Does YTM require that coupons are re-invested?

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longvista
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Does YTM require that coupons are re-invested?

Post by longvista »

Different sources, when defining YTM, say the following:
Wikipedia - "The YTM calculation formulates certain stability conditions of the security, its owner, and the market going forward: The owner reinvests all interest payments rather than spending them, to gain the benefit of compounded returns."

Investopedia - "Calculating the yield to maturity can be a complicated process, and it assumes all coupon or interest payments can be reinvested at the same rate of return as the bond."

Annette Thau (author of The Bond Book): "YTM calculations are based on the assumption that coupons are never spent; they are always reinvested. [...] coupons are reinvested at an interest rate equal to the yield-to-maturity."

However, the following paper by 3 researchers states:
This note addresses a common misconception, found in investment texts and popular investment education literature,
that in order to earn the yield to maturity on a coupon bond an investor must reinvest the coupon payments. We identify a
sample of text and professional sources making this claim, demonstrate that yield to maturity entails no assumption of
coupon reinvestment, discuss a cause for this confusion and offer a possible remedy
I am trying to verify their claim, but I am unable to do so. Let's take the following coupon bond:

Code: Select all

Maturity: 3 years
Principal: $1,000
Current Price: $1,000
Coupon Payment: $50
According to the HP 12C financial calculator, and Excel's RATE function, the YTM of this bond is 5%.

Let's verify whether we can get a 5% Annualized Return (as predicted by the 5% YTM).
So if I purchase the bond for $1,000, then I will get the following cashflows: $50, $50, $50, $1,000 = $1,150. This means I will earn a Total Return of $150, expressed as a percentage: $ 150 / $1,000 = 15%. The Annualized Return would be (1 + 15%) ^ (1 / 3) - 1 ≃ 4.77%.

The Annualized Return of 4.77% does not match the stated YTM of 5%.

Investopedia, Wikipedia and Annette Thau are saying that in order to earn the 5% YTM, we actually need to reinvest the coupons also at the 5% rate. The researchers say that they are wrong. Let's try reinvesting the coupons and see if we get to 5%.

Here is the table showing the interest payments we get when we reinvest the interest payments at 5%

Code: Select all

Interest payments               End of year 1  End of year 2  End of year 3
Bond 1                                     50             50             50
Bond 2 (from interest of bond 1)                         2.5            2.5
Bond 3 (from interest of bond 1)                                        2.5
Bond 4 (from interest of bond 2)                                      0.125
The sum of the interest payments received in this case would be: 50 + 50 + 50 + 2.5 + 2.5 + 2.5 + 0.125 = 157.625

This would mean that if we reinvest the coupon payments at 5%, we end up with $157.625 in interest income, instead of only $150. What's the Total Return? $157.625 / $1,000 = 15.7625%. The Annualized Return would then be (1 + 15.7625%) ^ (1 / 3) - 1 = 5%.

This matches the 5% YTM.

So now I am confused. The paper from the researchers says that "it is an erroneous assumption", that "in order to earn the yield to maturity on a coupon bond an investor must reinvest the coupon payments".

How is it erroneous? If I do not reinvest the coupon payments, my Annualized Return will be ~4.77%. If I do reinvest the coupons, my Annualized Return will be 5.00%, exactly equal to YTM.

Question: If the research paper is correct and achieving an Annualized Return of 5% does not require reinvesting coupons, then I should get a 5% Annualized Return when not reinvesting the coupons, but I only get 4.77%. Why is that?

Question: Perhaps the research paper is not correct, because the only way to get a 5% Annualized Return is to reinvest the coupons, contrary to what the research paper says. Is the research paper incorrect?
Last edited by longvista on Thu Nov 03, 2022 2:08 pm, edited 6 times in total.
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Beensabu
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Re: Does YTM require that coupons are re-invested?

Post by Beensabu »

Apparently, Excel has an IRR function that you can use that will make it all make sense.

viewtopic.php?p=6938456#p6938456
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Topic Author
longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

Beensabu wrote: Thu Nov 03, 2022 1:18 pm Apparently, Excel has an IRR function that you can use that will make it all make sense.

viewtopic.php?p=6938456#p6938456
I read the post, and Microsoft's documentation on the IRR function. I can understand on a high level that the IRR function is another way for calculating the YTM. But this doesn't help me to answer the questions stated in my OP.

Could you please clarify, what IRR helps to make sense of?
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Oicuryy
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Re: Does YTM require that coupons are re-invested?

Post by Oicuryy »

Copy and paste this starting in cell A1. No reinvesting required.

Code: Select all

=50/1.05
=50/1.05^2
=50/1.05^3
=1000/1.05^3
=SUM(A1:A4)
Ron
Money is fungible | Abbreviations and Acronyms
Topic Author
longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

Oicuryy wrote: Thu Nov 03, 2022 1:48 pm Copy and paste this starting in cell A1. No reinvesting required.

Code: Select all

=50/1.05
=50/1.05^2
=50/1.05^3
=1000/1.05^3
=SUM(A1:A4)
Ron
Hi Ron

I copied it, and the result is the PV of those cash flows, confirming that 5% is the YTM. But this does not answer the questions in my OP, which is basically - if the 3 researchers are correct and achieving an Annualized Return of 5% (YTM) does not require reinvesting coupons, then why do I only end up with ~ 4.77% if I don't reinvest the coupons?
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Beensabu
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Re: Does YTM require that coupons are re-invested?

Post by Beensabu »

longvista wrote: Thu Nov 03, 2022 1:56 pm the questions in my OP, which is basically - if the 3 researchers are correct and achieving an Annualized Return of 5% (YTM) does not require reinvesting coupons, then why do I only end up with ~ 4.77% if I don't reinvest the coupons?
It looks like the formula you are using to calculate annualized return might be incorrect?
"The only thing that makes life possible is permanent, intolerable uncertainty; not knowing what comes next." ~Ursula LeGuin
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longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

Beensabu wrote: Thu Nov 03, 2022 2:01 pm
longvista wrote: Thu Nov 03, 2022 1:56 pm the questions in my OP, which is basically - if the 3 researchers are correct and achieving an Annualized Return of 5% (YTM) does not require reinvesting coupons, then why do I only end up with ~ 4.77% if I don't reinvest the coupons?
It looks like the formula you are using to calculate annualized return might be incorrect?
That could be one reason, but because I am not so good at finding mistakes in my calculations as others are, could you please help me out and point out the mistake for me?
RetiredinKaty
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Re: Does YTM require that coupons are re-invested?

