Bond pricing theory

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Andrew321
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Bond pricing theory

Post by Andrew321 » Mon Feb 11, 2019 8:12 pm

Hello Bogleheads,
I am currently working through William Bernstein's The Four Pillars of Investing and am confused by his discussion about bond pricing.

He writes about a hypothetical annuity that pays $100 per year. Here are my questions:

1. What is the "current (market) interest rate" that determines the divisor to calculate bond value? Who or what determines it? Is it the Fed?
2. Although I understand that the value of a bond and the interest rate are inversely related, I'm not sure why. Why would someone pay less for a bond that gives $100 annually when interest rates are higher?
3. What's the difference between interest and yield? I have the terms conflated in my head, and I think that might be the cause of some of my confusion.
4. Why is the interest rate "virtually synonymous with inflation risk"? I think it has something to do with purchasing power, but that's a result of poorly developed intuition.
5. If anyone has a reading list to help me with bond theory, feel free to put the title in your response.

Thank you!

Andrew

bluquark
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Re: Bond pricing theory

Post by bluquark » Mon Feb 11, 2019 8:48 pm

I’m not an expert and might have a few points corrected by other Bogleheads, but here’s my understanding.
1. What is the "current (market) interest rate" that determines the divisor to calculate bond value? Who or what determines it? Is it the Fed?
The Fed sets the interbank interest rate which is a floor, as it’s a risk-free rate. When an institution issues bonds, it has a set amount to raise in mind and will accept a slightly increased rate to attract enough investors. And vice versa for bond purchasers. Collectively, this sets rates for different categories of bonds.
2. Although I understand that the value of a bond and the interest rate are inversely related, I'm not sure why. Why would someone pay less for a bond that gives $100 annually when interest rates are higher?
Say I have a bond fund yielding 1.1k$ annually. I can use 11 of your old bonds or I can use 10 of the new bonds that yield 110$ annually, to constitute my bond fund. New investors into my bond fund will pay the same amount to get in, regardless of whether it’s made out of the old bonds, the new bonds, or a mix.
3. What's the difference between interest and yield? I have the terms conflated in my head, and I think that might be the cause of some of my confusion.
Interest, yield, and dividend are basically synonyms. For historical reasons, different words developed for different asset classes.
4. Why is the interest rate "virtually synonymous with inflation risk"? I think it has something to do with purchasing power, but that's a result of poorly developed intuition.
The Fed can and will raise rates to control high inflation, or lower rates to control deflation. Moreover, long bonds lose value when inflation rises or when rates rise.

Thesaints
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Re: Bond pricing theory

Post by Thesaints » Mon Feb 11, 2019 9:11 pm

Andrew321 wrote:
Mon Feb 11, 2019 8:12 pm
1. What is the "current (market) interest rate" that determines the divisor to calculate bond value? Who or what determines it? Is it the Fed?
There is no such thing. Each bond has its own interest rate; families of bonds tend to have very similar, if not the same interest rate. It largely depends on who issued them and their maturity date. It is set by the ask/bid market activity. Only Savings Bonds, which are not marketable, have an interest rate set by the Treasury.
3. What's the difference between interest and yield? I have the terms conflated in my head, and I think that might be the cause of some of my confusion.
Yield is how much the bond will pay. It is not necessarily paid all under the form of periodical interest payments. For instance, Zero Coupon bonds do not pay any interest at all, yet they have their own yield.
You buy a $100 ZC for $90 and at maturity you get $100. This bond makes no interest payments but its yield would be about 11% (if it matures in 1 year), or about 1.1% (if it matures in 10 years).
Viceversa, a $100 bond that pays a $1 interest coupon every year has a 1% yield regardless of its maturity (if you purchased it for $100).
4. Why is the interest rate "virtually synonymous with inflation risk"? I think it has something to do with purchasing power, but that's a result of poorly developed intuition.
Not virtually synonymous. Interest rates are an ingredient in the formation of inflation. Higher rates tend to favor higher inflation, but it is not a simple, nor very tight relationship.

