Does YTM require that coupons are re-invested?

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km91
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Re: Does YTM require that coupons are re-invested?

Post by km91 »

dbr wrote: Tue Nov 08, 2022 1:05 pm So, my question would be what reason do financial professionals have to define and calculate YTM other than to estimate the total return? Or, is the concept useless except for that purpose? After all, an investor could very likely spend the cash rather than reinvest it and would have use for a calculation of the present value of the cash flows without reinvestment.
The YTM relates the cashflow structure of the bond to it's yield and price. If we want to convert a yield to a dollar price for the bond we have to use the YTM formula. I own a 30yr Treasury that pays 6% coupons and has 2 years to maturity that I want to sell. The YTM on the 2yr is 4.70% right now. To convert my bond to a dollar price I need to apply the YTM formula and solve for price. This application of the YTM formula makes no assumption about what I could've done with the coupons

If we know two of the characteristics of the bond, we can solve for the third. If we know yield and coupons we can solve for price, if we know price and coupons we can solve for yield, and if we know price and yield we can solve for coupons. This last one isn't very useful, there's not a lot of times in real life that we'd be given the yield and price and would want to find out what the coupon is. The useful applications are pricing the bond when we know it's yield, or estimating forward yield (or total return if we introduce an assumption around reinvestment) when we are given the price
acegolfer
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Re: Does YTM require that coupons are re-invested?

Post by acegolfer »

dbr wrote: Tue Nov 08, 2022 1:05 pm So, my question would be what reason do financial professionals have to define and calculate YTM other than to estimate the total return? Or, is the concept useless except for that purpose? After all, an investor could very likely spend the cash rather than reinvest it and would have use for a calculation of the present value of the cash flows without reinvestment.

My personal attitude toward that is that how things are defined and for what purpose the result is used are two different things and should not get mixed up. The Wiki article jumps over the pure definition to land on the use of estimating the return and in the process makes a false statement about what the definition of YTM actually is.
The above wiki states "Hence it is not an expected, or risk-adjusted rate." Unfortunately, ppl use YTM as the return estimate.

If you are asking me what's the purpose of bond yield, my answer is it measures the risk of a bond. Higher yield = riskier bond.
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Re: Does YTM require that coupons are re-invested?

Post by lws »

Great post.
Good definitions are very important.
We need them to have great discussions.
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Oicuryy
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Re: Does YTM require that coupons are re-invested?

Post by Oicuryy »

longvista wrote: Tue Nov 08, 2022 12:17 pm The critical point here is that the yield is not compounding - as @acegolfer puts it: It is a "non-compounded annualized return". However, CAGR is a compounded annualized return.

As per the research paper, the original intention of the YTM metric is to talk about the "non-compounded annualized return" (yield), not CAGR. Somewhere along the way, people got confused and started interpreting YTM in the sense of CAGR, instead of yield. As @dbr puts it, they might be confusing return with yield.
YTM is a compounded rate of return. Here is how that research paper puts it. (emphasis in original)
Figure 1 shows how the present value amounts that sum to the bond’s $1,000 price are earning a 5% compound rate of return in becoming the coupon or face value that each represents, e.g., the $43.19 part of the $1,000 price has earned 5% compounded for 3 years when the $50 coupon is received in year 3 ($43.19 x 1.053). Since each dollar of the bond’s price is earning a 5% compound rate of return it follows that by paying the price of the bond and receiving the promised coupons and face value at maturity that the investor earns the calculated YTM.
https://www.economics-finance.org/jefe/ ... lpaper.pdf

Both CAGR and YTM are compounded rates. The difference is in how they treat the coupons. CAGR treats the coupons as if they were all paid on the maturity date. YTM treats the coupons as being paid on the dates they were actually paid.

Ron
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Re: Does YTM require that coupons are re-invested?

Post by #Cruncher »

Oicuryy wrote: Tue Nov 08, 2022 5:31 pmYTM is a compounded rate of return.
Correct. This can be seen from the first table in my previous post which treated the bond as an analogous savings account. Here are the first three columns:

Code: Select all

Row     Col A      Col B      Col C      Col D  Formula in Col B copied right
  2      Rate       5.0%
  3  Withdraw       0.00      30.00      50.00
  4      Year    Balance
  5         0   1,000.00   1,000.00   1,000.00
  6         1   1,050.00   1,020.00   1,000.00  =B5*(1+$B$2)-B$3
  7         2   1,102.50   1,041.00   1,000.00  =B6*(1+$B$2)-B$3
  8         3   1,157.63   1,063.05   1,000.00  =B7*(1+$B$2)-B$3
  9      RATE     5.000%     5.000%     5.000%  =RATE(3,B3,-B5,B8,0)
Interest is paid on whatever the balance is in all three cases. That's what compounding amounts to. It's perhaps easiest to see in the middle case with a $30 annual withdrawal (i.e., coupon that is not reinvested). $1000 X 5% = $50 interest the first year. $1,020 X 5% = $51 interest the second year. And $1,041 X 5% = $52.05 interest the last year.
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Re: Does YTM require that coupons are re-invested?

Post by acegolfer »

#Cruncher wrote: Tue Nov 08, 2022 6:13 pm Interest is paid on whatever the balance is in all three cases.
So you are assuming coupons are reinvested at 5% YTM? Or, am I missing something? Curious in column B, how the $50 also earns 5% interest in yr-2.
km91
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Re: Does YTM require that coupons are re-invested?

Post by km91 »

acegolfer wrote: Tue Nov 08, 2022 6:34 pm
#Cruncher wrote: Tue Nov 08, 2022 6:13 pm Interest is paid on whatever the balance is in all three cases.
So you are assuming coupons are reinvested at 5% YTM? Or, am I missing something? Curious in column B, how the $50 also earns 5% interest in yr-2.
Column B is a savings account, by definition the interest is reinvested at the YTM, so YTM = realized return.

Column D mimics the cashflow of a bond. The interest is distributed to us and principal remains constant at $1000. The only way we can get our realized return = YTM is to find an investment for our interest coupon that also yields 5%

The YTM of each investment is the same and determined by the interest earned on the principal balance. Being able to roll interest into principal at a rate close to our starting YTM will determine how far our realized return drifts from the YTM
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Re: Does YTM require that coupons are re-invested?

Post by #Cruncher »

Referring to my previous post ...
km91 wrote: Tue Nov 08, 2022 6:56 pmColumn B is a savings account, by definition the interest is reinvested at the YTM, so YTM = realized return.

Column D mimics the cashflow of a bond. The interest is distributed to us and principal remains constant at $1000. The only way we can get our realized return = YTM is to find an investment for our interest coupon that also yields 5%
All three columns: B, C, and D represent both a bond and a savings account. While having to explain an analogy may mean it wasn't a good one in the first place, I'm going to try anyway. :wink:

For context, here's the table from the previous post.