Post by RetiredinKaty »

The yield to maturity is the profit you make on the trade at settlement, period. This assumes you hold the bond to maturity and there is no default. You can do whatever you want with the cash flows.

If you want a compounded annual rate of return equal to the bond's yield, you need to re-invest the coupon cash flows at the same yield. This is something completely different. Bond funds can routinely re-invest cash flows from coupons and from maturing bonds, but they are re-invested at the current market yield. Using YTM to estimate compounded growth is just a guess but it is probably as good as anything.
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longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

RetiredinKaty wrote: Thu Nov 03, 2022 2:20 pm The yield to maturity is the profit you make on the trade at settlement, period. This assumes you hold the bond to maturity and there is no default. You can do whatever you want with the cash flows.
What is the "profit you make on the trade at settlement"? My understanding is that when the bond matures, I will receive my final cash flow from that trade and now the trade is settled. This was also my assumption for the cash flow analysis above. To restate: the profit I receive without reinvesting coupons is $150, which gives an Annualized Return of ~4.77%..

If as you say, "the yield to maturity is the profit you make on the trade at settlement" (and I assume you are talking about Annualized Return), then YTM predicts I should earn 5% Annualized Return, but I only earned ~4.77% when I didn't reinvest coupons. Why?
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Beensabu
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Re: Does YTM require that coupons are re-invested?

Post by Beensabu »

longvista wrote: Thu Nov 03, 2022 2:03 pm
Beensabu wrote: Thu Nov 03, 2022 2:01 pm
longvista wrote: Thu Nov 03, 2022 1:56 pm the questions in my OP, which is basically - if the 3 researchers are correct and achieving an Annualized Return of 5% (YTM) does not require reinvesting coupons, then why do I only end up with ~ 4.77% if I don't reinvest the coupons?
It looks like the formula you are using to calculate annualized return might be incorrect?
That could be one reason, but because I am not so good at finding mistakes in my calculations as others are, could you please help me out and point out the mistake for me?
The Internal Rate of Return is the interest rate that makes the Net Present Value zero.

PV = -$1,000
Year 1: PV = $50 / 1.05 = $47.62
Year 2: PV = $50 / 1.05^2 = $45.35
Year 3: PV = $50 / 1.05^3 = $43.19
Year 3 (final payment): PV = $1,000 / 1.05^3 = $863.86

Those dollar amounts add up to $1000 (except a couple pennies because rounding).

-$1000 + $1000 = 0

So the IRR is 5%.

It looks like the formula you used is definitely floating around out there and is this:

Internal Rate of Return (IRR) = (Future Value ÷ Present Value)^(1 ÷ Number of Periods) – 1

And so perhaps that is where the disconnect between various authors/sources comes from?

So the question is, where did the formula that you are using come from?
"The only thing that makes life possible is permanent, intolerable uncertainty; not knowing what comes next." ~Ursula LeGuin
RetiredinKaty
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Re: Does YTM require that coupons are re-invested?

Post by RetiredinKaty »

After settlement, a bond held to maturity will have a yield of zero. For real bonds this is a real zero and for nominal bonds it is really zero. There is no growth without re-investment.

When a company considers a business investment, it develops estimates of capital costs (price), life of the project (maturity), net income (coupons), and salvage value (redemption for a bond). The company then calculates an internal rate of return for that project based on all of the estimates. Some or all of the net cash flow can be re-invested, or it can be distributed to the owners. This doesn't change the worthiness of the investment. Without re-investment there is no growth. Same is true for bonds.
km91
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Re: Does YTM require that coupons are re-invested?

Post by km91 »

longvista wrote: Thu Nov 03, 2022 2:33 pm If as you say, "the yield to maturity is the profit you make on the trade at settlement" (and I assume you are talking about Annualized Return), then YTM predicts I should earn 5% Annualized Return, but I only earned ~4.77% when I didn't reinvest coupons. Why?
YTM is not a compounding rate, it's the internal rate that sets the future value of the cashflows to current price. The paper and Excel's IRR and RATE functions are correct, yield to maturity is calculated without considering the reinvestment rate of the cashflows
FactualFran
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Re: Does YTM require that coupons are re-invested?

Post by FactualFran »

The annualized percent difference between the dollar value at maturity and the price paid will be equal to the yield if the coupon payments are reinvested at the same yield. However, what is done with the coupon payments does not affect the yield of the security.
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longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

Beensabu wrote: Thu Nov 03, 2022 2:44 pm
longvista wrote: Thu Nov 03, 2022 2:03 pm
Beensabu wrote: Thu Nov 03, 2022 2:01 pm
longvista wrote: Thu Nov 03, 2022 1:56 pm the questions in my OP, which is basically - if the 3 researchers are correct and achieving an Annualized Return of 5% (YTM) does not require reinvesting coupons, then why do I only end up with ~ 4.77% if I don't reinvest the coupons?
It looks like the formula you are using to calculate annualized return might be incorrect?
That could be one reason, but because I am not so good at finding mistakes in my calculations as others are, could you please help me out and point out the mistake for me?
The Internal Rate of Return is the interest rate that makes the Net Present Value zero.

PV = -$1,000
Year 1: PV = $50 / 1.05 = $47.62
Year 2: PV = $50 / 1.05^2 = $45.35
Year 3: PV = $50 / 1.05^3 = $43.19
Year 3 (final payment): PV = $1,000 / 1.05^3 = $863.86

Those dollar amounts add up to $1000 (except a couple pennies because rounding).

-$1000 + $1000 = 0

So the IRR is 5%.

It looks like the formula you used is definitely floating around out there and is this:

Internal Rate of Return (IRR) = (Future Value ÷ Present Value)^(1 ÷ Number of Periods) – 1

And so perhaps that is where the disconnect between various authors/sources comes from?
I do not understand the concept of IRR deeply, so could you please point out why it is relevant to the question at hand - i.e., how is it related to YTM and Annualized Return?
So the question is, where did the formula that you are using come from?
Just in case, I will clarify, that I am assuming that YTM means the same thing as Annualized Return. If that is not a correct assumption, somebody please correct me.

