SEC Yield, YTM, Distribution Yield, Current Yield

[Kevin M]

I own muni bond funds and direct CDs (purchased directly from banks and credit unions). The CDs are safer, since they have no credit risk and low term risk due to the early withdrawal option, but the muni bond funds might provide higher after-tax yields in return for taking some credit risk and more term risk. When posting about CDs here, I've often compared the yields to Treasuries of the same maturity, since both have essentially no credit risk. For my own education, I thought it would be interesting to take a look at some individual 5-year munis; since I'm a CA resident and own both the CA intermediate-term and long-term bond funds, I looked at CA munis.

Part of this investigation involved looking at yield to maturity (YTM) and current yield for the bonds, and it occurred to me that it might be interesting and even educational to some Bogleheads to compare yield to maturity (YTM) and current yield for the individual bonds to SEC Yield and distribution yield for a bond mutual fund that might own these bonds. I'll start with the individual bonds, since I think they're easier to understand, and then draw the parallels to the yields for the bond fund.

At first I looked at AAA muni bonds only, since I probably wouldn't consider owning individual bonds with a lower credit rating. But then I saw that about 75% of the bonds in the Vanguard California Intermediate-Term bond fund (VCADX, Admiral Shares) were rated AA, so I decided to look at AA bonds instead, even though I probably wouldn't buy them. I searched the Fidelity website and found 12 AA-rated CA muni bonds that were close to a 5-year term to maturity. Here are the averages of the relevant numbers for the 12 bonds:

Term to maturity: 4.97 years.
Duration: 4.53 years
Coupon: 3.64%
Price: 109.83
YTM: 1.57%
YTW: 1.25%
Current yield: 3.3%

YTW = yield to worst, which factors in the call options on the bonds; e.g., one bond has a coupon rate of 5% and a YTM of 2.4%, but a call provision makes it likely that the bond will be called before maturity, lowering the YTW to 0.9%. So YTW probably is the more relevant number if I were going to buy the bond.

Note that the current yield of 3.3% is much higher than the YTM of 1.57% and the YTW of 1.25%. Using the average numbers from above, the current yield is calculated by dividing the average coupon payment by the average price (I'll assume annual coupon payments for simplicity):

3.64/109.83 = 3.3%

The YTM is much lower because the bonds will all mature at a price of 100 (assuming no defaults), which is 9.83 less than the current average price. The loss of that 9.83 over the 5-year holding period reduces the YTM to 1.57% (not even factoring in the call provisions).

Now let's compare these numbers to VCADX. Here are the relevant average numbers as of 9/30/2014:

Stated maturity: 9.6 years
Duration: 4.9 years
Coupon: 4.1%
Price: 11.76
SEC yield: 1.65%
Average YTM: 1.7%
Distribution yield: 3.00%

Note that although the average maturity is much higher at 9.6 years, the average duration is only a little higher at 4.9 years vs. 4.5 years for the individual bonds. Note that the average coupon and YTM for the fund are in the same ballpark as for the bonds, and the distribution yield for the fund is close to the current yield for the bonds. This is a key take-away.

Since the average coupon rates for the bonds and the fund are quite a bit higher than the yields to maturity, the bonds are priced above face value (par), and current yield (bonds) and distribution yield (fund) are quite a bit higher than YTMs. The latter was shown in the calculation earlier by dividing coupon payment by current price.

The way Vanguard calculates distribution yield results in a number that is very close to the current yield calculation for a bond. If we divide the most recent distribution amount on 9/30/2014 by the price on 9/30/2014, and annualize it we get:

0.02890 / 11.76 / 30 x 365 = 2.99%

The only difference between this calculation and the Vanguard calculation resulting in 3.00% is that VG uses the average price during the month instead of the price on the date of the distribution (reinvest date):

Distribution Yield: The fund's current monthly income dividend per share, annualized (by dividing by the number of days in the month and multiplying by 365) and shown as a percentage of the fund's average NAV during the month.

Another thing to note is that the average YTM for the fund was pretty close to the SEC yield on 9/30/2014. SEC yield is basically an average YTM over the previous 30 days. So both SEC yield (as mandated by the SEC) and distribution yield (as calculated by Vanguard), involve some averaging over the previous month.

If the bond fund were to hold all of it's bonds to maturity and buy no more bonds, and none of the bonds defaulted or were called, the share price would gradually decrease, just as the average price of the individual bonds will gradually decrease. So other than the unknown rate at which the coupon payments could be reinvested, the YTM would be a good indicator for the total return over the holding period for the collection of bonds or the bond fund.

Of course the fund does not just hold a collection of bonds to maturity, but is constantly buying and selling bonds, and maintaining an average maturity and duration at an intermediate-term target. And this bond fund is actively managed, as are many other Vanguard bond funds, so without looking at the individual holdings over time, it's hard to say exactly what the Vanguard bond fund managers are doing. I've seen some puzzling relationships between SEC yield and distribution yield over time for some Vanguard bond funds, and have posted about that elsewhere.

Oh, and back to the original point of the investigation. Even with a combined federal and state marginal income tax rate of 50%, which is much higher than my marginal rate, the after-tax yield of a direct 5-year CD earning 2.52% (best I know of currently) beats the tax-free yield to worst of 1.25% for the AA muni bonds, even without accounting for the credit risk and higher term risk of the bonds.

Kevin

[Kevin M]

I thought it might be useful to summarize the OP without all of the detailed explanations and math.

- SEC yield for a bond fund is analogous to yield to maturity (YTM) for an individual bond. These yield measures factor in change in bond price in addition to coupon payments (dividend distributions).

- Distribution yield (as calculated by Vanguard) for a bond fund is analogous to current yield for an individual bond. Basically these yield measures are calculated by dividing coupon payment (annual) or dividend distribution by the bond price or fund share price (and annualizing).

- If the coupon rate of an individual bond is higher than the YTM, the bond price will be above face value, and the current yield will be higher than the YTM. Bond price will eventually fall to face value as the bond approaches maturity.

- Similarly, if the distribution yield of a bond fund is higher than the SEC yield (or average YTM), the average bond price is above average face value, and the average coupon rate is above YTM. Without additional sales or purchases, the average bond price of the current holdings would eventually fall to face value, causing the fund share price (NAV) to gradually decrease, but the ongoing purchases and sales of bonds also will affect NAV.

- A collection of 12 AA rated, 5-year California muni bonds has price and yield characteristics that are similar to the Vanguard California Intermediate-Term Tax-Exempt Bond fund. Examining the individual bonds can help in understanding the price and yield characteristics of the bond fund.

I also thought it might be helpful to show the bonds I used in the calculations in the OP, so here's a screenshot of them.



Also, I thought the following charts might help in understanding the relationship between bond price, yield to maturity, and current yield. These charts are based on the current yield curve.



Note for example in the left chart how price of the 2% coupon bond is about par (face value) at a term to maturity of 7 years, and in the right chart how current yield for the 2% coupon bond intersects with the yield curve at a term to maturity of about 7 years, at which point YTM currently is about 2% (the curve labeled "Yield treasury.gov" in the right chart is generated from the most recent yields to maturity from treasury.gov).

Note also how current yield is relatively stable compared to yield to maturity. This is analogous to how bond fund distribution yield is relatively stable compared to SEC yield, which you can verify by looking at the distribution yield history of a Vanguard bond fund, and comparing it to the SEC yield history.

Kevin

[Hank Moody]

Wonderful post, Kevin. I have one question for you.

I've often wondered, is the YTM for a bond fund calculated before or after fund expenses? I've always assumed it was before but I'd appreciate some clarification.

Thank you,
HM

[Kevin M]

Wonderful post, Kevin. I have one question for you.

I've often wondered, is the YTM for a bond fund calculated before or after fund expenses? I've always assumed it was before but I'd appreciate some clarification.

Thank you,
HM

Thanks Hank.

SEC Yield basically is average YTM over the previous 30 days minus expenses. This formula is mandated by the SEC, so all bond funds that publish yield figures must publish SEC Yield (aka 30-day yield). Or as Vanguard says if you hover your mouse over the "SEC yield" column header on the "Price & Yield" tab of the "Vanguard mutual funds" page:

SEC yield
A non-money market fund's SEC yield is based on a formula mandated by the Securities and Exchange Commission (SEC) that calculates a fund's hypothetical annualized income, as a percentage of its assets. A security's income, for the purposes of this calculation, is based on the current market yield to maturity (in the case of bonds) or projected dividend yield (for stocks) of the fund's holdings over a trailing 30 day period. This hypothetical income will differ (at times, significantly) from the fund's actual experience; as a result, income distributions from the fund may be higher or lower than implied by the SEC yield.