Code: Select all

Row     Col A      Col B      Col C      Col D  Formula in Col B copied right
  2      Rate       5.0%
  3  Withdraw       0.00      30.00      50.00
  4      Year   ----------- Balance ----------
  5         0   1,000.00   1,000.00   1,000.00
  6         1   1,050.00   1,020.00   1,000.00  =B5*(1+$B$2)-B$3
  7         2   1,102.50   1,041.00   1,000.00  =B6*(1+$B$2)-B$3
  8         3   1,157.63   1,063.05   1,000.00  =B7*(1+$B$2)-B$3
  9      RATE     5.000%     5.000%     5.000%  =RATE(3,B3,-B5,B8,0)
Here's a comparison of the bond and analogous savings account transactions:

Code: Select all

        Years            Bond                     Savings Account
-----   -----   -------------------------   ----------------------------
          0     You give issuer $1,000      You give bank $1,000
Col B   1 & 2   Issuer pays you nothing     You take nothing from bank
          3     Issuer pays you $1,157.63   You take $1,157.63 from bank
-----   -----   -------------------------   ----------------------------
          0     You give issuer $1,000      You give bank $1,000
Col C   1 & 2   Issuer pays you $30         You take $30 from bank
          3     Issuer pays you $1,093.05   You take $1,093.05 from bank
-----   -----   -------------------------   ----------------------------
          0     You give issuer $1,000      You give bank $1,000
Col D   1 & 2   Issuer pays you $50         You take $50 from bank
          3     Issuer pays you $1,050      You take $1,050 from bank
Note that in all three cases, the cash flows are identical for the bond and the savings account. Therefore, the returns are also identical for the bond and the savings account. So, if the return is compounded in the case of the savings account, it is also compounded in the case of the bond. That was the point of the analogy in the previous post.

Edit 11/9/22 7:45 AM. Corrected Year 3 in comparison table to include final year's interest for "Col C" and "Col D".
Last edited by #Cruncher on Wed Nov 09, 2022 7:47 am, edited 1 time in total.
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Re: Does YTM require that coupons are re-invested?

Post by InvestInPasta »

So if I put $1000 in a 5 years bond with an YTM of 10%, how much money will I have at maturity date?

$1500? Is it correct?
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acegolfer
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Re: Does YTM require that coupons are re-invested?

Post by acegolfer »

InvestInPasta wrote: Wed Nov 09, 2022 6:29 am So if I put $1000 in a 5 years bond with an YTM of 10%, how much money will I have at maturity date?

$1500? Is it correct?
yes, assuming it's a par bond with 10% coupon rate and you keep the coupons as cash (earning 0%). So the compounded annualized return (from terminal wealth after 5 yrs) would be (1500/1000)^(1/5) - 1 = 8.45% (less than 10% YTM).
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Re: Does YTM require that coupons are re-invested?

Post by JackoC »

acegolfer wrote: Tue Nov 08, 2022 2:42 pm
dbr wrote: Tue Nov 08, 2022 1:05 pm So, my question would be what reason do financial professionals have to define and calculate YTM other than to estimate the total return? Or, is the concept useless except for that purpose? After all, an investor could very likely spend the cash rather than reinvest it and would have use for a calculation of the present value of the cash flows without reinvestment.

My personal attitude toward that is that how things are defined and for what purpose the result is used are two different things and should not get mixed up. The Wiki article jumps over the pure definition to land on the use of estimating the return and in the process makes a false statement about what the definition of YTM actually is.
The above wiki states "Hence it is not an expected, or risk-adjusted rate." Unfortunately, ppl use YTM as the return estimate.

If you are asking me what's the purpose of bond yield, my answer is it measures the risk of a bond. Higher yield = riskier bond.
I'd say the reason the bond trading world came to* quote yields was mainly to allow a more apples to apples comparison of bonds of similar maturity and risk with different coupons. Just comparing their prices or coupons in isolation is meaningless. There are in recent times other ways to do that** but it's still a convenient shorthand. My point is not to quibble just to point out that estimating E[r] as a buy and hold investor isn't where the concept of YTM came from, although that might be obvious.

I'd also clarify, though not sure anyone is saying otherwise, for credit 'riskless' bonds YTM is also a reasonable proxy for expected return. At least compared to what you sometimes hear on this forum 'well in the past realized returns on rolling portfolios of bonds didn't track initial YTM that closely so I can assume whatever I want for expected return, and I like the past realized bond return starting from much higher rates than now'. Buzzer, wrong. YTM is a better estimate than that, relatively. If one wants the best estimate then secondary factors may enter in (principally, estimate the expected term premium, if estimated significantly non-zero, for rolling constant maturity portfolios, for which the question is most commonly asked). YTM is a reasonable estimate for the riskless or the riskless component of return, from which one could then subtract the expected value of credit losses or embedded issuer call/prepayment option value where applicable, besides expenses.

*some years ago The Economist had a great article in its 'Special Holiday Edition' about the UK bond market in the 19th century, the obvious arbitrages, obvious to 20th century bond traders that is. But a contemporary observer of then exceptional perception was quoted something like 'men of great understanding of bonds are obviously few', since the arbs persisted. Likewise there's more but not super obscure stuff in fixed income that's obvious now but wasn't even 30 yrs ago.
**derive zero coupon discount function for the risk category, or since there are so many gradations of risk, evaluate various bonds on a single zero coupon discount function, the interest rate swap curve, and see what the spread to the IRS floating index comes out for each. I would say this qualifies as a standard method for risky bonds, though it's more standard in markets where the (institutional) buyer would actually do that swap to convert a fixed risky bond to synthetic floating (raise funds floating, earn a credit/liquidity/other spread w/o interest rate risk).
Last edited by JackoC on Wed Nov 09, 2022 11:15 am, edited 1 time in total.
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Re: Does YTM require that coupons are re-invested?

Post by Oicuryy »

InvestInPasta wrote: Wed Nov 09, 2022 6:29 am So if I put $1000 in a 5 years bond with an YTM of 10%, how much money will I have at maturity date?

$1500? Is it correct?
Assuming five annual coupon payments of $100 and a final payment of $1000, then the bond will pay you a total of $1500. It does not matter how much you paid for the bond or what the YTM is. $1500 is all you will get from that bond.

You will get $1100 of that $1500 on the maturity date. What the other $400 has grown or shrunk to by the maturity date depends on what you did with those four coupon payments after you received them.

Ron
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JohnFromPNW
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Re: Does YTM require that coupons are re-invested?