So, if the YTM for a bond is 5%, then this would mean that the bond, if held to maturity (among other conditions), will have an Annualized Return of 5%.

The formula for calculating Annualized Return of an investment should be pretty straightforward, but taking from Investopedia:
Calculating the annualized performance of an investment or index using yearly data uses the following data points:

P = principal, or initial investment

G = gains or losses

n = number of years

AP = annualized performance rate

The generalized formula, which is exponential to take into account compound interest over time, is:

AP = ((P + G) / P) ^ (1 / n) - 1
Do you believe I am using an incorrect formula for calculating Annualized Return in my OP?

Or do you believe that a YTM of 5% does not mean that I am expected to make a 5% Annualized Return on my initial investment?
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longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

RetiredinKaty wrote: Thu Nov 03, 2022 2:51 pm After settlement, a bond held to maturity will have a yield of zero. For real bonds this is a real zero and for nominal bonds it is really zero. There is no growth without re-investment.
When you say "yield", are you talking about YTM? If not, which yield are you referring to? If yes, then what do you mean by "settlement" and what is the meaning of the fact that a bond, after settlement and held to maturity, will have a YTM of 0. It does not make sense to me.
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longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

km91 wrote: Thu Nov 03, 2022 3:03 pm
longvista wrote: Thu Nov 03, 2022 2:33 pm If as you say, "the yield to maturity is the profit you make on the trade at settlement" (and I assume you are talking about Annualized Return), then YTM predicts I should earn 5% Annualized Return, but I only earned ~4.77% when I didn't reinvest coupons. Why?
YTM is not a compounding rate, it's the internal rate that sets the future value of the cashflows to current price. The paper and Excel's IRR and RATE functions are correct, yield to maturity is calculated without considering the reinvestment rate of the cashflows
What is an "internal rate"? How is it different from a non-"internal" rate?

Also, I am not arguing against the fact that "yield to maturity is calculated without considering the reinvestment rate of the cashflows". I agree with that. What I can not seem to confirm, however, is the claim made by the paper that there is "a common misconception, found in investment texts and popular investment education literature, that in order to earn the yield to maturity on a coupon bond an investor must reinvest the coupon payments". Because according to my calculations, I do have to re-invest the coupon payments at the YTM, in order to have an Annualized Return of YTM (5% in this case).

You said also:
YTM is not a compounding rate
My understanding has been that if a bond has a 5% YTM, then when held to maturity (among other conditions), I will earn a 5% Annualized Return on my initial investment. This understanding is confirmed by this quote (https://www.robeco.com/en/key-strengths ... urity.html):
Expressed simply, the yield to maturity (YTM) of a bond is the annualized return that a bond investor would receive from holding the bond until maturity.
Are you saying that assumption is incorrect and that article is wrong? If so, what exactly does YTM tell us about a bond (not how it is calculated, but what does this particular metric show us)?
km91
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Re: Does YTM require that coupons are re-invested?

Post by km91 »

longvista wrote: Thu Nov 03, 2022 3:17 pm What is an "internal rate"? How is it different from a non-"internal" rate?

Also, I am not arguing against the fact that "yield to maturity is calculated without considering the reinvestment rate of the cashflows". I agree with that. What I can not seem to confirm, however, is the claim made by the paper that there is "a common misconception, found in investment texts and popular investment education literature, that in order to earn the yield to maturity on a coupon bond an investor must reinvest the coupon payments". Because according to my calculations, I do have to re-invest the coupon payments at the YTM, in order to have an Annualized Return of YTM (5% in this case).

You said also:
YTM is not a compounding rate
My understanding has been that if a bond has a 5% YTM, then when held to maturity (among other conditions), I will earn a 5% Annualized Return on my initial investment. This understanding is confirmed by this quote (https://www.robeco.com/en/key-strengths ... urity.html):
Expressed simply, the yield to maturity (YTM) of a bond is the annualized return that a bond investor would receive from holding the bond until maturity.
Are you saying that assumption is incorrect and that article is wrong? If so, what exactly does YTM tell us about a bond (not how it is calculated, but what does this particular metric show us)?
YTM is not the same as an annualized compounded growth rate. We can see that a 5% growth rate gets a final value of $1157 but the cashflows of the bond are only $1150. Every $1000 of face value purchased earns a cash yield of 5%, the face value of the bond doesn't grow
Last edited by km91 on Thu Nov 03, 2022 3:34 pm, edited 2 times in total.
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longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

FactualFran wrote: Thu Nov 03, 2022 3:06 pm The annualized percent difference between the dollar value at maturity and the price paid will be equal to the yield if the coupon payments are reinvested at the same yield. However, what is done with the coupon payments does not affect the yield of the security.
I'll try to re-word what you said:
When we take the dollar value (of all the cash flows from the bond), and we take the price we paid for the bond, and we subtract the dollar value from the price of the bond, then the result will be equal to the YTM if the coupon payments are reinvested at the same YTM.

However, YTM does not change when you do not reinvest the coupons at the same YTM.

Is this what you meant? If so, isn't there a contradiction between: "then the result will be equal to the YTM if the coupon payments are reinvested at the same YTM" and "However, YTM does not change when you do not reinvest the coupons at the same YTM"? If you think there's no contradiction, could you please elaborate?
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longvista
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

km91 wrote: Thu Nov 03, 2022 3:21 pm YTM is not the same as an annualized compounded growth rate. We can see that a 5% growth rate gets a final value of $1157 but the cashflows of the bond are only $1150. Every $1000 of face value purchased earns a cash yield of 5%, the face value of the bond doesn't grow
So are you saying that a 5% YTM does not mean that investing in the bond (and reinvesting coupons, among other conditions) gives me a 5% Annualized Return? If so, then what exactly does YTM show us? How is that metric useful for somebody looking at a bond's metrics?
km91
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Re: Does YTM require that coupons are re-invested?