The SEC yield for a money market fund is calculated by annualizing its daily income distributions for the previous seven days.

They don't mention expenses explicitly, but if you look up the definition of the SEC 30-day yield, you'll see that expenses are subtracted.

The average yield to maturity figure you can find if you look at "bond attributes" on Vanguard's mutual funds page is before expenses. All you see if you hover your mouse over the "Yield to maturity" column header is:

Yield to maturity

The rate of return an investor would receive if a security is held to its maturity date.

Also of interest, if you hover your mouse over "Avg. Maturity" (column header), it says:

Average Effective Maturity (taxable bond funds and balanced funds except Tax-Managed Balanced)
Average Effective Maturity is defined as the average length of time until fixed income securities held by a fund reach maturity and are repaid, taking into consideration that an action such as a call or refunding may cause some bonds to be repaid before they mature.

Average Stated Maturity (municipal bond funds and Tax-Managed Balanced)
Average Stated Maturity represents the average of the stated maturity dates for all fixed income securities held by the fund.

Average Maturity (money market funds only)
Average maturity represents the weighted average maturity of the fund's holdings using the date of the next interest rate adjustment for certain adjustable-rate securities held by the fund.

So for taxable bond funds, call provisions are factored in, but for muni bond funds they are not. I don't know the reason for the difference.

Kevin

[Hank Moody]

Ignoring call provisions, if I wanted to equate the expected YTM of a bond fund to that of a single bond, I would subtract the fund expenses from the bond fund?

[Kevin M]

Ignoring call provisions, if I wanted to equate the expected YTM of a bond fund to that of a single bond, I would subtract the fund expenses from the bond fund?

Well the SEC yield already has the expenses stripped out, but it's an average YTM over previous 30 days.

You can't see the average YTM today, but only for the last day of last quarter, so as of now only for 9/30/2014. Looking at an annual report or semiannual report, it is noted that YTM is before expenses. So yes, the YTM stated on the website or in a report is the average YTM of the bonds in the fund, before expenses.

You can look up SEC yield for 9/30/2014 and compare to reported YTM on that date. Since I used VCADX as an example, here are the numbers for that fund on 9/30/2014:

SEC yield: 1.65%
YTM: 1.7%

Admiral share fund expense ratio is 0.12%, so subtracting that from YTM on 9/30 gives 1.58%. The difference between this number and 1.65% must be due to the 30-day averaging to get SEC yield.

Another comment on this. I actually loaded the holdings of Vanguard Intermediate-Term Treasury fund into a spreadsheet, and calculated average YTM. What I find puzzling is that a simple average for YTM of holdings on 7/31/2014 is 1.78%, which agrees with the stated YTM of 1.8%, but the weighted YTM I compute is 1.86%.

I would think the weighted YTM would match the value reported by the fund in the semiannual report. It doesn't make sense not to compute some sort of weighted YTM, so they must be calculating weighted YTM differently than I am. Or maybe it has something to do with the options/futures the fund owns and/or the cash holdings. I simply multiplied the YTM of each bond by its relative weight (% of total holdings by market value) and summed the results. No options, futures or cash were included in my calculation. I cannot find anything in the semiannual report that provides any details on the YTM calculation.

EDIT: OTOH, my calculated weighted YTM is only 1-2 basis points too high to round down to the fund stated value of 1.8%. It would be nice though to know exactly how they're calculating it.

Kevin

[Hank Moody]

makes complete sense. thanks.

[Doc]

Another comment on this. I actually loaded the holdings of Vanguard Intermediate-Term Treasury fund into a spreadsheet, and calculated average YTM. What I find puzzling is that a simple average for YTM of holdings on 7/31/2014 is 1.78%, which agrees with the stated YTM of 1.8%, but the weighted YTM I compute is 1.86%.

A few days ago I was trying to get a handle on some of the minor discrepancies that have been noted in various threads recently. Some factors like odd coupon dates and annual versus semi-annual coupons are relatively small especially with today's current low rates. How you weight securities in a fund is another matter. Whether you just market rate or actually do a discounted cash flow analysis on the payments is a little more important. But what happens to that shorter term note when it matures? Do you just drop it from the calculation or do you assume it "rolls"? As an example I took a test portfolio with three coupon bonds - a five, a ten and a twenty three (don't ask). The results were an IRR of 2.52% and 2.84% for the maturing versus rolling assumptions respectively. That's a significant difference.

Of course bond funds don't behave like either of my model funds. So I don't know what significance this has on the question(s) at hand.

Our FI portfolio contains only easily marketable securities (not CD's). It is broken up into a number of parts for tax efficiency and ease of rebalancing. I try to keep the overall portfolio close to the Barcap 1-10 Government/Credit Index in terms of duration and sector breakdown. The return or IRR or YTM by whatever calculation method one uses is what it is. Having a more consistent methodology for estimating returns would not affect my investing decisions. Nevertheless I am enjoying these threads and will continue to put in my 200 bps from time to time.

[magellan]

The way Vanguard calculates distribution yield results in a number that is very close to the current yield calculation for a bond. If we divide the most recent distribution amount on 9/30/2014 by the price on 9/30/2014, and annualize it we get:

0.02890 / 11.76 / 30 x 365 = 2.99%

My understanding is that fund distributions are dependent on actual coupon payments received as well as IRS rules about how bond premiums are amortized.

This means that two different bond funds with the exact same holdings, but different acquisition dates and prices, could have different distribution yields.

Jim

[Kevin M]

My understanding is that fund distributions are dependent on actual coupon payments received as well as IRS rules about how bond premiums are amortized.

This means that two different bond funds with the exact same holdings, but different acquisition dates and prices, could have different distribution yields.

Jim

Good point. To be precise, this is what Vanguard says in their bond semiannual and annual reports:

Premiums and discounts on debt securities purchased are amortized and accreted, respectively, to interest income over the lives of the respective securities.

So premiums are amortized and discounts are accreted.

So working this through, with my sample CA muni bond portfolio, if purchased today at an average price of about 110, there would be about $10 per $100 par value that would need to be amortized over about 5 years (since average term to maturity is about 5 years). So that's about $2 per year per $100 face value that would be subtracted from investment income to account for the bonds falling from 110 now to 100 at maturity. Correct?

A fund that bought those bonds at par in the past would have a distribution yield of the average coupon divided by average price, so roughly 4/110 = 3.6% (ignoring expenses). But a fund that bought today would have a distribution yield of about (4 - 2) / 110 = 1.8%. Does that sound about right?

For comparison purposes, the VG CA muni fund, VCADX, has an average coupon of about 4% and a distribution yield of about 3%.

Kevin

[magellan]

A fund that bought those bonds at par in the past would have a distribution yield of the average coupon divided by average price, so roughly 4/110 = 3.6% (ignoring expenses). But a fund that bought today would have a distribution yield of about (4 - 2) / 110 = 1.8%. Does that sound about right?

It does and your example very nicely shows how fickle distribution yield can be and why it shouldn't be considered an indicator of the true 'earnings' that a bond fund is producing.

Depending on the prices a fund paid for its securities, its distribution yield can be all over the map. Also, if this were a taxable bond in a taxable account, 1.8% of that 3.6% dividend yield would be nothing more than a very tax inefficient return of capital.

[Doc]

A fund that bought those bonds at par in the past would have a distribution yield of the average coupon divided by average price, so roughly 4/110 = 3.6% (ignoring expenses). But a fund that bought today would have a distribution yield of about (4 - 2) / 110 = 1.8%. Does that sound about right?

Sounds right but I don't think it is. The problem is that the amortization/accretion effects have tax consequences even if there is no cash flow. I believe that funds have to distribute the taxable amount without regard to cash flow. This is why TIPS funds are often claimed to be "better" than a ladder because the funds distributions gives you the cash on hand to pay the tax on the OID even though the OID itself is effectively the same thing as accreting a market discount.

[magellan]

The problem is that the amortization/accretion effects have tax consequences even if there is no cash flow. I believe that funds have to distribute the taxable amount without regard to cash flow.

I can't exactly tell where you're agreeing with Kevin and where you're disagreeing. You both mention that distributions are NOT exactly tied to bond coupon payments because of tax consequences of premium/discount amortization/accretion. That sounds right to me.

I could be confused about this, but in my pre-boglehead days I owned a portfolio of individual bonds. Let me tell you, computing taxes due with an individual bond portfolio is not for the faint of heart. IMO, Vanguard's bond fund managers earn their ER just on this aspect alone. Still, unless I totally messed up my taxes back then, I think Kevin's example sounds about right.