Post by JohnFromPNW »

acegolfer wrote: Wed Nov 09, 2022 8:03 am
InvestInPasta wrote: Wed Nov 09, 2022 6:29 am So if I put $1000 in a 5 years bond with an YTM of 10%, how much money will I have at maturity date?

$1500? Is it correct?
yes, assuming it's a par bond with 10% coupon rate and you keep the coupons as cash (earning 0%). So the compounded annualized return (from terminal wealth after 5 yrs) would be (1500/1000)^(1/5) - 1 = 8.45% (less than 10% YTM).
For this example, YTM = IRR = 10.0%. So I guess the answer to the original question is "No, YTM does not require coupons are re-invested." However, YTM is not synonymous with Compound Annual Growth Rate.

I work in a business where we discuss "returns" primarily in terms of IRR. Initially outlay, cash inflow in the interim, and lump sum inflow at a defined future date. What happens with the cash flows in the interim is irrelevant for our purposes relative to the IRR, and therefore, by extension, would be irrelevant for the YTM. However, I don't really think of my portfolio in terms of IRR.
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

acegolfer wrote: Wed Nov 09, 2022 8:03 am
InvestInPasta wrote: Wed Nov 09, 2022 6:29 am So if I put $1000 in a 5 years bond with an YTM of 10%, how much money will I have at maturity date?

$1500? Is it correct?
yes, assuming it's a par bond with 10% coupon rate and you keep the coupons as cash (earning 0%). So the compounded annualized return (from terminal wealth after 5 yrs) would be (1500/1000)^(1/5) - 1 = 8.45% (less than 10% YTM).
Oicuryy wrote: Tue Nov 08, 2022 5:31 pm
longvista wrote: Tue Nov 08, 2022 12:17 pm The critical point here is that the yield is not compounding - as @acegolfer puts it: It is a "non-compounded annualized return". However, CAGR is a compounded annualized return.

As per the research paper, the original intention of the YTM metric is to talk about the "non-compounded annualized return" (yield), not CAGR. Somewhere along the way, people got confused and started interpreting YTM in the sense of CAGR, instead of yield. As @dbr puts it, they might be confusing return with yield.
YTM is a compounded rate of return. Here is how that research paper puts it. (emphasis in original)
Figure 1 shows how the present value amounts that sum to the bond’s $1,000 price are earning a 5% compound rate of return in becoming the coupon or face value that each represents, e.g., the $43.19 part of the $1,000 price has earned 5% compounded for 3 years when the $50 coupon is received in year 3 ($43.19 x 1.053). Since each dollar of the bond’s price is earning a 5% compound rate of return it follows that by paying the price of the bond and receiving the promised coupons and face value at maturity that the investor earns the calculated YTM.
https://www.economics-finance.org/jefe/ ... lpaper.pdf

Both CAGR and YTM are compounded rates. The difference is in how they treat the coupons. CAGR treats the coupons as if they were all paid on the maturity date. YTM treats the coupons as being paid on the dates they were actually paid.

Ron
When I see an expected return quoted somewhere for some kind of investment, my instinct is to calculate the total return that I would have after some time, if I invest in it. For example, if I see a 5%/year interest rate quoted for a $10,000 bank CD with a term of 3 years, I know that at the end of 3 years, I will have 3 * (5% * $10,000) + $10,000 = $11,500, so that's a total return of $1,500.

When I look at a bond on the market, I see its price, term, YTM and other details. My immediate instinct is to ask the same thing: if I invest in this, how big will my total return be? @InvestInPasta exemplifies this question well:
InvestInPasta wrote: Wed Nov 09, 2022 6:29 am So if I put $1000 in a 5 years bond with an YTM of 10%, how much money will I have at maturity date?
However, let's take a slightly different bond (where the price is different from the principal):

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50 / year
YTM: 10.32%
So how big is the total return at maturity, if I buy this bond? Yield To Maturity sounds like it would help me to calculate this. This seems to be confirmed by quotes like the following from Investopedia:
Yield to maturity (YTM) is the total return anticipated on a bond if the bond is held until it matures.
The first instinct is to apply the same calculation methodology that I applied to the CD:

Code: Select all

5 * ($800 * 10.32%) = $412.8 (coupon payments)
$412.8 + ($1,000 - $800) = $612.8 (total return)
Hmm, didn't work. I should actually get:

Code: Select all

$50 * 5 + ($1,000 - $800) = $450
I know that I can calculate the total return based on price, term, coupon amount and face value, but I want to calculate total return through YTM instead, because that's what I believe YTM should help me to find out.

I understand the definition of YTM, but I want to know its uses. I know that YTM can be used to have an apples-to-apples comparison between different bonds, but I am interested in YTM for calculating total return for a coupon bond where the coupons are not reinvested.

Question: How can I use YTM in order to calculate my total return on a coupon bond (when coupons are not reinvested)? In which formula do I need to plug in the YTM?

Question: If calculating total return in this way is not possible, what is the intended, original use of YTM - is it only the apples-to-apples comparison?

Question: If calculating total return with YTM is not possible, then why does Investopedia think that YTM is related to total return - is it because of their erroneous interpretation of YTM as the CAGR in the case of coupon reinvestment?

EDIT: Fixed wrong YTM number, pointed out by acegolfer.
Last edited by longvista on Mon Nov 21, 2022 2:12 pm, edited 3 times in total.
acegolfer
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Re: Does YTM require that coupons are re-invested?

Post by acegolfer »

longvista wrote: Mon Nov 21, 2022 12:30 pm

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50
YTM: 5.71%
Before I can answer, you must fix the error here.

YTM is not 5.71%. It's =rate(10,50,-800,1000)*2 = 15.96%.
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longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

acegolfer wrote: Mon Nov 21, 2022 12:37 pm
longvista wrote: Mon Nov 21, 2022 12:30 pm

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50
YTM: 5.71%
Before I can answer, you must fix the error here.

YTM is not 5.71%. It's =rate(10,50,-800,1000)*2 = 15.96%.
Thanks for pointing out the mistake. I should have used 50 as coupon for RATE, but used 10. However, my formula would be:

Code: Select all

=RATE(5, 50, -800, 1000) = 10.32%
I am not sure why you double the amount of years (from 5 to 10) and multiply by 2. Could you explain?
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longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

longvista wrote: Mon Nov 21, 2022 1:39 pm
acegolfer wrote: Mon Nov 21, 2022 12:37 pm
longvista wrote: Mon Nov 21, 2022 12:30 pm

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50
YTM: 5.71%
Before I can answer, you must fix the error here.