Post by km91 »

longvista wrote: Thu Nov 03, 2022 3:28 pm
km91 wrote: Thu Nov 03, 2022 3:21 pm YTM is not the same as an annualized compounded growth rate. We can see that a 5% growth rate gets a final value of $1157 but the cashflows of the bond are only $1150. Every $1000 of face value purchased earns a cash yield of 5%, the face value of the bond doesn't grow
So are you saying that a 5% YTM does not mean that investing in the bond (and reinvesting coupons, among other conditions) gives me a 5% Annualized Return? If so, then what exactly does YTM show us? How is that metric useful for somebody looking at a bond's metrics?
YTM tells us how much income the bond will generate per dollar of face value purchased

In the yield to maturity formula we are discounting the future cashflows back to present value. The ytm / irr is the discount rate that makes future value = present value. We are valuing future cashflows in today's dollars. With the annualized rate you present, we are valuing today's cashflows in terms of future dollars
Last edited by km91 on Thu Nov 03, 2022 3:50 pm, edited 1 time in total.
evancox10
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Re: Does YTM require that coupons are re-invested?

Post by evancox10 »

longvista wrote: Thu Nov 03, 2022 12:39 pm So if I purchase the bond for $1,000, then I will get the following cashflows: $50, $50, $50, $1,000 = $1,150. This means I will earn a Total Return of $150, expressed as a percentage: $ 150 / $1,000 = 15%. The Annualized Return would be (1 + 15%) ^ (1 / 3) - 1 ≃ 4.77%.

The Annualized Return of 4.77% does not match the stated YTM of 5%.
This formula you are using here to calculate a yield is just wrong, because it doesn’t take into account the timing of when the coupon payments are received. It just sums them all up and treats them the same as the final payment. It is an accurate calculation only for a zero coupon bond that pays only a final payment.

A bond that pays out the $X all at once at the end has a lower yield than one that pays out $X in total but part of $X comes early. This is because the first bond you had to have money invested longer than the second. Ergo it has a lower return or yield.

The reason why you get the “right” result when using this formula and reinvesting coupons is because you have modeled something like a bank account where interest stays in the account and then compounds, and then finally you withdraw it all at the end. This is a different series of cash flows than your bond, but does have the same IRR or YTM. Two different cash flows can have the same IRR. The important part is that the IRR or YTM calculation must account for when the returns are received. Receiving the same amount of money earlier implies a higher return or yield.
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Beensabu
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Re: Does YTM require that coupons are re-invested?

Post by Beensabu »

longvista wrote: Thu Nov 03, 2022 3:07 pm I do not understand the concept of IRR deeply, so could you please point out why it is relevant to the question at hand - i.e., how is it related to YTM and Annualized Return?

...

Do you believe I am using an incorrect formula for calculating Annualized Return in my OP?
I'm pretty sure (could be wrong) that: YTM = IRR = Annualized Rate of Return.

I'm also pretty sure that the formula you have found and are using to calculate annualized rate of return is the culprit behind the "YTM includes reinvestment of coupon payments" misunderstanding that is not right.

So the question is not "where did you find that formula?" but "where did that formula originate?" Because it's definitely out there everywhere now.

And if that formula does not come up w/ IRR in a way such that the interest rate that makes the net present value zero, then... was that formula simply supposed to be a quick and dirty way to get a close approximation back in the paper and pencil days?
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evancox10
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Re: Does YTM require that coupons are re-invested?

Post by evancox10 »

Beensabu wrote: Thu Nov 03, 2022 3:51 pm
longvista wrote: Thu Nov 03, 2022 3:07 pm I do not understand the concept of IRR deeply, so could you please point out why it is relevant to the question at hand - i.e., how is it related to YTM and Annualized Return?

...

Do you believe I am using an incorrect formula for calculating Annualized Return in my OP?
I'm pretty sure (could be wrong) that: YTM = IRR = Annualized Rate of Return.

I'm also pretty sure that the formula you have found and are using to calculate annualized rate of return is the culprit behind the "YTM includes reinvestment of coupon payments" misunderstanding that is not right.

So the question is not "where did you find that formula?" but "where did that formula originate?" Because it's definitely out there everywhere now.

And if that formula does not come up w/ IRR in a way such that the interest rate that makes the net present value zero, then... was that formula simply supposed to be a quick and dirty way to get a close approximation back in the paper and pencil days?
Not sure of the origin. I agree that YTM, IRR, and annualized rate of return more or less mean the same thing.

To elaborate with an example of where this formula fails, consider the following cash flow series:
-$1000, $1000, $50, $50, $50
(Year 0, year 1, year 2, year 3, year 4)

Because all amounts are summed without regard to when they were paid, using OP’s formula gives the same “return” as the normal bond with following cash flow series:

-$1000, $50, $50, $50, $1000

Finance would rightly say that the first investment has a higher yield or rate of return than second. (Wouldn’t you rather have the first, all else equal?) A yield calculation that can’t account for this is not very useful and not telling you much.
Last edited by evancox10 on Thu Nov 03, 2022 4:06 pm, edited 1 time in total.
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Oicuryy
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Re: Does YTM require that coupons are re-invested?

Post by Oicuryy »

longvista wrote: Thu Nov 03, 2022 12:39 pm Let's verify whether we can get a 5% Annualized Return (as predicted by the 5% YTM).
So if I purchase the bond for $1,000, then I will get the following cashflows: $50, $50, $50, $1,000 = $1,150. This means I will earn a Total Return of $150, expressed as a percentage: $ 150 / $1,000 = 15%. The Annualized Return would be (1 + 15%) ^ (1 / 3) - 1 ≃ 4.77%.
YTM is a rate of return. Rate is amount per unit of time. The problem with your calculation is that you did not account for the different times of the cash flows. You treated all the cash flows as if they had occurred at the end of three years.

YTM is a version of internal rate of return. Internal rate of return is the discount rate that makes the sum of the present values of the cash flows equal to zero. The present value calculation of each cash flow accounts for the timing of that cash flow.

You cannot expect the bond issuer to pay you two more years of interest on the $50 you took out two years before maturity. So of course you will get less interest if you do not reinvest the coupons than if you do. But the rate of interest (interest per unit of time) will be the same either way.

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

Post by dbr »

The mystery remains why the incorrect statement about reinvestment ever appeared and is so persistent. It might be that the concept of an internal rate of return with cash flows is not intuitive as to what it really means. A naive concept of how much money I would have if the cash flows were reinvested is intuitive. The problem is what are we supposed to think has been done with the coupon payments. YTM is just precisely a number concocted to not have to know that.