Ben Mathew
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Re: Bond pricing theory

Post by Ben Mathew » Mon Feb 11, 2019 9:36 pm

Andrew321 wrote:
Mon Feb 11, 2019 8:12 pm
1. What is the "current (market) interest rate" that determines the divisor to calculate bond value? Who or what determines it? Is it the Fed?
The Interest rate is the price for future $ that is determined by the supply and demand of all market participants, just like the price for bananas is determined by the supply and demand of all banana producers and consumers. So you, me, and the Fed--we all set the interest rate. The Fed is just one particularly large market participant who affects interest rates much more than you and me.
2. Although I understand that the value of a bond and the interest rate are inversely related, I'm not sure why. Why would someone pay less for a bond that gives $100 annually when interest rates are higher?
When interest rates are higher, it takes less $ now to compound out to $100 later. So $100 later is worth less $ now.
3. What's the difference between interest and yield? I have the terms conflated in my head, and I think that might be the cause of some of my confusion.
It's the same.
4. Why is the interest rate "virtually synonymous with inflation risk"? I think it has something to do with purchasing power, but that's a result of poorly developed intuition.
The interest rate is not "virtually synonymous with inflation risk," but is very much affected by it.

Suppose you and I want to strike a deal: I give you one apple now and you give me 1.2 apples next year. If the price of apples today is $1, and there is no inflation, the deal we would strike in $ is:

"I will give you $1 now and you will give me 1.2 apples * $1/apple = $1.20 next year."

So the nominal interest rate for my loan to you is 20%.

But if inflation is 10%, then the price of apples next year will be $1.10. So the deal we would strike will be:

"I will give you $1 now and you will give me 1.2 apples * $1.10/ apple = $1.32 next year."

Now the nominal interest rate is 32%.

So higher inflation requires that you pay me back more $ next year, i.e. a higher nominal interest rate, because $ next year is worth less in terms of real goods.

Thesaints
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Re: Bond pricing theory

Post by Thesaints » Mon Feb 11, 2019 9:38 pm

Unfortunately, only past inflation is known... :)

Ben Mathew
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Re: Bond pricing theory

Post by Ben Mathew » Mon Feb 11, 2019 9:54 pm

Thesaints wrote:
Mon Feb 11, 2019 9:38 pm
Unfortunately, only past inflation is known... :)
Yes, I should have said "expected inflation"

venkman
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Re: Bond pricing theory

Post by venkman » Mon Feb 11, 2019 10:17 pm

Andrew321 wrote:
Mon Feb 11, 2019 8:12 pm
3. What's the difference between interest and yield? I have the terms conflated in my head, and I think that might be the cause of some of my confusion.
The interest paid out on a bond is also known as the coupon. For most bonds, the coupon rate is fixed as a percentage of the bond's face value when the bond is issued, and never changes. A $1,000 bond with a 3% coupon will pay out $30/year to the bond owner for the life of the bond.

Yield can have several different meanings. Current yield refers to the coupon rate divided by the current value of the bond. If you bought the 3% bond above for $980, the current yield would be 3.06%. Yield to maturity includes the total cash flow one receives from the bond. For the previously mentioned bond, your total return would include all the interest you receive, plus the $20 gain you get when the bond matures and you get back the full $1000 face value. If the bond had 5 years till maturity, you would get back a total of $1150 (face value plus 5 years of interest), for a total return of $170 on your $980 investment. That works out to a yield to maturity of 3.25%.

dbr
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Re: Bond pricing theory

Post by dbr » Tue Feb 12, 2019 9:57 am

A technical detail about some terms is that some are quantities in dollars, such as the interest paid by a bond, and some are ratios or rates such as the interest rate which is the ratio of payment to face value of the bond or yield which is the ratio of payment to the current value of the bond (with some additional definitions which can be complex).

pward
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Re: Bond pricing theory