YTM is not 5.71%. It's =rate(10,50,-800,1000)*2 = 15.96%.
Thanks for pointing out the mistake. I should have used 50 as coupon for RATE, but used 10. However, my formula would be:

Code: Select all

=RATE(5, 50, -800, 1000) = 10.32%
I am not sure why you double the amount of years (from 5 to 10) and multiply by 2. Could you explain?
I updated the bond info to be clear that the coupon payment is yearly, not semi-annual. So that should bring the YTM to 10.32%.
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Re: Does YTM require that coupons are re-invested?

Post by acegolfer »

longvista wrote: Mon Nov 21, 2022 12:30 pm When I look at a bond on the market, I see its price, term, YTM and other details. My immediate instinct is to ask the same thing: if I invest in this, how big will my total return be? @InvestInPasta exemplifies this question well:
InvestInPasta wrote: Wed Nov 09, 2022 6:29 am So if I put $1000 in a 5 years bond with an YTM of 10%, how much money will I have at maturity date?
However, let's take a slightly different bond (where the price is different from the principal):

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50 / year
YTM: 10.32%
In Investinpasta example, the coupon rate is 10% equal to 10% YTM because it's a par bond. So the annual coupon pmt should be $100 not $50.
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longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

acegolfer wrote: Mon Nov 21, 2022 2:17 pm
longvista wrote: Mon Nov 21, 2022 12:30 pm When I look at a bond on the market, I see its price, term, YTM and other details. My immediate instinct is to ask the same thing: if I invest in this, how big will my total return be? @InvestInPasta exemplifies this question well:
InvestInPasta wrote: Wed Nov 09, 2022 6:29 am So if I put $1000 in a 5 years bond with an YTM of 10%, how much money will I have at maturity date?
However, let's take a slightly different bond (where the price is different from the principal):

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50 / year
YTM: 10.32%
In Investinpasta example, the coupon rate is 10% equal to 10% YTM because it's a par bond. So the annual coupon pmt should be $100 not $50.
Yeah, in my example, the coupon payment also differs - it's $50 instead of $100. I guess it does not make much of a difference - the questions that I raised still remain the same. Do you know the answers?
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Re: Does YTM require that coupons are re-invested?

Post by jeffyscott »

longvista wrote: Mon Nov 21, 2022 12:30 pmQuestion: How can I use YTM in order to calculate my total return on a coupon bond (when coupons are not reinvested)? In which formula do I need to plug in the YTM?
Borrowing #cruncher's analogy, this is like asking:
How can I use my savings account interest rate to calculate my total return when I am withdrawing money from the account every 6 months?

And, AFAIK, that can't be answered without knowing the amounts that are to be withdrawn (i.e. coupons).
Question: If calculating total return in this way is not possible, what is the intended, original use of YTM - is it only the apples-to-apples comparison?
Yes, I'd assume the use is to tell you what the yields of bonds are in a consistent manner.
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Re: Does YTM require that coupons are re-invested?

Post by Kevin M »

longvista wrote: Mon Nov 21, 2022 12:30 pm I understand the definition of YTM, but I want to know its uses. I know that YTM can be used to have an apples-to-apples comparison between different bonds, but I am interested in YTM for calculating total return for a coupon bond where the coupons are not reinvested.

Question: How can I use YTM in order to calculate my total return on a coupon bond (when coupons are not reinvested)? In which formula do I need to plug in the YTM?
The internal rate of return (IRR) does not depend on what is done with the coupons--it's strictly a discounted cash flow thing. The IRR will be approximately equal to the initial yield if you hold to maturity. Whenever I say "yield" without a qualifier, I mean yield to maturity.

You don't plug YTM into anything. You just use it to estimate the IRR of the investment. The only way people frame the investor's personal return is to assume all coupon payments are reinvested at the original yield, in which case you personal return would equal the IRR, but this is not realistic. With low coupon rates, reinvestment of the coupons won't have a large impact on personal return.
longvista wrote: Mon Nov 21, 2022 12:30 pm Question: If calculating total return in this way is not possible, what is the intended, original use of YTM - is it only the apples-to-apples comparison?
See above.
longvista wrote: Mon Nov 21, 2022 12:30 pm Question: If calculating total return with YTM is not possible, then why does Investopedia think that YTM is related to total return - is it because of their erroneous interpretation of YTM as the CAGR in the case of coupon reinvestment?
I don't rely on Investopedia as a credible source, but CAGR would approximately equal YTM if all coupons were reinvested at the original yield. It's not exact because of the semi-annual compounding convention used to quote bond yields.

Kevin
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

Kevin M wrote: Mon Nov 21, 2022 2:47 pm The internal rate of return (IRR) does not depend on what is done with the coupons--it's strictly a discounted cash flow thing. The IRR will be approximately equal to the initial yield if you hold to maturity. Whenever I say "yield" without a qualifier, I mean yield to maturity.
Firstly, thanks for clarifying what you mean by "yield". A lot of time is wasted due to ambiguity and people talking past each other, simply because of ambiguous terminology.
Kevin M wrote: Mon Nov 21, 2022 2:47 pm You don't plug YTM into anything. You just use it to estimate the IRR of the investment. The only way people frame the investor's personal return is to assume all coupon payments are reinvested at the original yield, in which case you personal return would equal the IRR, but this is not realistic. With low coupon rates, reinvestment of the coupons won't have a large impact on personal return.


I can understand why reinvesting the coupons at YTM is not realistic and as a consequence, getting a CAGR equal to IRR is also not realistic. Then it seems that the only clear use for YTM is for being able to compare different bonds. For example, if you look at all bonds with the same term and credit quality, and you find one whose YTM is higher than the other bonds, then you have found a good deal. However, YTM does not tell you much about the personal return you can expect from that bond.

Is that a correct rephrasing and summary?

Also, what would then be a good way of calculating the personal return you can expect from the bond - total return, based on price, face value, coupon payments and term?
Kevin M wrote: Mon Nov 21, 2022 2:47 pm I don't rely on Investopedia as a credible source, but CAGR would approximately equal YTM if all coupons were reinvested at the original yield. It's not exact because of the semi-annual compounding convention used to quote bond yields.

Kevin
I brought Investopedia more as an example to illustrate the confusion around YTM, rather than an authoritative source. Regardless, I have found many inaccuracies in Investopedia and am always careful when reading it. Do you have a better source to recommend?
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

jeffyscott wrote: Mon Nov 21, 2022 2:45 pm
longvista wrote: Mon Nov 21, 2022 12:30 pmQuestion: How can I use YTM in order to calculate my total return on a coupon bond (when coupons are not reinvested)? In which formula do I need to plug in the YTM?
Borrowing #cruncher's analogy, this is like asking:
How can I use my savings account interest rate to calculate my total return when I am withdrawing money from the account every 6 months?