There are other examples. It is commonly said that CAGR is the geometric average of the returns. That is not true. CAGR is the return in the geometric average of the gains calculated by gain = 1 + R%/100. It is another case of a statement made and then the formula used is not what the statement said was being done. This is also like saying things like risk is the standard deviation of periodic returns when risk is all kinds of things but the standard deviation of periodic returns really is just the standard deviation of periodic returns.

Sorry for sidetracking the conversation, but I think several posters have elucidated that the referenced paper is right and authors have written things that are not right.
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Re: Does YTM require that coupons are re-invested?

Post by dbr »

I was going to add that this whole issue tends to be related to anxiety regarding how certain will my wealth be at maturity of a bond. The problem here is that the rate at which we will be able to reinvest the coupons introduces uncertainty. Assuming that rate to be the coupon rate on the bond is an artificial finesse around reinvestment risk. That is always true when there are cash flows in or out. Note people often cite the advantage in certainty that comes with a zero coupon bond. An opposite point of view appears if we take the purpose of the bond to be just exactly that it does produce predictable cash flows. That, of course, is also a finesse in that money received in the future is of uncertain value depending on the assumed discount rate.
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Re: Does YTM require that coupons are re-invested?

Post by Soon2BXProgrammer »

Oicuryy wrote: Thu Nov 03, 2022 4:04 pm
longvista wrote: Thu Nov 03, 2022 12:39 pm Let's verify whether we can get a 5% Annualized Return (as predicted by the 5% YTM).
So if I purchase the bond for $1,000, then I will get the following cashflows: $50, $50, $50, $1,000 = $1,150. This means I will earn a Total Return of $150, expressed as a percentage: $ 150 / $1,000 = 15%. The Annualized Return would be (1 + 15%) ^ (1 / 3) - 1 ≃ 4.77%.
YTM is a rate of return. Rate is amount per unit of time. The problem with your calculation is that you did not account for the different times of the cash flows. You treated all the cash flows as if they had occurred at the end of three years.

YTM is a version of internal rate of return. Internal rate of return is the discount rate that makes the sum of the present values of the cash flows equal to zero. The present value calculation of each cash flow accounts for the timing of that cash flow.

You cannot expect the bond issuer to pay you two more years of interest on the $50 you took out two years before maturity. So of course you will get less interest if you do not reinvest the coupons than if you do. But the rate of interest (interest per unit of time) will be the same either way.

Ron
I think the only thing i'd add to this point is..

a zero coupon bond, with a 5% YTM, trades at a discount and has a duration equal to the maturity of the bond.

a bond with a 10% coupon, and a 5% YTM, trades at a premium, and has a duration that is less then the maturity of the bond.

(Some notes on various duration calculation methods https://corporatefinanceinstitute.com/r ... /duration/ )

The implication to this is that the zero coupon bond has your money for longer, and is more price sensitive to interest rate changes in comparison to a bond with a large coupon.

The YTM is the same, they have different durations and different sensitivities to interest rate changes.

YTM while a great metric, isn't the end all be all of bond metrics.
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Re: Does YTM require that coupons are re-invested?

Post by jjj_22 »

longvista wrote: Thu Nov 03, 2022 12:39 pm Let's verify whether we can get a 5% Annualized Return (as predicted by the 5% YTM).
Here you’re saying annualized return and YTM are the same. But the paper you reference explicitly explains that they are not:
“with a normal bond […] even though it pays a given interest rate — say 10% — the holder cannot be assured that a compounded 10% return will be realized. For that rate to materialize, each semi-annual coupon must be reinvested at 10% as it is received … This terminal or accumulated value perspective requires knowledge of the rate earned on reinvested coupons but is a distinctly different concept than YTM.
(Emphasis added)
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Re: Does YTM require that coupons are re-invested?

Post by toddthebod »

Every time this topic comes up, people talk past each other because some people interpret YTM to (incorrectly) equate to CAGR. YTM is agnostic to what you do with your coupon payments. CAGR is not.
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Re: Does YTM require that coupons are re-invested?

Post by Beensabu »

toddthebod wrote:Every time this topic comes up, people talk past each other because some people interpret YTM to (incorrectly) equate to CAGR.
There you go, OP!

The formula you were using is for CAGR (compound annual growth rate), not annualized return.

You can do a search for "CAGR vs. annualized return" to see how they are different. Pretty much, annualized return does not take compounding into account. Therefore, it does not include reinvestment of coupon payments.

And I've posted this at risk of losing my darn internet yet again. So I think having a conclusion to wrap it all up is important. But yeah... if I get jammed again, I'm done.
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Re: Does YTM require that coupons are re-invested?

Post by jeffyscott »

toddthebod wrote: Thu Nov 03, 2022 6:21 pm Every time this topic comes up, people talk past each other because some people interpret YTM to (incorrectly) equate to CAGR. YTM is agnostic to what you do with your coupon payments. CAGR is not.
This post (and the discussion that preceded it) helped me understand that difference and what YTM actually means, perhaps it can do the same for some others:
viewtopic.php?p=6695078#p6695078
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Re: Does YTM require that coupons are re-invested?

Post by grabiner »

There are two equivalent definitions of YTM for a bond with a fixed series of payments. For both models, $X is the current value of the bond, and Y% is the yield to maturity.

1. The present value of each payment, discounted at a rate of Y%, adds up to $X.

2. If you own the bond, and can reinvest each coupon at a rate of Y%, then your total return when the bond matures will also be Y%.

Both calculations convert each payment to its adjusted value on a given date; the difference is whether you use the present date or the maturity date. For example, for a 10-year bond worth $1000 with a 4% yield, formula 1 says that you could invest $1000 in zero-coupon bonds yielding 4% to get the same payments as the bond, while formula 2 says that if you reinvest all the coupons at 4%, you will have $1000 * (1.02^20) = $1486 in ten years. (The reason for 1.02^20 rather than 1.04^10 is that bond yields are quoted based on semiannual rates. For example, if the bond is at par, it will pay $20 every six months, which is slightly better than paying $40 every year since you can invest the mid-year payments.)
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Re: Does YTM require that coupons are re-invested?

