jeffyscott wrote: Sat May 14, 2022 2:11 pm
#Cruncher wrote: Sat May 14, 2022 12:50 pm... I'll also assume a flat yield curve; e.g., the same yields for 1-year as for 10-year bonds.
Wouldn't it actually be only the 10 and 9 year yields that need to be the same, since that's all that the fund owns/trades?
I'm also computing the value of the single bond each year 10, 9, ..., 2, and 1.
LMK5 wrote: Sat May 14, 2022 8:38 pmYou stated: "After ten years the return for the single bond is naturally 2%. For the "fund" it's 1.0%.". Wouldn't the 2% and 1% returns be average annual total returns and not cumulative returns?
Yes, I'm computing annual returns; Compound Annual Growth Rates (CAGR) to be precise.
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Re: Assessing Recovery Time for Bond Funds

Post by MIretired »

Very interesting, #cruncher!
Contrary to my misguided analysis:
A 1yr bill has zero duration risk when measured in 1 yr. intervals.
But a 2 yr. bill has (2-0)/2 yrs. avg. duration risk not assuming coupons.
I think that's correct.

Edit: But in fact it's way too interesting for me to fathom at this time.

Edit: using limits, it's (1/3 2 yr + 1/3 1 yr + 1/3 0 yr) / 3 avg duration, = (2 + 1 + 0)/3 = 1 yr avg duration, correct?

Edit:And thanks dbr and nisiprius and others.
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