And, AFAIK, that can't be answered without knowing the amounts that are to be withdrawn (i.e. coupons).
Question: If calculating total return in this way is not possible, what is the intended, original use of YTM - is it only the apples-to-apples comparison?
Yes, I'd assume the use is to tell you what the yields of bonds are in a consistent manner.
Based on what you and Kevin M are saying, it seems that if I want to know my total return from holding the bond in question, YTM is actually quite irrelevant. I just need to look at the following:

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50
Based on this I can calculate my Total Return:

Code: Select all

Total Return = 5 * $50 + ($1,000 - $800) = $450
And even CAGR:
CAGR = ($1,250/$800)^(1/5) - 1 = 9.34%
Again, YTM is not relevant here. Is this correct?
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Re: Does YTM require that coupons are re-invested?

Post by vineviz »

longvista wrote: Mon Nov 21, 2022 3:25 pm However, YTM does not tell you much about the personal return you can expect from that bond.

Is that a correct rephrasing and summary?
No.

YTM tells you a great deal about the personal return you can expect.

It doesn't tell you EXACTLY the personal return you will ACTUALLY receive, but it's close enough that there's little value in trying to find a more accurate estimate.
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

Oicuryy wrote: Tue Nov 08, 2022 1:17 pm longvista,

Welcome back. Please go to Portfolio Visualizer's FAQ and read their definitions of Compound Annualized Growth Rate (CAGR), Time-Weighted Rate of Return (TWRR) and Money-Weighted Rate of Return (MWRR).

https://www.portfoliovisualizer.com/faq#definitions

Note the line in their definition of CAGR that says, "In the presence of cashflows other return metrics such as time-weighted rate of return (TWRR) and money weighted rate of return (MWRR) are preferred." (my emphasis)

The coupon payments from a bond are cash flows. Yield to maturity (YTM) is a money weighted rate of return (MWRR).

Ron
Thanks for pointing this resource out to me. I was not aware that Portfolio Visualizer has such succinct definitions available. Nevertheless, the deeper meaning and relevance of TWRR and MWRR is currently eluding me, and I need to do some more reading on this.

Does anyone know of any resources which defines all, or most, finance related terms in a clear and most importantly correct way (in light of the whole YTM confusion), that we could always reference and point to?
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

vineviz wrote: Mon Nov 21, 2022 3:42 pm
longvista wrote: Mon Nov 21, 2022 3:25 pm However, YTM does not tell you much about the personal return you can expect from that bond.

Is that a correct rephrasing and summary?
No.

YTM tells you a great deal about the personal return you can expect.

It doesn't tell you EXACTLY the personal return you will ACTUALLY receive, but it's close enough that there's little value in trying to find a more accurate estimate.
I can understand that it does not give an exact answer, but how then do we have to interpret the YTM if we want to know my personal return?:

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50 / year
YTM: 10.32%
I.e., in order to get a "close enough" result for our personal return, should we interpret YTM as CAGR, or as a non-compounding annual yield, or something else?

Also, if we just calculate the personal return based on the price, face value, coupons, term - it seems to me to be an exact result, and as such, better than YTM. What am I misunderstanding here?
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Re: Does YTM require that coupons are re-invested?

Post by dbr »

longvista wrote: Mon Nov 21, 2022 3:25 pm
Also, what would then be a good way of calculating the personal return you can expect from the bond - total return, based on price, face value, coupon payments and term?
There is an inherent ambiguity when talking about the return on an investment when there are cash flows. A standard option is to explicitly allow for the cash flows and calculate internal rate of return or time weighted return in the usual way.

A different option would be to calculate the total return of what amounts to a portfolio of the bond plus the retained cash flows. To do that you are required to specify what happens to the retained cash flows. This is necessary in order to know the final value of the bond plus retained cash flows. One assumption is that the cash flows are retained as exactly that, cash which is still on hand at the maturity of the bond. So the return is

(Final Value - Beginning Value)/(Beginning Value)

where the beginning value is the price of the bond and the final value is the face value of the bond plus the sum of the cash flows. Usually one then annualizes this in the normal way. This means one has calculated a CAGR for the system but not for then bond itself. Usually one does not report a CAGR when there are cash flows, but formally there is no reason a person can't include cash flows except that the result is then not a report about investment performance but rather a report about your personal financial history. The statement in Portfolio Visualizer that IRR/TWRR is "preferred" is a bit disingenuous for sure.

You can also hypothesize investment of the cash flows when received and calculate the final value of the cash flows under whatever return you were able to obtain. This is either an actual backward looking result or at least a hypothetical backward looking result. One hypothesis we have already discussed is reinvestment of the cash flows at the YTM.

A different way to posit this is to state that you can't compute an accurate total return for your bond when you buy it because you don't know yet what will be done with the cash flows.

Still a different perspective is to get away from worrying about calculating a number of some kind and just think through what your position will be in the future, spent, reinvested, or cash in the wallet of your money. You don't have to reduce this to a defined financial term.
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Re: Does YTM require that coupons are re-invested?

Post by vineviz »

longvista wrote: Mon Nov 21, 2022 3:49 pm
vineviz wrote: Mon Nov 21, 2022 3:42 pm
longvista wrote: Mon Nov 21, 2022 3:25 pm However, YTM does not tell you much about the personal return you can expect from that bond.

Is that a correct rephrasing and summary?
No.

YTM tells you a great deal about the personal return you can expect.

It doesn't tell you EXACTLY the personal return you will ACTUALLY receive, but it's close enough that there's little value in trying to find a more accurate estimate.
I can understand that it does not give an exact answer, but how then do we have to interpret the YTM if we want to know my personal return?:

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50 / year
YTM: 10.32%
I.e., in order to get a "close enough" result for our personal return, should we interpret YTM as CAGR, or as a non-compounding annual yield, or something else?

Also, if we just calculate the personal return based on the price, face value, coupons, term - it seems to me to be an exact result, and as such, better than YTM. What am I misunderstanding here?
You can never know exactly what your personal return will be when there are future cash flows involved. It’s impossible to do.

I might have missed it earlier, but I’m not sure what personal financial planning decisions would depend materially in having a MORE accurate estimate of future bond returns than YTM provides.
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Re: Does YTM require that coupons are re-invested?

Post by Kevin M »

longvista wrote: Mon Nov 21, 2022 3:38 pm Based on what you and Kevin M are saying, it seems that if I want to know my total return from holding the bond in question, YTM is actually quite irrelevant. I just need to look at the following:

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50
Based on this I can calculate my Total Return:

Code: Select all

Total Return = 5 * $50 + ($1,000 - $800) = $450
And even CAGR:
CAGR = ($1,250/$800)^(1/5) - 1 = 9.34%
Again, YTM is not relevant here. Is this correct?
YTM is very relevant. First, consider a Treasury bill or STRIPS, where the coupon is 0%. In this case, CAGR will be very close to initial yield (to maturity), since there are no coupons to reinvest.