Post by FactualFran »

longvista wrote: Thu Nov 03, 2022 3:22 pm Is this what you meant? If so, isn't there a contradiction between: "then the result will be equal to the YTM if the coupon payments are reinvested at the same YTM" and "However, YTM does not change when you do not reinvest the coupons at the same YTM"? If you think there's no contradiction, could you please elaborate?
There is no contradiction because YTM and investment return are not the same. As the quote in the opening post from Wikipedia puts it: "The owner reinvests all interest payments rather than spending them, to gain the benefit of compounded returns". The investment return includes that compounding; YTM does not. The investment return depends on the return on the interest payments; the YTM does not.
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Re: Does YTM require that coupons are re-invested?

Post by Oicuryy »

Image

That is from the help page for the excel PRICE function. The excel YIELD function uses a trial and error method to find the value of yld that makes the right side of the equation equal to the current price of the bond. That matches grabiner's definition 1.

Would someone please post the formula to calculate yield using his definition 2.

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

Post by grabiner »

Oicuryy wrote: Fri Nov 04, 2022 12:21 am Image

That is from the help page for the excel PRICE function. The excel YIELD function uses a trial and error method to find the value of yld that makes the right side of the equation equal to the current price of the bond. That matches grabiner's definition 1.

Would someone please post the formula to calculate yield using his definition 2.
Multiply both sides of the equation by (1+yield/frequency)^(number of payments). This corresponds to doing all calculations in maturity-date dollars at the yield, rather than in present dollars.
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Re: Does YTM require that coupons are re-invested?

Post by dbr »

grabiner wrote: Thu Nov 03, 2022 10:01 pm
2. If you own the bond, and can reinvest each coupon at a rate of Y%, then your total return when the bond matures will also be Y%.

So, if you impose a certain condition on a calculation that is not YTM then you get a number that is equal to the YTM. That is the origin of the problem. This is the fact that if you calculate IRR with no cash flows then the number you get is equal to CAGR, etc. The whole point of IRR is what you get when there are cash flows.

It is slight of hand to say that the calculation of YTM ASSUMES the cash flows are reinvested when in fact it explicitly does not if you use the actual definition of YTM and not a fake substitute. The whole point of YTM is what you get without assuming anything about reinvesting.
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Re: Does YTM require that coupons are re-invested?

Post by Oicuryy »

grabiner wrote: Fri Nov 04, 2022 7:38 am Multiply both sides of the equation by (1+yield/frequency)^(number of payments). This corresponds to doing all calculations in maturity-date dollars at the yield, rather than in present dollars.
Thanks, I should have thought of that. I knew that the rate that makes the present values of the cash flows sum to zero also makes the future values sum to zero.

It is still not right to say an investor must reinvest dividends to earn the YTM rate. Either way it's calculated, YTM is the dollar-weighted rate of return the investor will get if the coupons are not reinvested.

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

Post by longvista »

Beensabu wrote: Thu Nov 03, 2022 9:19 pm
toddthebod wrote:Every time this topic comes up, people talk past each other because some people interpret YTM to (incorrectly) equate to CAGR.
There you go, OP!

The formula you were using is for CAGR (compound annual growth rate), not annualized return.

You can do a search for "CAGR vs. annualized return" to see how they are different. Pretty much, annualized return does not take compounding into account. Therefore, it does not include reinvestment of coupon payments.
I actually could not find a difference between CAGR and Annualized Return. It appears that their formulas are essentially the same.

Did you perhaps mean to say "Annual Rate of Return" instead of "Annualized Return"? If you did indeed want to say "Annualized Return", could you please point me to a source which shows that Annualized Return is defined differently than CAGR?
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Re: Does YTM require that coupons are re-invested?

Post by patrick »

This is largely a question of definition.

If you keep the coupons in cash stuffed in your mattress, then the combined annual return (of the bond in combination with the cash stuffed in the mattress) will not match the YTM. However, the YTM is still the correct return for the bond itself.
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

toddthebod wrote: Thu Nov 03, 2022 6:21 pm Every time this topic comes up, people talk past each other because some people interpret YTM to (incorrectly) equate to CAGR. YTM is agnostic to what you do with your coupon payments. CAGR is not.
So if I understand you correctly and what you say is true, then it appears that sources such as Wikipedia, Investopedia and Annette Thau are incorrectly interpreting YTM to equate to CAGR.

If that is so, then I assume that YTM is actually showing the expected Annual Rate of Return. At least Wall Street Prep (https://www.wallstreetprep.com/knowledg ... urity-ytm/) is defining it as such: "The Yield to Maturity (YTM) represents the expected annual rate of return earned on a bond under the assumption that the debt security is held until maturity.".

First, let's clarify the meaning of Annual Rate of Return. My understanding of Annual Rate of Return is that if we invest $1,000 for 3 years and the Annual Rate of Return is 5%, then every year, I would get a return of $50, and my Total Return would be $150.

Question: Is this understanding of Annual Rate of Return correct?

If that's the case, and YTM is indeed actually referring to Annual Rate of Return, instead of CAGR, then it would make sense.

Taking again the bond in my example:

Code: Select all

Maturity: 3 years
Principal: $1,000
Current Price: $1,000
Coupon Payment: $50
The YTM is 5%. If we don't interpret this as the expected CAGR, but instead interpret it as the Annual Rate of Return, then put differently, a YTM of 5% for my bond would mean: 5% earned on the initial investment (current price of $1,000), for 3 years (until maturity).

So the Total Return would be $50 * 3 = $ 150.

If that's indeed the case, that would explain why there is confusion regarding YTM.

Question: Does YTM actually promise an Annual Rate of Return, instead of CAGR?

Question: Assuming that the answer to the previous question is "yes", is this the likely reason why many sources get it wrong?
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Re: Does YTM require that coupons are re-invested?