The long and short of it is that bond coupon payments and taxable interest income don't have to be equal when premium and discount bonds are involved.

This article explains how it works better than I could.

Jim

[Electron]

I also hold VCADX and enjoyed reading your analysis. I have often wondered if the SEC Yield posted for Vanguard bond funds is very useful to the average investor. At times it seems to change more than expected with changes in NAV and might even be deceptive at times.

One example recently has been Vanguard High Yield Corporate VWEHX.

05-15-14 - NAV $6.14 - SEC Yield 3.88%

10-17-14 - NAV $6.01 - SEC Yield 4.98%

The change in SEC Yield from 3.88% to 4.98% is quite dramatic and represents a 28% increase. It would seem unlikely that the fund holdings experienced any major changes during that five month period. If anyone has any thoughts please pass them along. The latest dividend was in line with previous months. One interesting thing about this fund is that the dividend payout has been in a significant long term decline. The dividend was $0.084 in 1989 and has steadily declined to about $0.027 today.

[Doc]

1) in reply to Doc

I can't exactly tell where you're agreeing with Kevin and where you're disagreeing.

2)

My understanding is that fund distributions are dependent on actual coupon payments received as well as IRS rules about how bond premiums are amortized.

3)

A fund that bought those bonds at par in the past would have a distribution yield of the average coupon divided by average price, so roughly 4/110 = 3.6% (ignoring expenses). But a fund that bought today would have a distribution yield of about (4 - 2) / 110 = 1.8%. Does that sound about right?

I agree with Magellan statement 2 but I don't think that is the same as Kevin statement 3. Kevin says average coupon but Magellan says coupons plus amortization (and accretion by extension.) But maybe I misread what Kevin is saying.

I'm not sure how this affects the distribution yield if at all.

[magellan]

A fund that bought those bonds at par in the past would have a distribution yield of the average coupon divided by average price, so roughly 4/110 = 3.6% (ignoring expenses). But a fund that bought today would have a distribution yield of about (4 - 2) / 110 = 1.8%. Does that sound about right?

Kevin says average coupon but Magellan says coupons plus amortization (and accretion by extension.)

Ok, I think I understand. In Kevin's example, the 3.6% DY case assumes that the bond was bought at par, so there is no amortization of the premium possible.

The rules about amortization and accretion have to do with the price paid when the bond was purchased. So in the case of the bond bought at par and now worth 110 (the 3.6% DY case), the coupon cash flow will exactly equal taxable interest income (and the distribution) because there's no premium to amortize. It's only in the other case, where the bond was bought at a premium, that amortization may be done.

I think the IRS is saying that if an investor can choose between two similarly risky bonds, say each yielding 2% with the same duration, the fact that one has a 2% coupon and sells at par, while another has a 5% coupon and sells at a premium, shouldn't cause different levels of taxable income to have to be reported by the investor. The risk and nature of the income from those two investments is similar, so its tax treatment should also be similar.

OTOH, the fact that an investor who bought that same 5% coupon bond a few years ago at par has to declare the full coupon payment as income is still fair, because the bond was actually yielding 5% at the time of purchase, the investor expected to earn 5% on the bond, so the investor should pay interest income taxes on the full 5% for the life of the bond. Maybe that's not the logic the IRS uses, but that's how I see it.

Jim

[Kevin M]

Ok, I think I understand. In Kevin's example, the 3.6% DY case assumes that the bond was bought at par, so there is no amortization of the premium possible.

Exactly. That was just an example using some numbers to illustrate Jim's original point, and make sure we all agree on how it works.

Kevin

[Kevin M]

I also hold VCADX and enjoyed reading your analysis. I have often wondered if the SEC Yield posted for Vanguard bond funds is very useful to the average investor. At times it seems to change more than expected with changes in NAV and might even be deceptive at times.

I actually have done quite a bit of investigation on this. Here are some links to posts on this as far as back as 3+ years.

Bogleheads • View topic - sec yield vs distribution yield

Bogleheads • View topic - SEC Yield, Distribution Yield and NAV over 18 months

Here's one that started out as a CD vs. Bond fund thread, but morphed into an exploration of SEC yield vs. Price:

Bogleheads • View topic - CD vs. Bond Fund: what if no early withdrawal?

And here's a chart from that reply that makes your point very clear, with some of the explanatory comment included:

Notice that there are daily percentage price changes of as much as about +/- 0.75% for zero change in SEC yield, and generally a wide range of % price changes for various changes in SEC yields.

A discussion of the chart continued here: Bogleheads • View topic - Distribution Yield versus SEC Yield

One thing that was pointed out in that thread is that SEC yield is a 30-day average, so we should expect some noise in a plot of daily prices and SEC 30-day (average) yields. Forum member tfb provided an example to help understand it. However, I'm not convinced that's the full answer.

I've done some subsequent investigation of this recently using 30-day average price, and the results still look pretty flaky. I had planned on getting to that in this thread at some point, but have been focusing more on yield curve investigations lately. I redid some other charts using 30-day average price, but I don't think I redid the one above.

In one of those old threads I looked at Short-Term Tax-Exempt fund due to the very short duration (about 1 year), thinking we should be able to see some sort of convergence over a 1-year period (equal to duration) when considering price, SEC yield, and distribution yield. I've added the data to my spreadsheet for the months since that original post on the topic, so now have much more than one year of yield and price data for a fund with a duration of about one year. Will share it at some point.

Kevin

[Doc]

Ok, I think I understand. In Kevin's example, the 3.6% DY case assumes that the bond was bought at par, so there is no amortization of the premium possible.

Exactly. That was just an example using some numbers to illustrate Jim's original point, and make sure we all agree on how it works.

Kevin

OK, I thought Kevin was making a more general statement.

I was thinking of a fund that holds only zero coupon bonds with a 2% YTM. That fund has a taxable yield of about 2% but an actual coupon of zero. But it pays out the 2%. The underlying principle is that the fund operates on the accrual basis but the taxpayer for the most part is on a cash basis. The fund has to satisfy both so it pays out the 2% even if it has to sell assets to do it.

[davidsorensen32]

Oh, and back to the original point of the investigation. Even with a combined federal and state marginal income tax rate of 50%, which is much higher than my marginal rate, the after-tax yield of a direct 5-year CD earning 2.52% (best I know of currently) beats the tax-free yield to worst of 1.25% for the AA muni bonds, even without accounting for the credit risk and higher term risk of the bonds.
Kevin

I just checked Oct 31 distributions from my VCADX holdings and they are 3% annualized (i.e. I take the distributions on Oct 31 and multiply by 12). I've held the position over multiple years (I forgot how long but < 5yrs if memory serves right) and all dividends have been redistributed since I began the position. Since I'm a resident of CA, all distributions are tax free. Therefore I love your analysis but disagree with your conclusion above.

[Kevin M]

I think we're all on the same page now. I'll try to make it even more clear with this example, which is similar to the example I already gave, but instead I'll just use a single bond. I won't bother doing actual bond math to verify the numbers, since the principle is the same even if the numbers are off some.

Investor/fund A bought a 10-year bond at par (price = 100) 5 years ago with coupon of 4%. Since the bond was purchased at par, there is no premium to amortize or discount to accrete. Therefore investor A pays taxes on the 4% coupon each year. Fund A that only owns this one bond distributes the 4% coupon (less expenses) as "dividend distributions". So for every $100 of face value, investor A receives $4 annually in coupon payments, and fund A distributes $4 per $100 face value (ignoring expenses).

Investor B buys the same bond today at a price of 110. Of course coupon still is 4%, and now the bond matures in 5 years. This bond is referred to as a premium bond because it is selling at a premium to par (price higher than 100). Since the bond will mature at par value of 100, there will be a $10 loss per $100 of face value over 5 years, for an annual loss of $2 per $100 face value due to premium amortization.

Investor B chooses to amortize the premium and deduct the annual amortized premium amount of $2 per $100 face value from the 4% coupon payments ($4 per $100 face value), resulting in a net taxable interest income of $2 per $100 face value. Similarly, Fund B subtracts the annual amortized premium amount from annual investment income, reducing the annual dividend distribution amount to $2 per $100 face value (ignoring expenses), and reports a distribution yield of 2% (ignoring expenses).

So at a price of 110, Fund A reports a distribution yield of 4/110 = 3.6%, and Fund B reports a distribution yield of 2/110 = 1.8% (all ignoring expenses). This is an example of Jim's original point that two funds with the exact same holdings could report different distribution yields depending on when they bought the bonds (i.e., at what prices they bought the bonds).