Now, say I'm comparing a bill to a note of same maturity, and the note has a higher yield. If the coupon is small, then the return will be close to the initial yield, since a small coupon is not very different than 0 coupon. If the coupon is large and maturity is five years, then the coupon payments are a more significant part of the return. Usually very high coupon bonds have lower yields at the maturities I'm buying, so I typically don't buy them anyway.

The problem with your CAGR calculation is that you are not discounting the cash flows based on the time of the payment. This is what IRR does, but then you're back to your question again. The earlier coupon payments are more valuable because they are discounted less than further out payments. You are valuing the first coupon payment the same as the last, which isn't typically the way we calculate yields and returns. Just think of the impact of inflation on the coupon payments, and you should see that the earlier payments are more valuable than the later payments in terms of purchasing power (assuming positive inflation).

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Re: Does YTM require that coupons are re-invested?

Post by #Cruncher »

Kevin M wrote: Mon Nov 21, 2022 4:43 pm
longvista wrote: Mon Nov 21, 2022 3:38 pm... it seems that if I want to know my total return from holding the bond in question, YTM is actually quite irrelevant. I just need to look at the following:

Code: Select all

Maturity: 5 years
Principal: $1,000
Current Price: $800
Coupon Payment: $50
Based on this I can calculate my Total Return:

Code: Select all

Total Return = 5 * $50 + ($1,000 - $800) = $450
And even CAGR:
CAGR = ($1,250/$800)^(1/5) - 1 = 9.34%
Again, YTM is not relevant here. Is this correct?
YTM is very relevant. ... The problem with your CAGR calculation is that you are not discounting the cash flows based on the time of the payment.
Longvista is correct that yield-to-maturity (YTM) isn't relevant in computing his "Total Return" and resultant compound annual growth rate (CAGR). However, I agree with you, Kevin, that YTM is indeed relevant for properly evaluating bonds because it doesn't ignore the timing of interest payments like his "Total Return" does.

I'll try to illustrate this with two actual Treasury issues both maturing 11/15/2027. One has a 2-1/4% coupon and the other a 6-1/8% coupon. Prices are from Tuesday's WSJ Treasury Quotes. [1]

Code: Select all

Row           Col A       Col B     Col C  Formula in Col B copied to Col C
  2      Face value       1,000
  3      Settlement  11/23/2022
  4         Matures  11/15/2027
  5           Years           5
  6          Coupon      2.250%    6.125%
  7       Ask price      92.180   109.313
  8     YTM (YIELD)      3.998%    4.041%  =YIELD($B3,$B4,B6,B7,100,2,1) [2]
  9      YTM (RATE)       4.01%     4.03%  =RATE($B5,100*B6,-B7,100,0)   [3]
 10            Cost      921.80  1,093.13  =$B2*(B7/100)
 11  "Total Return"      190.70    213.13  =$B5*$B2*B6+$B2-B10
 12            CAGR       3.83%     3.63%  =((B10+B11)/B10)^(1/$B5)-1
Longvista's CAGR on the bottom row shows the 2-1/4% bond with a 0.2% point better return, while their yields-to-maturity are practically identical. Most investors would consider his approach -- which assumes coupons are stashed in a non-interest bearing account for five years -- to be unrealistic.
  1. The WSJ Treasury Quotes shows prices for notes and bonds in a strange way. For example, it shows the Ask Price for the 2-1/4% issue as "92.0560". I believe this means 92 + 5/32 + 6/256, which is the "92.180" I show on row 7.
  2. The Excel YIELD function calculates the yield-to-maturity precisely using the settlement and maturity dates and semi-annual interest payments. Using the ask prices it produces the same yields as the WSJ Quotes.
  3. The Excel RATE function, as I use it here, assumes for simplicity exactly a five year term and annual interest payments.
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Re: Does YTM require that coupons are re-invested?

Post by acegolfer »

OP.

I suggest you don't consider YTM as ex-post annualized return. That's simply not what YTM is for.
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Re: Does YTM require that coupons are re-invested?

Post by dbr »

acegolfer wrote: Wed Nov 23, 2022 8:19 am OP.

I suggest you don't consider YTM as ex-post annualized return. That's simply not what YTM is for.
Yes, there is no reason different measures can't mean different things and be useful for a different purpose.

But let's not forget we have already resolved the question that started this thread, the answer to which is that calculation of the YTM involves no information or assumptions about what is done with the cash flow from the bond. This is contrary to erroneous statements made in many publications regarding the calculation of YTM. There is nothing wrong with something being a matter of definition and something else having a different definiton.
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Re: Does YTM require that coupons are re-invested?

Post by acegolfer »

dbr wrote: Wed Nov 23, 2022 9:01 am But let's not forget we have already resolved the question that started this thread, the answer to which is that calculation of the YTM involves no information or assumptions about what is done with the cash flow from the bond. This is contrary to erroneous statements made in many publications regarding the calculation of YTM. There is nothing wrong with something being a matter of definition and something else having a different definiton.
I believe OP already noted that he understood this part. But in the last 2 days, he asked about whether YTM = total return. I thought he already knew the answer but was surprised when he asked the question again.
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Re: Does YTM require that coupons are re-invested?

Post by dbr »

acegolfer wrote: Wed Nov 23, 2022 9:06 am
dbr wrote: Wed Nov 23, 2022 9:01 am But let's not forget we have already resolved the question that started this thread, the answer to which is that calculation of the YTM involves no information or assumptions about what is done with the cash flow from the bond. This is contrary to erroneous statements made in many publications regarding the calculation of YTM. There is nothing wrong with something being a matter of definition and something else having a different definiton.
I believe OP already noted that he understood this part. But in the last 2 days, he asked about whether YTM = total return. I thought he already knew the answer but was surprised when he asked the question again.
Far and away the most puzzling thing about this thread for me is how to account for the bizarre misstatements by investing professionals about YTM and investment of cash flows. That is a question we have not answered.
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Re: Does YTM require that coupons are re-invested?

Post by acegolfer »

dbr wrote: Wed Nov 23, 2022 9:15 am Far and away the most puzzling thing about this thread for me is how to account for the bizarre misstatements by investing professionals about YTM and investment of cash flows. That is a question we have not answered.
I use critical thinking and try to interpret their statements within their context. Some statements require untold assumptions.
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Re: Does YTM require that coupons are re-invested?

Post by BrooklynInvest »

Someone please correct me -

The YTM of a bond held to maturity is a combination of the stated yield from that bond (with those yields reinvested at that YTM) PLUS/MINUS any price impact.