Post by pward » Tue Feb 12, 2019 11:16 am

Andrew321 wrote:
Mon Feb 11, 2019 8:12 pm
1. What is the "current (market) interest rate" that determines the divisor to calculate bond value? Who or what determines it? Is it the Fed?
Bonds operate in a free market just like stocks do. So, just like stocks, yields and pricing are determined by supply and demand. The fed can influence this through changing the overnight rate, buying treasuries, and/or selling treasuries on the open market. But the fed can't fully control the rates of a free market, which is why their target interest rates come as a range, not an exact number. When it comes to bonds other than treasuries, they are going to have more risk, and the open market will decide how much premium is required to make it worth the risk.
Andrew321 wrote:
Mon Feb 11, 2019 8:12 pm
2. Although I understand that the value of a bond and the interest rate are inversely related, I'm not sure why. Why would someone pay less for a bond that gives $100 annually when interest rates are higher?
Generally speaking in a perfect world it evens out. If someone is going to take on less yield, they need to profit an equal amount in the price difference to keep things equal. Although, that's an overly simplistic view. Just like above, these things are done in the open market, so it's a bit more nuanced. It's not just current interest rate, but the speculated future interest rates that effect this. Also bond maturity comes into play, the longer a bonds maturity, the greater it reacts to an interest rate change because the longer the change would presumably benefit or penalize a bond. If interest rates go down by 1% it won't effect an existing 1 year treasury bill that much at all. But an existing 30 year treasury could appreciate 30% or more off of that small change.
Andrew321 wrote:
Mon Feb 11, 2019 8:12 pm
3. What's the difference between interest and yield? I have the terms conflated in my head, and I think that might be the cause of some of my confusion.
Generally they are used interchangeably. Though to be precise, the bond holder collects a yield, and a bond ower pays interest.
Andrew321 wrote:
Mon Feb 11, 2019 8:12 pm
4. Why is the interest rate "virtually synonymous with inflation risk"? I think it has something to do with purchasing power, but that's a result of poorly developed intuition.
This is because the Fed uses the interest rates primarily to control inflation/deflation so that they hit their target range. Generally, over a super long term timeframe, federal interest rates tend to average out to about the same as the reported inflation. Also, going back to the free market aspect, the higher the interest rate, the greater the perceived risk of inflation is in the market. If 10 year bond was paying 3% interest, would you want to buy it if you believed that inflation was going to increase to 5% over that timespan? No, you would demand higher rates, as would other investors, and that would effectively drive interest rates up because purchasers would stop buying until the rates rose to a level perceived to be fair.

Valuethinker
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Re: Bond pricing theory

Post by Valuethinker » Tue Feb 12, 2019 11:36 am

Andrew321 wrote:
Mon Feb 11, 2019 8:12 pm
Hello Bogleheads,
I am currently working through William Bernstein's The Four Pillars of Investing and am confused by his discussion about bond pricing.

He writes about a hypothetical annuity that pays $100 per year. Here are my questions:

1. What is the "current (market) interest rate" that determines the divisor to calculate bond value? Who or what determines it? Is it the Fed?
Very indirectly. The Fed sets the Discount Rate which is the rate at which banks can borrow from the Fed. That has an influence on all other interest rates because it is used as a basis when different market participants (dealers in bonds, basically) charge interest to each other -- the Fed Funds rate.

https://fred.stlouisfed.org/series/FEDFUNDS
https://www.bankrate.com/glossary/f/federal-funds-rate/


So when the Fed raises the Discount Rate it is seen as signal that the Fed is tightening, and interest rates rise for all maturities and for between private borrowers and lenders - the market makes this happen by driving down bond prices, thus raising their Yield To Maturity. The Fed may also engage in Open Market Operations (buying and selling of securities) to move market interest rates, if they ignore the Fed tightening signal.
2. Although I understand that the value of a bond and the interest rate are inversely related, I'm not sure why. Why would someone pay less for a bond that gives $100 annually when interest rates are higher?
Opportunity cost - what else can I do with the money? If interest rates rise you now have better alternatives to give you that $100 of interest. The price of the security has to adjust (fall) to reflect that.

If I lend you $100k at 5% on a 30 year fixed rate mortgage, then if interest rates go to 6% I am going to be a loser as a lender - I could have lent that money to Joe over there at 6%, instead it is stuck being lent to you. Thus if I try to sell that loan to someone else, they are not going to pay $100k for it. Although of course if interest rates *fall* that fixed payment of $5k p.a. is going to look attractive to someone who can only get 3% lending it to another householder.

That interest rate risk (volatility) is one of the reasons why, using the Federal Agencies (Fannie Mae/ Freddie Mac/ GNMA + some others) mortgage lenders often "securitize" their mortgages, selling them off to the Agencies who repackage them as bonds (Mortgage Backed Securities) and sell them on to investors. The dreaded CDOs of "The Big Short" fame and the Financial Crisis were even more imaginative repackagings (into "tranches") of US residential mortgages' bonds - a financial weaponization of a long established type of investment instrument, to cause financial mass destruction (it seemed like a good idea at the time ;-)).
3. What's the difference between interest and yield? I have the terms conflated in my head, and I think that might be the cause of some of my confusion.{/quote]
I lend you $100 for 1 year and you pay me back $103. That $3 is the interest.