Investor C (aka Doc) buys 5-year a zero coupon bond today at a price of 90. This is referred to as a discount bond because it is priced below par value of 100. In the case of a zero, it is referred to as a pure discount bond, because the coupon rate is 0%, so all return is purely from the accretion of the discount. The discount of 10 accretes to interest income over 5 years at an annual rate of 10/5 = 2%. Doc pays taxes on annual interest income of $2 per $100 face value.

So back to the OP, I think of current yield as coupon rate divided by current bond price. So using this definition, a zero-coupon bond has a current yield of 0%. However, if a zero-coupon bond fund reported distribution yield, I assume it would report a non-zero value. I just checked and verified this: PIMCO 25+ Year Zero Coupon U.S. Treasury Index ETF reports a distribution yield of 3.42% (and incidentally SEC yield of 2.94%).

So as usual, a zero coupon bond provides a great way to clarify a bond concept. In this case it makes it very clear that current yield for an individual bond, at least the way I think about it, is not directly analogous to distribution yield for a bond fund. It is much more analagous however if we think of current yield as including the amortization/accretion of bond premium/discount, and not just the coupon payment.

Investopedia actually defines "adjusted/modified current yield" as including the adjustment for premium or discount: Advanced Bond Concepts: Yield and Bond Price | Investopedia. So I'd modify the OP to state that adjusted current yield for a bond is analogous to distribution yield for a bond fund (as reported by Vanguard).

Kevin

[Kevin M]

Oh, and back to the original point of the investigation. Even with a combined federal and state marginal income tax rate of 50%, which is much higher than my marginal rate, the after-tax yield of a direct 5-year CD earning 2.52% (best I know of currently) beats the tax-free yield to worst of 1.25% for the AA muni bonds, even without accounting for the credit risk and higher term risk of the bonds.
Kevin

I just checked Oct 31 distributions from my VCADX holdings and they are 3% annualized (i.e. I take the distributions on Oct 31 and multiply by 12). I've held the position over multiple years (I forgot how long but < 5yrs if memory serves right) and all dividends have been redistributed since I began the position. Since I'm a resident of CA, all distributions are tax free. Therefore I love your analysis but disagree with your conclusion above.

How can you disagree with the conclusion? It's just math. Re-read my statement carefully. The collection of muni bonds has a yield-to-worst of 1.25%, meaning that unless interest rates increase enough so that the bonds with high coupons aren't called, you are guaranteed to earn 1.25% if you hold the bonds to maturity and if there are no defaults. You are guaranteed to earn 1.26% on the 5-year CD at a combined marginal tax rate of 50%. Since 1.26% > 1.25%, the CD has a higher guaranteed after-tax return even at a 50% combined marginal rate.

Also remember that with the bonds there is reinvestment risk and default risk. With the CD there is none (other than the risk of the bank being shut down and your money being returned by the FDIC, so in a sense there is some call risk).

Now the collection of the CDs is not the same as the fund. The individual CDs will approach par value at maturity, so you definitely must account for the premium amortization in your return calculations. The fund does not have a maturity date, and we don't really know the details of what the bond managers will do in terms of actively managing the holdings based on their forecasts. However we do know that any bond premiums will be amortized and subtracted from investment income.

Therefore, other than generating alpha through active management and getting better prices after costs, the only way the fund can produce a better outcome than the collection of bonds is by taking more risk. I hold the fund because I'm willing to take the risk with a portion of my fixed income in return for somewhat higher after-tax yields, and the possibility of further capital gains due to yet lower future interest rates (the upside of term risk).

I too have owned VCADX for quite awhile (probably also about five years), but I don't see what that has to do with estimating returns going forward.

Having distribution yield much higher than SEC yield is a factor in not just dumping all of my muni bond funds now or before now, but I admit there is some risk in this. So far the risk has been mostly paying off, but at some point the tide will turn (at least I hope so, since higher rates would be good longer term), and we may suffer some significant capital losses.

Kevin

[Electron]

I actually have done quite a bit of investigation on this. Here are some links to posts on this as far as back as 3+ years.

Thanks for the links and comments. That is interesting seeing the short term variation in bond fund SEC Yield relative to NAV. One might guess that the variation is less with bond index funds than with actively managed bond funds. Over the 30 day SEC period an actively managed fund could buy and sell individual bonds and change the composition of the portfolio to some degree. In addition, bonds could occasionally mature and be replaced with other securities.

I'm still quite surprised by the data I presented on Vanguard High Yield Corporate VWEHX as shown below. That change in SEC yield is very large compared with the change in NAV.

05-15-14 - NAV $6.14 - SEC Yield 3.88%

10-17-14 - NAV $6.01 - SEC Yield 4.98%

However, I will note that High Yield funds and High Yield ETFs apparently had large net outflows in the August through October period. If VWEHX experienced net redemptions, they likely sold their most liquid and lower yielding bonds which could have increased the SEC yield to some degree.

[Kevin M]

Electron, maybe I will grab the data for VWEHX, do the analysis, and show some charts for SEC yield and Price, with and without averaging price over 30 days. It's on my list.

I own that fund too, but I haven't paid much attention to it lately. After the investigations I've done already, I'm just not surprised to see a screwy relationship between the price and SEC yield from time to time.

Kevin

[ogd]

05-15-14 - NAV $6.14 - SEC Yield 3.88%

10-17-14 - NAV $6.01 - SEC Yield 4.98%

However, I will note that High Yield funds and High Yield ETFs apparently had large net outflows in the August through October period. If VWEHX experienced net redemptions, they likely sold their most liquid and lower yielding bonds which could have increased the SEC yield to some degree.

Part of this is a result of the 30 day average. Mid-October the SEC yield started heading down, so the present 4.71% might be a better correspondent to the price in mid-October.

But what's left is still pretty large. Perhaps it's a change in the credit composition of the fund (which would be an active decision regardless of outflows), or perhaps it's a "riding the yield curve" effect.

[Electron]

Thanks Kevin and ogd for the comments on Vanguard High Yield. The charts show three waves of selling this year after price highs in late June. The lows were approximately 8/05, 9/29, and 10/15. It looks like the second two waves of selling occurred within 30 day time frames and that may partly explain the high SEC yields.

Here are two very interesting links on recent outflows. It looks as though High Yield funds held assets of $176.3B and had record outflows of $12.6B over four weeks.

http://www.forbes.com/sites/spleverage/ ... h-outflow/

http://www.businessinsider.com/high-yie ... ent-2014-8

It's possible that Vanguard picked up some bargains during the selloff but I don't know if that is the case. I read earlier this year that active funds may be able to trade against the big High Yield ETFs depending on their own cash flows. Index ETFs must buy or sell at market price when speculators move in or out. An active fund can take advantage of that at times and buy or sell individual securities at attractive prices.

[Electron]

Here is some data on Vanguard High Yield Corporate to see if SEC Yield was able to predict future returns. SEC Yield data is from Vanguard while Total Return figures are from Morningstar.

Vanguard High Yield Corporate - VWEHX

SEC Yield 1 Year Ago 4.44% - 1 Year Total Return 6.15%
SEC Yield 3 Years Ago 6.57% - 3 Year Total Return 8.32%
SEC Yield 5 Years Ago 7.43% - 5 Year Total Return 9.41%
SEC Yield 10 Years Ago 5.77% - 10 Year Total Return 6.63%
SEC Yield 15 Years Ago 9.23% - 15 Year Total Return 6.35%

The nature of High Yield bonds probably results in the return being less predictable than what might be seen for a Total Bond index.

On a side note, I just reviewed the latest Annual Report on Vanguard Money Market funds. It's not a very exciting read with current yields at 0.01%. One surprise is that Vanguard is absorbing considerable expenses to be able to pay out that low yield.

[Kevin M]

I'm still quite surprised by the data I presented on Vanguard High Yield Corporate VWEHX as shown below. That change in SEC yield is very large compared with the change in NAV.

05-15-14 - NAV $6.14 - SEC Yield 3.88%
10-17-14 - NAV $6.01 - SEC Yield 4.98%

I've done some investigation of this, and although not yet complete, will post what I've got so far in the next reply. Before presenting the data, here are some thoughts on the conceptual understanding of the relationship between price and yield.

First consider a single bond. Price is the present value of the discounted cash flows, and YTM is the single discount rate that makes the discounted cash flows equal to price. The cash flows consist of coupon payments and principal repayment at the end of the holding period. The discount function for each cash flow is a function of time to maturity and discount rate. So the only variables that affect the relationship between price and YTM (single discount rate) are time to maturity (part of the discount function) and coupon rate (determines the cash flows before maturity). (Coupon frequency also is a variable, but conceptually we can just assume annual coupon payments. Face value also is a variable in the equation, but that is a known constant.)