Post by longvista »

km91 wrote: Thu Nov 03, 2022 3:21 pm YTM is not the same as an annualized compounded growth rate. We can see that a 5% growth rate gets a final value of $1157 but the cashflows of the bond are only $1150. Every $1000 of face value purchased earns a cash yield of 5%, the face value of the bond doesn't grow
km91 wrote: Thu Nov 03, 2022 3:03 pm YTM is not a compounding rate, it's the internal rate that sets the future value of the cashflows to current price. The paper and Excel's IRR and RATE functions are correct, yield to maturity is calculated without considering the reinvestment rate of the cashflows
km91 wrote: Thu Nov 03, 2022 3:40 pm YTM tells us how much income the bond will generate per dollar of face value purchased

In the yield to maturity formula we are discounting the future cashflows back to present value. The ytm / irr is the discount rate that makes future value = present value. We are valuing future cashflows in today's dollars. With the annualized rate you present, we are valuing today's cashflows in terms of future dollars
So basically, it looks like you are saying the same thing as toddthebod:
Every time this topic comes up, people talk past each other because some people interpret YTM to (incorrectly) equate to CAGR. YTM is agnostic to what you do with your coupon payments. CAGR is not.
So to verify whether I understand your point correctly, you are saying that a YTM of 5% actually means that for every year, until maturity, you will earn 5% on your initial investment? So that if I invest $800 and YTM is 5% and term is 4 years, then I will earn 5% * 4 = $ 200 in total, in contrast to the (incorrect) interpretation of earning a 5% compounding rate of return (CAGR / Annualized Return).

Is that what you are saying?
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Re: Does YTM require that coupons are re-invested?

Post by patrick »

Going back to the original question:
So now I am confused. The paper from the researchers says that "it is an erroneous assumption", that "in order to earn the yield to maturity on a coupon bond an investor must reinvest the coupon payments".

How is it erroneous? If I do not reinvest the coupon payments, my Annualized Return will be ~4.77%. If I do reinvest the coupons, my Annualized Return will be 5.00%, exactly equal to YTM.

Question: If the research paper is correct and achieving an Annualized Return of 5% does not require reinvesting coupons, then I should get a 5% Annualized Return when not reinvesting the coupons, but I only get 4.77%. Why is that?

Question: Perhaps the research paper is not correct, because the only way to get a 5% Annualized Return is to reinvest the coupons, contrary to what the research paper says. Is the research paper incorrect?
The research paper is surely correct that returns earned from the bond itself are 5% regardless of what you do with the coupons.

Your 4.77% figure comes from analyzing a larger portfolio. At the start, the portfolio includes only the bond. Later on, your portfolio is part bond, part cash earning 0% interest. It should be no surprise that mixing a 5% bond with 0% cash lowers the return below 5%.

That said, I wouldn't call someone incorrect merely for choosing a different one of these two perspectives.
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Re: Does YTM require that coupons are re-invested?

Post by JackoC »

patrick wrote: Fri Nov 04, 2022 1:01 pm This is largely a question of definition.

If you keep the coupons in cash stuffed in your mattress, then the combined annual return (of the bond in combination with the cash stuffed in the mattress) will not match the YTM. However, the YTM is still the correct return for the bond itself.
Yeah I think it's entirely a semantic question. As was correctly stated already, YTM itself is just the (single) rate at which you must discount all the future cash flows of the bond to arrive at its current market PV. If the YTM of a given bond is 4% today for me, it's 4% for you and 4% for the other person regardless of whether I plan to sell it tomorrow, you plan to hold it to maturity reinvesting the coupons and the other person intends to hold to maturity but consume the coupons as they come in.

However, the closely related inference that the expected return ('CAGR') of a credit riskless bond held to maturity, keeping all the cashflows invested, *is* the YTM *does* assume the coupons are reinvested at the YTM. And 'reinvestment risk' is a real and useful practical concept, relative to the assumption E[r]=YTM. A low coupon below par bond has less reinvestment risk, a high coupon above par bond has more, a zero coupon bond has none, all assuming no default and that the investment horizon is the bond maturity date. IMO the thing OP read saying, 'YTM doesn't assume reinvestment' while true on a narrow semantic basis is confusing and misleading practically.
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Re: Does YTM require that coupons are re-invested?

Post by Beensabu »

longvista wrote: Fri Nov 04, 2022 12:49 pm Did you perhaps mean to say "Annual Rate of Return" instead of "Annualized Return"?
I don't even know anymore, to be honest. :)

Ah, the joys of terminology...

I think the main thing is that the formula you have used is for calculating CAGR and only takes initial and final values into account, whereas YTM/IRR considers periodic cash flows as of the time they are actually received.
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Re: Does YTM require that coupons are re-invested?

Post by km91 »

Beensabu wrote: Fri Nov 04, 2022 2:17 pm
longvista wrote: Fri Nov 04, 2022 12:49 pm Did you perhaps mean to say "Annual Rate of Return" instead of "Annualized Return"?
I don't even know anymore, to be honest. :)

Ah, the joys of terminology...

I think the main thing is that the formula you have used is for calculating CAGR and only takes initial and final values into account, whereas YTM/IRR considers periodic cash flows as of the time they are actually received.
CAGR is a backward looking measure of total return. We invested in a bond with initial value of $x, it paid coupons of y%, we may or may not have done anything with the coupons, and now have a final value of $z. We don't know at what rate coupons will be reinvested at so we can only calculate CAGR after the fact

YTM is a forward looking measure of return. We invest in a bond with initial value of $x, we know it will pay coupons of y%, and will repay par at maturity. We know that this bond will generate a stream of income equivalent to the ytm on a forward looking basis without needing to make any assumptions around reinvestment of the cashflows. We use ytm so we can measure the expected returns of all bonds against one another on a comparable basis
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Re: Does YTM require that coupons are re-invested?

Post by toddthebod »

longvista wrote: Fri Nov 04, 2022 1:04 pm
toddthebod wrote: Thu Nov 03, 2022 6:21 pm Every time this topic comes up, people talk past each other because some people interpret YTM to (incorrectly) equate to CAGR. YTM is agnostic to what you do with your coupon payments. CAGR is not.
So if I understand you correctly and what you say is true, then it appears that sources such as Wikipedia, Investopedia and Annette Thau are incorrectly interpreting YTM to equate to CAGR.
The Wikipedia article is not even self-consistent:
It is the (theoretical) internal rate of return (IRR, overall interest rate): the discount rate at which the present value of all future cash flows from the bond (coupons and principal) is equal to the current price of the bond.
Note that there is no mention of what you do with the future cash flows.
longvista wrote: Fri Nov 04, 2022 1:04 pm First, let's clarify the meaning of Annual Rate of Return. My understanding of Annual Rate of Return is that if we invest $1,000 for 3 years and the Annual Rate of Return is 5%, then every year, I would get a return of $50, and my Total Return would be $150.