For a bond bought below par YTM would be higher than the stated yield because the investor is redeeming at par which is higher than purchase price.

For a bond bought above par YTM would be lower than the stated yield for the same reason in reverse.

Today's YTM on my hypothetical bond is a reasonable measure of my total return from capital gain/loss and income - as of today. But it will fluctuate because the price of the bond will fluctuate even though the stated yield is fixed.

Assuming the income is reinvested at that YTM isn't terrible since YTM of a single bond is likely very similar to that of the bond sector for similar duration and credit quality.
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Re: Does YTM require that coupons are re-invested?

Post by dbr »

acegolfer wrote: Wed Nov 23, 2022 9:20 am
dbr wrote: Wed Nov 23, 2022 9:15 am Far and away the most puzzling thing about this thread for me is how to account for the bizarre misstatements by investing professionals about YTM and investment of cash flows. That is a question we have not answered.
I use critical thinking and try to interpret their statements within their context. Some statements require untold assumptions.
It is probably true that in the end the whole thing falls in the category of "no harm, no foul." No one is publishing erroneous YTMs.
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Re: Does YTM require that coupons are re-invested?

Post by vineviz »

dbr wrote: Wed Nov 23, 2022 9:15 am Far and away the most puzzling thing about this thread for me is how to account for the bizarre misstatements by investing professionals about YTM and investment of cash flows. That is a question we have not answered.
I don't think the misstatements are "bizarre", in so far as it is quite easy to take a true statement about YTM vs. total return and turn it into a "misstatement" simply by making some pretty subtle changes to wording: changes that seem might seem innocuous at first glance, even to someone who could articulate the difference if they thought about it a little harder.
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Re: Does YTM require that coupons are re-invested?

Post by dbr »

vineviz wrote: Wed Nov 23, 2022 10:34 am
dbr wrote: Wed Nov 23, 2022 9:15 am Far and away the most puzzling thing about this thread for me is how to account for the bizarre misstatements by investing professionals about YTM and investment of cash flows. That is a question we have not answered.
I don't think the misstatements are "bizarre", in so far as it is quite easy to take a true statement about YTM vs. total return and turn it into a "misstatement" simply by making some pretty subtle changes to wording: changes that seem might seem innocuous at first glance, even to someone who could articulate the difference if they thought about it a little harder.
Yes, true. My objection is more one of formalism. By that I mean making a statement about the definition and then presenting the definition and seeing that the definition does not agree with the statement. The formula for YTM does not include any terms that refer to what is done with the cash flows received, so why would one say that it does.

But, you are right, it is beating a dead horse at this point.
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Re: Does YTM require that coupons are re-invested?

Post by desertgoose »

dbr wrote: Wed Nov 23, 2022 9:15 am
Far and away the most puzzling thing about this thread for me is how to account for the bizarre misstatements by investing professionals about YTM and investment of cash flows. That is a question we have not answered.
It isn't bizarre at all. It is 100% factually correct to say that the YTM is the return one would have at maturity if the bond is held to maturity and all interim cash flows are reinvested at the YTM. You can quibble that it isn't the definition of YTM. But it is absolutely a mathematical attribute of YTM.
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Re: Does YTM require that coupons are re-invested?

Post by FactualFran »

BrooklynInvest wrote: Wed Nov 23, 2022 9:31 am Someone please correct me -

The YTM of a bond held to maturity is a combination of the stated yield from that bond (with those yields reinvested at that YTM) PLUS/MINUS any price impact.

For a bond bought below par YTM would be higher than the stated yield because the investor is redeeming at par which is higher than purchase price.

For a bond bought above par YTM would be lower than the stated yield for the same reason in reverse.

Today's YTM on my hypothetical bond is a reasonable measure of my total return from capital gain/loss and income - as of today. But it will fluctuate because the price of the bond will fluctuate even though the stated yield is fixed.

Assuming the income is reinvested at that YTM isn't terrible since YTM of a single bond is likely very similar to that of the bond sector for similar duration and credit quality.
A suggestion: replace the term "stated yield" with "interest rate".
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Re: Does YTM require that coupons are re-invested?

Post by vineviz »

FactualFran wrote: Wed Nov 23, 2022 3:49 pm
BrooklynInvest wrote: Wed Nov 23, 2022 9:31 am Someone please correct me -

The YTM of a bond held to maturity is a combination of the stated yield from that bond (with those yields reinvested at that YTM) PLUS/MINUS any price impact.

For a bond bought below par YTM would be higher than the stated yield because the investor is redeeming at par which is higher than purchase price.

For a bond bought above par YTM would be lower than the stated yield for the same reason in reverse.

Today's YTM on my hypothetical bond is a reasonable measure of my total return from capital gain/loss and income - as of today. But it will fluctuate because the price of the bond will fluctuate even though the stated yield is fixed.

Assuming the income is reinvested at that YTM isn't terrible since YTM of a single bond is likely very similar to that of the bond sector for similar duration and credit quality.
A suggestion: replace the term "stated yield" with "interest rate".
Agreed. Or "coupon yield" or "coupon rate".
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Re: Does YTM require that coupons are re-invested?

Post by michaelingp »

desertgoose wrote: Wed Nov 23, 2022 3:42 pm
dbr wrote: Wed Nov 23, 2022 9:15 am
Far and away the most puzzling thing about this thread for me is how to account for the bizarre misstatements by investing professionals about YTM and investment of cash flows. That is a question we have not answered.
It isn't bizarre at all. It is 100% factually correct to say that the YTM is the return one would have at maturity if the bond is held to maturity and all interim cash flows are reinvested at the YTM. You can quibble that it isn't the definition of YTM. But it is absolutely a mathematical attribute of YTM.
Thank you desertgoose for this succinct summary. Lest anyone think this thread is has no practical use, and is just a swirl of conflicting information, it has actually answered an important question for me. I have invested now and then in BulletShares, but they've always confused me because the YTM they advertise has not matched what I've actually received when they mature. I now realize that a big part of this is the fact that I have not re-invested the interest payouts, or considered what the interest on them would be. Thank you Bogleheads!
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Re: Does YTM require that coupons are re-invested?

Post by ThereAreNoGurus »

Can somebody point to an online calculator or provide a formula for calculating yield to maturity without the re-investment of bond coupons?
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Re: Does YTM require that coupons are re-invested?