Notice that nowhere in the above paragraph was default risk (aka credit risk) mentioned. A change in perceived default risk of the bond results in changes to both price and YTM, but the mathematical relationship between price and YTM holds. Default risk is not an independent variable in the equation that relates price and YTM; it affects both, but it does not affect them independently. In addition to looking at the formula that relates price and yield (describe in previous paragraph), you can look at the PRICE and YIELD spreadsheet functions to see that there is no parameter that represents default/credit risk.

If we think of a fund as roughly a constant maturity, constant coupon bond, then the average price/yield relationship should hold over time for the fund regardless of changes in default/credit risk of the holdings. An increase/decrease in perceived default risk results in a increase/decrease in average YTM/price, but the relationship between YTM and price holds. So as with an individual bond, default risk is not an independent variable in the equation that relates price and YTM; it affects both, but it does not affect them independently. Therefore, a change in perceived default risk of fund holdings, in and of itself, has no effect on the price/YTM relationship.

For a constant maturity, constant coupon bond, active bond management strategies, such as any yield-curve management strategy, are irrelevant, since by definition the term to maturity does not change; i.e., the bond occupies the same x-axis position (term to maturity) on the yield curve over time.

However, for an actively managed fund especially, we cannot use the constant maturity, constant coupon analogy, since changes in the fund holdings over time result in changes to average maturity and average coupon. Nevertheless, the price/yield relationship applies to each bond in the fund, and therefore should apply to the average price/yield relationship for all fund holdings, as long as the averaging is done correctly.

So for any bond held in the fund at the beginning and end of the holding period, t1 and t2, we can simply plug in the YTM and term to maturity for t1 and t2, along with the values that do not change such as coupon rate and face value, to get prices p1 and p2. Bonds held at t1 but not at t2, or vice versa, obviously will have an impact on the average fund values for YTM and price at t1 and t2. Still, it seems that there should be a way to calculate prices for the fund at t1 and t2 using some sort of averages of YTM, coupon rate and term-to-maturity of fund holdings at t1 and t2. What does not seem to work quite right is to plug average YTM, average coupon, and average term to maturity into the PRICE formula, which I will show in my next post.

If anyone can suggest a calculation that does work, please do. I may yet come up with it, since I can load quarterly fund holdings into a spreadsheet, and do some analysis. Note that average YTM (not just SEC yield), average coupon, and average maturity, and holdings all are reported quarterly (in annual and semiannual reports, and in quarterly N-Q SEC filings). Note that SEC Yield <> Average YTM, since the former is an average over 30 days, and the latter is a weighted, effective average as of the reporting date (i.e., for a single day).

Kevin

[Electron]

If anyone can suggest a calculation that does work, please do. I may yet come up with it, since I can load quarterly fund holdings into a spreadsheet, and do some analysis. Note that average YTM (not just SEC yield), average coupon, and average maturity, and holdings all are reported quarterly (in annual and semiannual reports, and in quarterly N-Q SEC filings).

Thanks for your efforts on Vanguard High Yield Corporate. I hope the fund holds mostly conventional bonds as there are many types and I'm not sure what effect that might have. The fund could also invest some assets in derivatives. Here is a list of various types of High Yield bonds.

http://www.investinginbonds.com/learnmo ... =19&id=199

In my post above I showed SEC yield and subsequent Total Returns for VWEHX. Here is the same type of data for the Vanguard Total Bond Market Index fund. Data is from Vanguard and Morningstar. The favorable climate for bonds was undoubtedly a factor in the Total Return figures generally exceeding the SEC yields.

Vanguard Total Bond Market Index fund - VBMFX

SEC Yield 1 Year Ago 2.03% - 1 Year Total Return 4.31%
SEC Yield 3 Years Ago 2.28% - 3 Year Total Return 2.36%
SEC Yield 5 Years Ago 3.33% - 5 Year Total Return 4.06%
SEC Yield 10 Years Ago 3.97% - 10 Year Total Return 4.55%
SEC Yield 15 Years Ago 6.63% - 15 Year Total Return 5.31%

[Kevin M]

In my post above I showed SEC yield and subsequent Total Returns for VWEHX. Here is the same type of data for the Vanguard Total Bond Market Index fund.

The relationship between initial yield and holding-period return is a completely different topic than the relationship between yield and price. Return for any bond is a function of coupon rate, reinvestment rate, start price and end price. Return for a bond fund will be related to the averages of these values over the holding period. I'm going to stay on the price/yield topic for awhile, and see if we can make some progress on sorting it out.

Kevin

[Kevin M]

So here is some analysis of the relationship of price and yield for VWEHX (since Electron brought this fund up).

I start these analyses by pulling data for the longest period for which Vanguard shows distribution yields, currently back to 5/31/2013. I do this since I'm also interested in looking at the relationship between distribution yield, SEC yield and price over time. Here I'll just focus on SEC yield and price. All analysis here is for daily price and SEC yield from 5/31/2013 through 11/5/2014, which encompasses the period highlighted by Electron as having surprising price vs. SEC yield behavior.

First let's look at how well the duration rule of thumb works if we just look at daily SEC yield and daily price. The duration rule of thumb is:

dP/P = -D * dY (1)

where dP/P is the percent price change, D is modified duration, and dY is change in yield. I'm labeling this as equation 1. We can also write this as (equation 2):

dP/P + D * dY = 0 (2)

Therefore we can evaluate how well the duration rule of thumb works by calculating daily values of dP/P and dY, using the latest modified duration as an approximation of D, and calculating the value of equation 2. We then plot this value on the y-axis with date on the x-axis. To the extent the duration rule works as advertised, we might expect to see a flat line at y-intercept 0, perhaps with some slight variation due to changing duration. Here's what we actually see for the period examined:



So quite a bit of variance from a flat line at y=0.

Next, since SEC yield is YTM average over previous 30 days, let's average price over previous 30 days to see if it helps:



OK, that looks better (I kept the same y-axis scale so the improvement is obvious). Note that the first 30 days don't represent 30-day average prices since I didn't pull data for 30 days before 5/31/2013, but for the purposes of this investigation, that's irrelevant, since we're investigating a period since May 2014.

Note that even after averaging price over previous 30 days, we still see significant deviation from 0 over the time period highlighted by Electron of 5/15/2014 - 10/17/2014, especially toward the more recent end of that range. So let's drill down to that time range, and look at actual values of dP/P and -D * dY, which according to equation 1 should be equal:



So from this it's clear that -D * dY varies much more than dP/P in both directions over several periods between mid-July and mid-October. So consistent with Electron's observations, SEC yield varies much more relative to price (even after averaging over 30 days) than one would expect from the duration rule of thumb. Remember, if the rule held perfectly, the red and blues lines would be coincident (one would be directly on top of the other).

I'm going to pause here, and in the next post, dig into the numbers some more.

Kevin

[Electron]

Here is some analysis of the relationship of price and yield for VWEHX.

Thanks very much for this analysis which looks like a considerable amount of work. I'm starting to think we may never fully understand the large changes in SEC yield that were seen during that period.

I'll pass along a few comments that might be of interest. In the Annual Report, Vanguard defines 30 Day SEC Yield a little differently than what is seen on their website. In particular, the calculation uses YTM for bonds but actual income for asset backed securities. There could also be effects from rounding. Note that $0.01 represents a significant percentage with NAV in the area of $6.00. Rounding effects would also occur with the SEC yield.

The Morningstar site provides some interesting data on the portfolio held by VWEHX. They show Cash at 4.02%, Bonds at 91.26%, and Other at 4.72%. Only 79.64% of the fund is invested in the United States.

Here are the top sectors as shown by Morningstar.

Corporate Bonds - 84.07%
Bank Loans - 4.13%
Preferred Stocks - 1.11%
Convertibles - 0.46%
Commercial MBS - 0.33%

There may also be derivatives in the portfolio and bonds with call provisions. It's also not clear if the portfolio holds any bonds of other types such as Pay-in-Kind.

Lastly, the Morningstar Fund Analysis tab mentions that 15% of the fund is relatively liquid to handle difficult times in the High Yield Market. The SEC yield could rise quite a bit if lower yielding securities were sold leaving the less liquid and higher yielding securities in the portfolio. Those securities would now represent a higher percentage of total assets.

[Doc]

Price is the present value of the discounted cash flows, and YTM is the single discount rate that makes the discounted cash flows equal to price. The cash flows consist of coupon payments and principal repayment at the end of the holding period. The discount function for each cash flow is a function of time to maturity and discount rate.