I buy a bond that has a coupon per $100 of face value of $2.50. That is the interest it pays.

However I buy the bond at say $95. So when the bond matures I get a $5 capital gain as well as the $2.50 pa coupon. Thus the YIeld To Maturity takes into account both the $2.50 pa but also the $5 capital gain. Since it's a gain I know the yield is above 2.5% (the coupon) -- for a 10 year bond (guessing) it would be around 2.75%.
4. Why is the interest rate "virtually synonymous with inflation risk"? I think it has something to do with purchasing power, but that's a result of poorly developed intuition.
If I buy a bond yielding 3.0% with a 10 year maturity:

- part of my return is a forecast of inflation for 10 years (say 2%)
- another part of my return is my expected real return (about 1%)
- there's some bit for other risks including risk of default (credit risk), liquidity risk (can't sell the bond in the secondary market before maturity), currency risk (if the investor is not in the same home currency as the security pays out) - for a US Treasury bond this should be very small - almost default risk free (we hope!), highly liquid, most investors (even non USA) would want to be paid in USD

(the biggest single risk I have ignored is risk of interest rates falling and having to reinvest the coupons at a lower yield to maturity. The YTM calculation assumes you can just reinvest at the same rate as when you bought the bond).

Now it's perfectly possible I am wrong, and in 10 years it turns out my inflation was 3%, and I made a zero real return.

"interest rate virtually synonymous with inflation risk" is assuming that the long run real returns on US Treasury securities are 0%, I think. That's true for Money Market instruments (less than 1 year to maturity). For US Treasury bonds, I believe the long term real returns are higher, around 1%. Bernstein seems to be ignoring that.

However it's also true that if inflation expectations move up, the Central Bank tends to increase interest rates - bond prices fall. So in that sense, interest rate risk (the risk of fluctuating interest rates, changing the price of bonds) is inflation risk.

5. If anyone has a reading list to help me with bond theory, feel free to put the title in your response.

Thank you!

Andrew

Larry Swedroe has a book on bonds.

Annette Thau (? on spelling?) has a bond book which is frequently quoted here.

It's wise not to try to get too clever on bonds. Higher yield and higher risk in bonds are pretty much perfectly correlated - there really are no free lunches. High Yield bonds (junk, or sub investment grade) have higher yields but also higher risk. Ditto Leveraged Loan funds (which tend to have floating interest rates). Emerging Market bonds, etc.

Thus, I would suggest that you either hold a US Treasury Intermediate Term bond fund or Vanguard Total Bond Market. There's all kinds of reasons why the latter is not "optimal" (Larry Swedroe goes through them) but in the long run it has served investors well - unpleasant fluctuations have not been too unpleasant and performance has been good. The Expense Ratios on bonds *really* count in fund returns. A portfolio of US Treasury bonds will yield around 2.7%, so a 0.5% charge (50 basis points in the jargon) will eat 1/5th of your returns.

The other alternative is TIPS. TIPS protect you against unexpected inflation. They are thus the safest security out there.

In my example above, the "breakeven inflation rate" (derived from comparing the 10 year US Treasury (straight bond) with the 10 year TIPS yield) is 3% - 1% = 2%. If inflation is above 2% over that period, the TIPS bond will outperform the straight Treasury. If below, the Treasury bond will outperform.

For a lot of reasons, the optimal fixed income portfolio is up to 50% TIPS bonds. However at current TIPS yields that can be ignored (US TIPS are paying about 1% real yield).

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patrick013
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Re: Bond pricing theory

Post by patrick013 » Tue Feb 12, 2019 1:58 pm

High demand high prices, low demand lower prices. Today spreads are lower than average and the observation would be high demand then and sales volume increasing.

Treasury term premium estimate

Andrew321 wrote:
Mon Feb 11, 2019 8:12 pm

If anyone has a reading list to help me with bond theory, feel free to put the title in your response.
...enabling readers to gain an inside view of the key factors affecting US interest rates...

Fed Watching: Making Sense of Market Information : Brian Kettell
age in bonds, buy-and-hold, 10 year business cycle

Topic Author
Andrew321
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Re: Bond pricing theory

Post by Andrew321 » Sat Feb 16, 2019 6:33 pm

Thank you all very much! I appreciate your expertise. The resources and explanations have made the pricing issue clear to me.

Best,
Andrew

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