Question: Is this understanding of Annual Rate of Return correct?
I don't know. The SEC says:
The annual rate of return is the percentage change in the value of an investment.
Investor.gov says:
An annual rate of return is the profit or loss on an investment over a one-year period.
I don't see how that applies to this discussion.
longvista wrote: Fri Nov 04, 2022 1:04 pm Taking again the bond in my example:

Code: Select all

Maturity: 3 years
Principal: $1,000
Current Price: $1,000
Coupon Payment: $50
The YTM is 5%.
We agree on this. You have earned your yield-to-maturity when you receive those coupon payments and principal back, regardless of what you do with those coupons. If you reinvest them at the same rate because you want to earn a CAGR of 5%/year, you will have $1,157.63 at the end of three years. Nowhere in your description or in any brokerage quote or in any yield calculation is there some claim to ending up with $1,157.63.
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Re: Does YTM require that coupons are re-invested?

Post by km91 »

longvista wrote: Fri Nov 04, 2022 1:04 pm Question: Does YTM actually promise an Annual Rate of Return, instead of CAGR?

Question: Assuming that the answer to the previous question is "yes", is this the likely reason why many sources get it wrong?
YTM promises exactly what it describes; a yield. In it's simplest form, yield = income / principal invested. If a stock has a dividend yield of 5% we will expect it to generate a stream of dividend payments equal to 5% of the invested amount. If a bond has a YTM of 5% we will expect it to generate a stream a coupon and price appreciation payments equal to 5% of the invested amount. If a property has a rental yield of 5% we will expect it to generate a stream of rental income equal to 5% on the invested amount.

CAGR is a backward looking metric, it is unknowable at the time when we make the investment without making assumptions. A simplifying assumption that is often made is that if we are able to reinvest coupons at the YTM, YTM = CAGR. The quoted YTM of a bond makes no assumptions about the reinvestment rate of coupons. It solely considers the the yield on principal invested in the security
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Re: Does YTM require that coupons are re-invested?

Post by #Cruncher »

km91 wrote: Thu Nov 03, 2022 3:03 pm... Excel's IRR and RATE functions are correct, yield to maturity is calculated without considering the reinvestment rate of the cashflows
Correct. Maybe this will be clearer with a savings account analogy -- but then again, maybe not. :wink: A savings account pays its interest rate on whatever the balance happens to be. The rate doesn't change depending on our withdrawals. Assume one pays 5% and we start out with a $1,000 balance. The following table shows the balance each year with five assumed annual withdrawals: None, $30, $50, $100, and $367.21. On row 9 the RATE function confirms a 5% rate of return in all five cases.

Code: Select all

Row     Col A     Col B     Col C     Col D     Col E     Col F
  2      Rate      5.0%
  3  Withdraw      0.00     30.00     50.00    100.00    367.21
  4      Year  -------------------- Balance -------------------  Formula in Col B copied right
  5         0  1,000.00  1,000.00  1,000.00  1,000.00  1,000.00
  6         1  1,050.00  1,020.00  1,000.00    950.00    682.79  =B5*(1+$B$2)-B$3
  7         2  1,102.50  1,041.00  1,000.00    897.50    349.72  =B6*(1+$B$2)-B$3
  8         3  1,157.63  1,063.05  1,000.00    842.38      0.00  =B7*(1+$B$2)-B$3
  9      RATE      5.0%      5.0%      5.0%      5.0%      5.0%  =RATE(3,B3,-B5,B8,0)
Now look at the corresponding cash flows assuming we withdraw the balance at the end of year 3. On row 16, the IRR function again confirms a 5% rate of return in all five cases.

Code: Select all

 11      Year ------------------- Cash Flow -------------------  Formula in Col B copied right
 12         0 -1,000.00 -1,000.00 -1,000.00 -1,000.00 -1,000.00
 13         1      0.00     30.00     50.00    100.00    367.21  =B3
 14         2      0.00     30.00     50.00    100.00    367.21  =B3
 15         3  1,157.63  1,093.05  1,050.00    942.38    367.21  =B3+B8
 16       IRR      5.0%      5.0%      5.0%      5.0%      5.0%  =IRR(B12:B15)
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Re: Does YTM require that coupons are re-invested?

Post by dbr »

toddthebod wrote: Fri Nov 04, 2022 3:25 pm
The Wikipedia article is not even self-consistent:
It is the (theoretical) internal rate of return (IRR, overall interest rate): the discount rate at which the present value of all future cash flows from the bond (coupons and principal) is equal to the current price of the bond.
Note that there is no mention of what you do with the future cash flows.
Yes, it would seem there is a failure to understand how to talk about rate of return when the timing is messed up by having cash flows coming and going at different times. That is why there is no average return (CAGR) when there are cash flows, but we can try to talk about something that has to do with yield. The fact that YTM is defined so that what one does with the coupon payments is not considered is the whole point because what is done with the payments has nothing to do with buying, holding, and redeeming the bond. One could do anything with them with different results depending on what that is. It would be like trying to make the IRR on a portfolio with contributions depend on what you were earning on the contributions before you invested them. If a person does want to know how wealthy they will be at the end, then that does depend on what they do with the payments.
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Re: Does YTM require that coupons are re-invested?

Post by jeffyscott »

#Cruncher wrote: Fri Nov 04, 2022 3:35 pm
km91 wrote: Thu Nov 03, 2022 3:03 pm... Excel's IRR and RATE functions are correct, yield to maturity is calculated without considering the reinvestment rate of the cashflows
Correct. Maybe this will be clearer with a savings account analogy -- but then again, maybe not. :wink: A savings account pays its interest rate on whatever the balance happens to be.
Very good analogy, each time you get a coupon payment that's a (forced) withdrawal from the "savings account" (bond) of $X. So the "savings account" would no longer pay interest on that $X that has been withdrawn from it. But it continues to pay the same interest rate on the remaining balance.

The higher the coupon, the greater the forced withdrawals are. A zero coupon has no forced withdrawals.
And so it goes, And so it goes, And so it goes, And so it goes, But where it's goin' no one knows
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