Actually the present value is a function of cash flow and time from now not the time to maturity. If you use time to maturity you have the future value formulation and have to make the reinvestment assumptions which defeats the credit risk assumption. So everyone should use the present value formulation like they do in B-School and we can all agree on what Kevin meant to say. (Caveat: I never went to B-School.)

If anyone can suggest a calculation that does work, please do.

I think you need to use all the cash slows from all the bonds in the fund and apply the present value equation to the entire set. Using an average of the YTM of each bond may not work depending on the shape of the yield curve. (I'm thinking the second derivative effect.) Converting each bond to its zero equivalent might be a reasonable approximation. I did a test case several days ago in which I "reinvested" each bond at its original coupon. (The reinvestment assumption applied to each bond individually.) It made a significant difference. The model used the total cash flows from each bond and and I used Excel's IRR function to calculate the IRR or YTM in bond speak, but I only had six bonds in my "fund" so the calculation was less daunting than for a real fund. On the first question of how to weight the bonds in the averaging process the weighed average gave a YTM of 2.18%, the actual cash flow method had a YTM of 2.52% and the "rolling note" method gave a YTM of 2.84%. While these are significantly different I purposely created a "fund" with a very wide maturity and coupon range in its portfolio. FWIW I decided that the weighted average method was "good enough" for our 0-10 portfolio with ~ 10 funds &/or individual notes.

[Electron]

Here is a possible explanation for the variations seen in Kevin's charts.

The links I posted earlier mentioned a record $12.6B outflow over four weeks in the $176.3B High Yield market. That represents over 7% of assets coming out of High Yield funds. If Vanguard High Yield had outflows of several percent or more I'd expect the SEC Yield to rise since the fund holds 15% of assets in lower yielding reasonably liquid securities.

Investors continue to buy the dips so significant inflows would follow the outflows. In this case the portfolio would have been seeing noticeable day to day changes and the SEC yield would be moving up and down despite the 30 day averaging.

[Kevin M]

The links I posted earlier mentioned a record $12.6B outflow over four weeks in the $176.3B High Yield market. That represents over 7% of assets coming out of High Yield funds. If Vanguard High Yield had outflows of several percent or more I'd expect the SEC Yield to rise since the fund holds 15% of assets in lower yielding reasonably liquid securities.

But you'd also expect to see the fund price fall, and the fund price % change should be even larger than expected based on duration because price is not averaged over 30 days but SEC yield is. So I don't think this explains large moves in SEC yield relative to price change.

Investors continue to buy the dips so significant inflows would follow the outflows. In this case the portfolio would have been seeing noticeable day to day changes and the SEC yield would be moving up and down despite the 30 day averaging.

Yes, but again the price should change even more as explained above. The puzzle is not that we see relatively large, daily changes in price or yield, but that we see relatively large changes in one with respect to the other.

I am basically restating and clarifying what I wrote in the first 3 or 4 paragraphs of this reply up-thread, so you might want to re-read that and make sure you understand the reasoning.

To summarize, buying or selling bonds affects the overall fund price and YTM; it does not change the relationship between the two.

Now if comparing SEC yield, which is a 30-day average YTM, to price, then yes, for a large, single-day change in price/YTM, we would expect to see a larger change in daily price, since price is not averaged over 30 days but YTM is. However, after averaging price over the same 30-day period, we should expect the large differences to go away. Indeed, we see from the plot using average price over 30 days that the large, unexpected variances between price change and YTM change are significantly reduced.

So, averaging price over 30 days gets us part of the way there, but we still see some unexpected variance in expected the price/yield relationship that is yet to be explained. As I said, we can take this further, and I already have to some extent, and will follow up with more number crunching later.

Kevin

[Kevin M]

Price is the present value of the discounted cash flows, and YTM is the single discount rate that makes the discounted cash flows equal to price. The cash flows consist of coupon payments and principal repayment at the end of the holding period. The discount function for each cash flow is a function of time to maturity and discount rate.

Actually the present value is a function of cash flow and time from now not the time to maturity.

Yes, thanks for the correction; time from now until time of the cash flow. So the discount function using YTM is:

d(y,n) = 1/(1+y)^n

where y is YTM per compounding period and n is number of compounding time periods from now. So for semiannual compounding,

d(y,n) = 1/(1 + y/2)^2n

where y is YTM (annual) and n = number of years from now.

If anyone can suggest a calculation that does work, please do.

I think you need to use all the cash slows from all the bonds in the fund and apply the present value equation to the entire set.

I may get back to you on this when I get to the point of calculating values for the fund holdings. It has to be something that is feasible to do for several hundred holdings. If it's just a matter of entering a formula in one or a few cells for each bond (resulting in one or a few columns for all holdings), then it should be doable.

Perhaps we can sink our teeth into it when I present the details on doing some preliminary analysis on the fund holdings, since then we will have a more concrete picture of the available data and what we can do with it.

Thanks,

Kevin

[Doc]

I may get back to you on this when I get to the point of calculating values for the fund holdings. It has to be something that is feasible to do for several hundred holdings. If it's just a matter of entering a formula in one or a few cells for each bond (resulting in one or a few columns for all holdings), then it should be doable.

I think you can get "close enough" if you convert each bond to its equivalent zero and then use the zeros. It is still a lot of work and I'm not sure how big the benefit will be.

[Kevin M]

Now if comparing SEC yield, which is a 30-day average YTM, to price, then yes, for a large, single-day change in price/YTM, we would expect to see a larger change in daily price, since price is not averaged over 30 days but YTM is.

I realized that I had not yet shown a chart that shows the large daily change in price relative to daily change in SEC yield before averaging price over 30 days. Here is that chart for VWEHX:



Note that without averaging price, we see generally larger daily percent changes in price (red) relative to daily changes in SEC yield x duration (blue) than vice versa, as expected.

Here is the same chart after averaging price over 30 previous days, using the same vertical scale for direct comparison:



In contrast to the first chart, note that price and yield changes tend to track more as expected, but now the 30-day average percent price changes (red) are relatively small compared to the changes in SEC yield (30-day average YTM) x duration (blue).

So by averaging price over previous 30 days, the focus of the investigation switches from "unexpectedly" large price change relative to SEC-yield change to unexpectedly large SEC yield change relative to price change.

Kevin

[Kevin M]

So by averaging price over previous 30 days, the focus of the investigation switches from "unexpectedly" large price change relative to SEC-yield change to unexpectedly large SEC yield change relative to price change.

But it turns out this is not the case for the particular data shared by Electron:

Code: Select all

Date        Price    Pavg30  SEC Yld
---------
5/15/2014    6.14    6.125    3.88
10/17/2014   6.01    6.012    4.98
             -----   -----    ----
change      -2.12%  -1.84%    1.10
theoretical         -4.84%  

Note that in this case, averaging price actually resulted in a smaller percent change in price (from -2.12% to -1.84%), which is even further from the theoretical price change of -4.84% based on the duration rule of thumb (-4.4 x 1.1 = -4.84).

I don't know how to investigate further for those particular dates, since I don't know how to get any more details on fund holdings or characteristics (e.g., average coupon, average maturity) for those dates. To go further, I think we need to look at changes between quarterly reporting dates, since we can get fund holdings and characteristics for quarterly reporting dates. That's next.

Kevin

[Electron]

The puzzle is not that we see relatively large, daily changes in price or yield, but that we see relatively large changes in one with respect to the other.

Averaging price over 30 days gets us part of the way there, but we still see some unexpected variance in expected the price/yield relationship that is yet to be explained.

My point was that the duration of the fund may change on many individual days especially in times of volatility. When I mentioned that 15% of Vanguard High Yield is in relatively liquid issues, I was suggesting that significant outflows on a given day could result in the fund selling mostly cash and lower yielding shorter maturity investments to meet redemptions. That would result in the average duration of the portfolio increasing and the YTM would also change with the new portfolio composition separate from any effect from price changes.

That could explain why the charts shown earlier did not display as anything close to a flat line. There was a lot of volatility in the High Yield Market over several months and the percentage in shorter maturity and lower duration liquid assets may have changed daily. The portfolio manager would have been hard at work dealing with alternating outflows and inflows trying to manage the liquid and not so liquid portions of the portfolio.

The same type of chart for a bond index fund should look much better than for an actively managed fund. The duration should be much more stable for the bond index fund in the short term. Convexity could also come into play for any fund probably with minor effect in most cases.