Post by JayB »

ThereAreNoGurus wrote: Thu Nov 24, 2022 11:58 pm Can somebody point to an online calculator or provide a formula for calculating yield to maturity without the re-investment of bond coupons?
Fidelity Price/Yield Calculator

As Fidelity explains,
The Price/Yield Calculator is designed to help you calculate the estimated yield or price of a bond. If the calculation type is Yield and a price is entered, the page calculates and displays:

Accrued Interest
Total Cost
Yield to Maturity
Yield to Call
If the calculation type is Price and a yield is entered, the page calculates and displays:

Accrued Interest
Total Cost
Price
The following bond types will be supported:

Corporate Bonds
Municipal
Treasury
Agency/GSE
Fixed Rate
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Re: Does YTM require that coupons are re-invested?

Post by ThereAreNoGurus »

JayB wrote: Fri Nov 25, 2022 1:01 am
ThereAreNoGurus wrote: Thu Nov 24, 2022 11:58 pm Can somebody point to an online calculator or provide a formula for calculating yield to maturity without the re-investment of bond coupons?
Fidelity Price/Yield Calculator

As Fidelity explains,
The Price/Yield Calculator is designed to help you calculate the estimated yield or price of a bond. If the calculation type is Yield and a price is entered, the page calculates and displays:
Thanks for the link. I tried that calculator. It returns a yield to maturity and it appears to me that calculation is based on re-investment of the bond coupons. In other words it appears to be a standard YTM calculator.
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Re: Does YTM require that coupons are re-invested?

Post by JayB »

ThereAreNoGurus wrote: Fri Nov 25, 2022 1:32 am
JayB wrote: Fri Nov 25, 2022 1:01 am
ThereAreNoGurus wrote: Thu Nov 24, 2022 11:58 pm Can somebody point to an online calculator or provide a formula for calculating yield to maturity without the re-investment of bond coupons?
Fidelity Price/Yield Calculator

As Fidelity explains,
The Price/Yield Calculator is designed to help you calculate the estimated yield or price of a bond. If the calculation type is Yield and a price is entered, the page calculates and displays:
Thanks for the link. I tried that calculator. It returns a yield to maturity and it appears to me that calculation is based on re-investment of the bond coupons. In other words it appears to be a standard YTM calculator.
I plugged in the price, coupon, maturity date, and par value for some bonds listed at Fidelity brokerage and the quoted YTMs match the calculator's YTMs. The calculator never asks for $ values at maturity with coupons reinvested, so I don't see how you can form a conclusion that the calculator incorporates re-investment of bond coupons in computing YTM. The only terminal $ values that the calculator apparently deals with are Call Prices and Par Values.
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ThereAreNoGurus
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Re: Does YTM require that coupons are re-invested?

Post by ThereAreNoGurus »

JayB wrote: Fri Nov 25, 2022 7:14 am I plugged in the price, coupon, maturity date, and par value for some bonds listed at Fidelity brokerage and the quoted YTMs match the calculator's YTMs. The calculator never asks for $ values at maturity with coupons reinvested, so I don't see how you can form a conclusion that the calculator incorporates re-investment of bond coupons in computing YTM. The only terminal $ values that the calculator apparently deals with are Call Prices and Par Values.
I conclude that the coupons are re-invested by the calculator because Fidelity's definition of YTM says that's what YTM is (same with Vanguard and Schwab and every other calculator I could find online).

Fidelity's definition of YTM appears on this page: https://www.fidelity.com/learning-cente ... tes-yields
And this is the key portion:
But the bond's yield to maturity in this case is higher. It considers that you can achieve compounding interest by reinvesting the $1,200 you receive each year. It also considers that when the bond matures, you will receive $20,000, which is $2,000 more than what you paid.
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JayB
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Re: Does YTM require that coupons are re-invested?

Post by JayB »

ThereAreNoGurus wrote: Fri Nov 25, 2022 9:53 am
JayB wrote: Fri Nov 25, 2022 7:14 am I plugged in the price, coupon, maturity date, and par value for some bonds listed at Fidelity brokerage and the quoted YTMs match the calculator's YTMs. The calculator never asks for $ values at maturity with coupons reinvested, so I don't see how you can form a conclusion that the calculator incorporates re-investment of bond coupons in computing YTM. The only terminal $ values that the calculator apparently deals with are Call Prices and Par Values.
I conclude that the coupons are re-invested by the calculator because Fidelity's definition of YTM says that's what YTM is (same with Vanguard and Schwab and every other calculator I could find online).

Fidelity's definition of YTM appears on this page: https://www.fidelity.com/learning-cente ... tes-yields
And this is the key portion:
But the bond's yield to maturity in this case is higher. It considers that you can achieve compounding interest by reinvesting the $1,200 you receive each year. It also considers that when the bond matures, you will receive $20,000, which is $2,000 more than what you paid.
The expanation by Fidelity that you that you provide a link for above has already been established as mistaken when it come to reinvesting coupons to attain the quoted YTM. There are many many mistaken accounts across the web and in textbooks about how YTM is calculated, including from reputable sources; this has been the central discussion point of this thread. What matters is how the bonds are actually quoted in ask and bid YTM quotations on a brokerage site like Fidelity. Fidelity's Price/Yield calculator YTM results match the way their bond Brokerage quotes YTMs.
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ThereAreNoGurus
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Re: Does YTM require that coupons are re-invested?

Post by ThereAreNoGurus »

JayB wrote: Fri Nov 25, 2022 10:17 am The expanation by Fidelity that you that you provide a link for above has already been established as mistaken when it come to reinvesting coupons to attain the quoted YTM. There are many many mistaken accounts across the web and in textbooks about how YTM is calculated, including from reputable sources; this has been the central discussion point of this thread. What matters is how the bonds are actually quoted in ask and bid YTM quotations on a brokerage site like Fidelity. Fidelity's Price/Yield calculator YTM results match the way their bond Brokerage quotes YTMs.
I've read this thread. Perhaps I missed or mis-understood the posts showing that the brokerage firms' YTM calculations do not compound coupon payments. Can you point me to a post that shows this or makes that claim?

I didn't gather that was the "central point of this thread." I thought the main point was, what was YTM to begin with... Ie., whether coupon interest should be compounded or not when calculating YTM. And I gathered that including compound interest was included in the definition of YTM. I did not see a post demonstrating that the YTM calculations provided by the major brokerages did not include compounded interest of coupons. I'd certainly appreciate it if you or somebody else posted a link to the post that draws that conclusion.

Edited to add:

I re-read one of #Cruncher's posts on this page. It may have answered Longvista's question on how to calculate YTM without compounding coupon rates. I'll re-read it.

However I'm still under the impression that the online calculators do compound coupon payments.

Further edited to add:

Upon a more careful read of #Cruncher's post above, I'm assuming row 12 provides the formula for calculating a YTM without compounding coupon payments:
12 CAGR 3.83% 3.63% =((B10+B11)/B10)^(1/$B5)-1
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