[Kevin M]

Electron, I think you're still missing the point. By definition, YTM and price are related mathematically, so with the other variables fixed (e.g., coupon rate, term to maturity), you cannot have a change in one without a corresponding change in the other. Similarly, duration is directly related to YTM mathematically, and duration is not a variable in the formula that relates price and YTM. So duration is not an independent variable in the price/yield relationship, but a dependent variable related to yield. Just look at the spreadsheet formulas for YIELD, PRICE and DURATION to confirm all of this for yourself.

So your statement about YTM changing "separate from any effect from price change" is a non sequitur. There is no such thing as YTM change without price change without a change in one of the other variables in the relationship.

It doesn't matter what the fund sold unless it changes one of the other variables, like average coupon rate or term to maturity, unless they are reporting YTM based on a subset of fund assets. Now I guess this could be the case for the average effective YTM they report, but I doubt it.

With respect to SEC yield, the requirement for the calculation is to include "dividends and interest earned during the period" (source: www.sec.gov/about/forms/formn-1a.pdf. Note that it does not say "dividends and interest only on bonds". So it seems that the SEC yield calculation includes dividends and interest from all holdings. If you read the source you'll see the details of how YTM of each obligation is included in "dividends and interest".

With regard to price, all assets are included in the calculation of Net Asset Value (NAV), which is what fund price is on a per share basis. So by definition, price does not exclude any assets such as cash.

Therefore, it does not seem possible that any change in fund holdings could affect YTM or SEC yield without also affecting price, without a change in one of the other variables in the price/yield relationship. It makes no difference whether the assets bought/sold are liquid or illiquid. They still have a price and YTM that are directly, mathematically related.

One thing that does affect the price/YTM relationship is coupon rate. At a fixed YTM a lower coupon rate results in a lower price. So it could be that fund composition changes in a way that changes the average coupon rate in a way that causes YTM and price to change in a way that we wouldn't predict based on the duration rule of thumb.

Of course average term to maturity also affects both YTM and price, so a change in fund composition that resulted in a different average term to maturity also could result in changes that deviate from the duration rule of thumb.

Whatever affects YTM also affects duration, so these are inextricably linked.

Again, if you just look at the PRICE and YIELD spreadsheet functions, you see the variables that affect them both, and so the answer to any apparent discrepancy between price and YTM must be explained by changes in these variables. If you hold all variables fixed except price, yield change is directly, inversely related to price change. If you hold all variables fixed except yield, price change is directly, inversely related to yield change. So one of the other variables must be changing for the price/yield to change in a way other than predicted by the duration formula (accounting for convexity, which I don't think is a big factor here, but perhaps it is).

So it seems to me that the answer to the apparent puzzle must be in a change in something like average coupon rate or average term to maturity, which is the direction I'm headed in the investigation.

Kevin

[Electron]

Therefore, it does not seem possible that any change in fund holdings could affect YTM or SEC yield without also affecting price, without a change in one of the other variables in the price/yield relationship.

Here is my thinking on the subject.

Consider a bond fund that has 50% of assets in High Yield bonds and 50% of assets in money market instruments earning a very low yield. On a day when all bond prices are unchanged, investors redeem half of the shares in the fund and the portfolio manager decides to sell all the money market securities. The fund NAV remains unchanged, the dollar value of my shares remains the same, but the YTM of the fund now increases substantially. The duration of the fund also increases substantially. The daily dividends that I accrue will now essentially double.

I don't know about the large redemption in the fund. Bonds drop in price the next day and the drop in fund NAV seems unusually large since I track a fund that until now had a similar asset allocation.

The duration rule of thumb chart can only show a flat line if we now incorporate the increased duration on new data.

[Kevin M]

Consider a bond fund that has 50% of assets in High Yield bonds and 50% of assets in money market instruments earning a very low yield. On a day when all bond prices are unchanged, investors redeem half of the shares in the fund and the portfolio manager decides to sell all the money market securities. The fund NAV remains unchanged, the dollar value of my shares remains the same, but the YTM of the fund now increases substantially.

OK, I see what you're saying. So the net assets of the fund drop by x%, but the number of shares also drops by x%, so the NAV per share does not change (given the assumption of no change in bond prices), but the YTM increases because lower-yielding assets were sold. That's a good one, and should be something we can dig into when looking at changes between quarterly reporting periods, since number of shares outstanding is reported as well as all the other numbers of interest.

If you look at an annual or semiannual report, you see "capital share transactions" dollar amounts in the "Statement of Changes in Net Assets". "Financial Highlights" which shows NAV at beginning of period and end of period "for a share outstanding throughout each period", and these values are only affected by Investment Operations and Distributions (not capital share transactions). So we should be able to factor in the change in number of shares into the analysis.

When I look at quarterly reported holdings, I calculate yield and price based on face value and market value of individual holdings. So if I do the averaging correctly, we should see the price/yield relationship hold, since value per share doesn't come into it. If I get this far, we an see if that's the case, and if so, how much effect change in number of shares has on NAV per share vs. average YTM.

The duration of the fund also increases substantially. The daily dividends that I accrue will now essentially double.

We don't even have to consider duration for the dividend part. But this is a somewhat different topic, because dividends aren't based on YTM, but on net investment income. You would see this in higher distribution yield. Dividends are from net investment income, which is coupon payments (and any other interest payments) plus amortized/accreted premiums/discounts. If lower-interest securities are sold, coupon and interest payments + amortized/accreted premiums/discounts will be higher per share.

I don't know about the large redemption in the fund. Bonds drop in price the next day and the drop in fund NAV seems unusually large since I track a fund that until now had a similar asset allocation.

OK. This is why we need to look at quarterly reports to see what is really going on. Quarterly reports will show us higher YTM, higher duration, change in number of shares, etc.

The duration rule of thumb chart can only show a flat line if we now incorporate the increased duration on new data.

So let's look at quarterly changes and see if we can explain it all!

Kevin

[Electron]

So let's look at quarterly changes and see if we can explain it all!

Quarterly changes may not yield much useful information. The links I posted earlier mentioned record outflows in the High Yield market. We saw sharp declines in NAV and then rebounds as investors bought the dips. A typical High Yield fund probably had net redemptions on some days and net purchases on other days. We would need information on the daily changes in the portfolio. The cash and lower yielding liquid percentage of the portfolio could have changed frequently meaning that the YTM and duration of the fund also changed frequently. Vanguard High Yield intentionally has about 15% of the fund in cash and liquid instruments to handle difficult periods. That information is available on Morningstar for premium subscribers which I get at no cost through T. Rowe Price.

If my theory is correct, we could get much better results on your duration rule of thumb chart for a bond index fund such as VBMFX. In this case, the portfolio should continue to closely track the index on a daily basis. I'll plan to download that data in the next day or two and take a look.

[Kevin M]

So let's look at quarterly changes and see if we can explain it all!

Quarterly changes may not yield much useful information.

Not for daily changes between quarters, but for changes from one quarterly reporting date to another we should hopefully be able to figure out if the price/yield relationship seems to hold, and if not, why not.

For example, for holding period starting 7/31/2014 and ending 9/30/2014 (dates for which we can currently see average YTM, average coupon, average maturity, etc., the price change was -1.16%, the average 30-day price change was -1.07%. The change in SEC yield was +49 basis points, and change in average YTM was +40 basis points. Using SEC yield change and duration of 4.4 gives theoretical rule-of-thumb price change of -2.16%, and using average YTM change gives -1.76%. Average duration was 4.3 and 4.4 respectively for these two dates, and using the average of the two in the rule of thumb only lower the theoretical price change % by 2 basis points to -1.74%.

So even using average effective YTM (instead of the 30-day average SEC yield), there is a discrepancy of 60 basis points between theoretical using duration rule of thumb and actual price change. We should be able to explain that by looking at changes in coupon rates, maturities, number of shares etc. If we can, then that should give us a clue as to how much each of these changes accounts for the apparent discrepancy using the duration rule of thumb.

For example, average coupon did not change between the two dates, but average time to maturity increased from 5.00 to 5.30. When I do a price calculation factoring in the change in average TTM, I get a theoretical price change of -1.56%, so this gets us 20 basis points closer to actual, with only 40 basis points left to explain.

If we can find two successive quarters where the apparent price/yield change discrepancy is relatively large, that would be even better.

Anyway, the input on the effect of changes in number of shares, combined with the yields on assets bought and sold, on the per share NAV is a good addition to the variables to consider, so thanks for bringing that to the table.

Kevin

[Electron]

Kevin - Thanks for the quarterly analysis and comments. I hope you are able to learn more with another set of successive quarters. Also glad I was able to contribute something to the discussion.

The hypothetical portfolio example I presented where YTM and duration increased didn't actually need a fund redemption and change in the number of outstanding shares. The portfolio manager could have simply replaced some or all of the money market securities with high yield bonds. He could also have sold some lower duration bonds and replaced them with longer duration bonds. If a lot of cash came into the fund from new purchases on a given day, the YTM and duration would decrease unless the new money was invested in bonds before the close of the market.

This afternoon I was able to exactly replicate the duration rule of thumb chart that you presented in this thread. I then created a similar chart for the Vanguard Total Bond Market Index fund hoping to see better results since that fund tracks an index with no active management. The positive and negative spikes do seem somewhat more uniform with smaller maximum excursions. Perhaps that is a good result since the duration is higher in the Bond Index fund. I think the results might be even better if NAV and SEC Yield were provided with another significant digit. There are many days where NAV or SEC Yield are unchanged but that would not be the case if another significant digit were provided.

[Kevin M]

The hypothetical portfolio example I presented where YTM and duration increased didn't actually need a fund redemption and change in the number of outstanding shares. The portfolio manager could have simply replaced some or all of the money market securities with high yield bonds. He could also have sold some lower duration bonds and replaced them with longer duration bonds. If a lot of cash came into the fund from new purchases on a given day, the YTM and duration would decrease unless the new money was invested in bonds before the close of the market.

Yes indeed. In re-reading our posts, I see that we're actually in complete agreement, except that I neglected to note that your scenarios all involve a one-day (or very quick) change in portfolio composition, which of course can have a one-day (or very quick) effect on average yield to maturity but not on overall fund value (and therefore not on share price). To put it another way, the price/yield relationship holds for a given set of securities, but if one set of securities is swapped for another, the overall portfolio yield can change, but of course for a swap there is no change in overall portfolio value on that day. Of course the YTM/duration of the new portfolio will be reflected in subsequent price changes.

This is why we can only dig into these things in more detail when we have the quarterly numbers.

Also let's remember that the effect on change in SEC yield will be muted by the 30-day averaging. So with 29 days of no changes followed by a one-day shift from all bonds into into all cash (or vice versa), for example, there still will be 29 days of all bond (or all cash) YTM included in the 30-day yield.

Again, we can see the actual average YTM (not averaged over 30 days) in the quarterly reports, so should be able to see fairly clearly the changes in portfolio characteristics that account for the quarterly yield/price changes.

This afternoon I was able to exactly replicate the duration rule of thumb chart that you presented in this thread. I then created a similar chart for the Vanguard Total Bond Market Index fund hoping to see better results since that fund tracks an index with no active management. The positive and negative spikes do seem somewhat more uniform with smaller maximum excursions. Perhaps that is a good result since the duration is higher in the Bond Index fund. I think the results might be even better if NAV and SEC Yield were provided with another significant digit. There are many days where NAV or SEC Yield are unchanged but that would not be the case if another significant digit were provided.

All good points. A bond index fund should not be making big shifts in distributions of yields or maturities within its target ranges.

By contrast, an actively managed fund is quite likely to do so. In another thread I showed the change in distribution of holdings by term to maturity for the Vanguard Intermediate-Term Treasury fund (actively managed), and they were significant. Although the fund holdings remain in the 3-10 year range, they are quite concentrated in certain subsets of that range, and the concentration changes over time as the fund managers' views change on yield curve prognostication, etc. Of course sometimes that helps and sometimes it hurts, as they acknowledge in their annual and semiannual reports.

Incidentally, the Treasury funds are good candidates for analysis of quarterly numbers, since they have fewer than 100 holdings, and it's easy to copy/paste the holdings from the quarterly reports into a spreadsheet. It's much messier with VWEHX, not so much because of the number of holdings but because of the formatting, which requires some manual editing to get into a form that can be analyzed.

Kevin

[Electron]

Kevin - Thanks for the additional comments and information. My interest in this thread evolved basically into three related items.

First, I wanted to understand why the duration rule of thumb chart did not display a straight line. Secondly, I wanted to mention that Vanguard High Yield maintains a cash position and liquid bond position to deal with significant redemptions in the fund. Lastly, the duration and average maturity of an actively managed bond fund can change daily. This is especially true for bond funds that invest in less liquid securities and maintain a liquidity buffer.

I don't know what Vanguard High Yield might see on a day of large redemptions, but suppose it could be 1% of assets. If the redemption was handled using the cash buffer, I believe duration and average maturity would both increase about 1%. That could create a 1% spike in the duration rule of thumb chart! NAV would see larger daily changes with the higher actual duration, but our formula uses a fixed duration of 4.4 years.

It also appears that having only two significant digits for NAV and SEC Yield prevents the duration rule of thumb chart from having a lot of value for a mutual fund. The rounding becomes an issue. Consider a case where actual NAV is $6.035 and the reported NAV can be $6.03 or $6.04. That one cent difference translates to a 0.17% spike on the rule of thumb chart.

I checked the quarterly holdings you mentioned on the Vanguard site. It's quite interesting looking through the various issues. I also noticed quite a few bonds with a 0% coupon.

[Kevin M]

I don't know what Vanguard High Yield might see on a day of large redemptions, but suppose it could be 1% of assets. If the redemption was handled using the cash buffer, I believe duration and average maturity would both increase about 1%. That could create a 1% spike in the duration rule of thumb chart! NAV would see larger daily changes with the higher actual duration, but our formula uses a fixed duration of 4.4 years.

I did some analysis of this using my 12-bond portfolio.

I added a cash holding with a price of 100 (percent of par, to put it in same terms as bond prices), which comes out to a weight of about 7% of holdings (a little less than average bond weight). I assume the cash is earning 0.1% with a term of 30 days, so YTM for cash is 0.10% and duration is 0.08 years.

I then calculated all the numbers (e.g., average weighted YTM, duration, price, etc.) with the cash, then subtracted the values without cash from the values with cash (i.e., as if all cash had been used to redeem shares). For the muni bonds, this resulted in these changes:

Code: Select all

                            YTM     Duration  Price
                            -----   --------  -----
Weighted Average with cash  1.46%   4.22      109.46
Weighted Average no cash    1.57%   4.54      110.18
no cash - with cash         0.10%   0.31        0.72
no-cash  / with-cash - 1    7.07%   7.44%       0.66%

So it depends what you mean by "an increase of about 1%". In the analysis above, a cash redemption of about 7% of portfolio value results in a YTM increase of only 10 basis points (0.10 percentage points). It's the percentage point change that we're looking at in the duration rule of thumb.

If you're speaking in terms of relative percent change, then you're in the ballpark, since we see the relative % increase in YTM is 1.57/1.46 - 1 = 7.07%, but that's not what we're looking at in the charts.

Of course there also was a decrease in total fund value of about 7%, but I guess if that all went to redeeming shares we wouldn't see a per share price change on that day. For example, if we think of bond par price of 100 as representing $100 of value at $1/share, then with the cash portfolio value is $1,418 for 1,418 shares, and without cash value is $1,317 for 1,317 shares, since we redeemed 100 shares to distribute $100 to shareholders.

So I don't think the average price change on the day of the liquidation shown above has any meaning.

Since duration increased by 0.31 years to 4.22 to 4.54 years, we would see price change on subsequent days (assuming no more portfolio changes) be correspondingly larger by about an absolute 31 basis points (0.31 percentage points), and a relative 7.5% (4.54/4.22 - 1).

Incidentally, I also modified the values in my min-fund to be more representative of a high-yield fund (higher YTM and coupon), and the only difference is that YTM increased by about 30 basis points instead of 10 basis points (relative increase of about 7.4% vs. 7.1% for muni) . Increase in duration was about the same.

See any flaws?

Kevin

[Electron]

I did some analysis of this using my 12-bond portfolio.

That analysis looks fine to me. The 1% spike I mentioned was incorrect. I was attempting to show how much an actual duration change in a bond fund might display on the duration rule of thumb chart. The actual effect appears to be quite small. The assumption is that fund price changes relative to rate changes would be slightly larger if actual duration increased.

It is possible to do some interesting things with the duration rule of thumb spreadsheet and chart. I temporarily made NAV constant for 20 dates and SEC Yield constant for 20 dates. Now the duration rule of thumb chart displays a nice flat line for those 20 dates.

To see the possible error from the rounding in NAV, I now alternated values up and down by one cent. The NAV in my case was $6.02, $6.03, $6.02, $6.03 and so on. Assume that true NAV is $6.025 and that with tiny day to day changes the NAV would round one way or the other. The duration rule of thumb chart now displays spikes of 0.17%. That value would differ depending on NAV but I believe it represents a limitation in the chart for mutual funds where NAV is rounded to the nearest cent.

Using this same technique in the spreadsheet, you can also change other values or combinations to see the effect.