Bond prices falling

[Alaric]

Can anyone tell me where I have gone wrong? Or what I've misunderstood?

It says present value in the definition. So you have to discount.

Compound Interest theory

A payment of 100 in n periods time at a rate of interest of i per period has a present value of 100 / (1+i) ^ n .

If you just add up the payments, you are using an interest rate of zero.

[GoSeigen]

In this thread, one or two posters have mentioned the term duration. I am vaguely familiar with what it means.....I think it indicates "how long it will take to make your original investment back" (correct me if I'm wrong). However, on reading some bumpf about the term on the internet, I'm somewhat confused by one of the definitions I found.

On this site: https://www.thestreet.com/topic/46361/duration.html it says:

Duration measures the time it takes to recover half the present value of all future cash flows from the bond. The discount rate for calculating the present value of the cash flows is the bond's yield. So as a bond's price and yield change, so does its duration.

For example, a bond with 10 years till maturity and a 7% coupon trading at par to yield 7% has a duration of 7.355 years. At a yield of 6% (price 107 14/32), its duration is 7.461 years. At a yield of 8% (price 93 7/32), its duration is 7.246 years.

I tried to apply some maths, and couldn't quite apply the results to the above. I firstly assumed a fixed yield of 7% and sited the stated duration of 7.355 years. I then multiplied those two figures to see that in that time:

7 * 7.355 = 51.49

(i.e. ~51 units due to cash flows)

But surely if the above bond has 10 years left at 7%, then the sum of the future cash flows is 10 * 7 = 70. However the italicised definition states that the duration will recover half the future cash flows, but half of 70 is 35, and the above 51.49 is most certainly not equal to this.

Where as 51.49 is approximately half of 100, which would have been the value of my original investment, so it would appear that I was along the right lines with my original understanding, but the internet definition I found screws up my thoughts by it's reference to the phrase all future cash flows from the bonds.

Can anyone tell me where I have gone wrong? Or what I've misunderstood?

many thanks
Matt

Matt, Wikipedia covers it pretty well, though the maths may look daunting if unfamiliar:

https://en.wikipedia.org/wiki/Bond_duration

There are slightly different measures of duration, but essentially they are a weighted average of the time remaining for the cashflows to be paid. Note that it includes all cashflows including any principal payment. For a perpetual bond it is approximately the inverse of the yield (not coupon!).

There are two main rules of thumb that are useful:
1. If you match duration to the required maturity date of your investment you maximise the probability that you will receive the yield priced into the bond. (IIRC) This is why people say match the duration to your maturity date.
2. Duration is also a measure of volatility of a bond (or share!) i.e. price sensitivity to yield changes. Approximately, for every x% change in yield the bond's price will move d.x% where d is the modified duration of the bond. e.g. a 1% change in yield of a 1-year bond moves its price 1% whereas a 1% yield change in equities yielding 4% moves their price 25%.


HTH

GS

[hiriskpaul]

Yield curves are fundamental to pricing future cash flows. If you are going to receive a series of payments, a yield curve can be used to calculate a present value for those payments. Each point on the curve gives you the interest rate to use in the discounting of cash flows on that date. In the case of a gilt it is reasonable to expect the present value of all future cash flows to equal the current market value of the gilt.

There are 2 main ways of constructing yield curves. One is to use interbank rates and swap rates. These give fixed points on the yield curve. To get intermediate points interpolation is used, these days it is usually cubic spline interpolation. The other way of constructing a curve is more complicated and involves finding a best fit curve to cash flows from a set of government bonds.

Wikipedia maybe a good source of information on yield curves. Or maybe not, it can be too terse.

Bond duration has a precise mathematical definition and is actually related to the change (to first order) in the price of a bond resulting from a small change in yield. It has certain properties including the one you mention about the time to get half your money back, but that is an approximation. Again, Wikipedia might be a good starting point.

You would not normally expect credit spreads to vary with gilt yields, so if gilt yields rise by 1%, investment grade corporate bond yields would be expected to rise 1%. This is just the expected behaviour though and markets often do not behave in ways people expect them to!

[hiriskpaul]

I see Alaric and GS replied before me. I agree with their comments and on the error in your calculations Alaric is right. You need to calculate present values of the cash flows using the yield to maturity of the bond as the discount factor. Do that and you will see a much closer match.

[colin]

Forgive my naivety but surely if gilt yields rise, then the spread to gilts narrows? I.e. corporate bond yields are not as advantageous as they were previously upon gilt yields.

Yes that's about it, the cycle works like this... during a recession investors worry that some companies won't be able to pay the money they owe on their bonds, so the prices of those bonds fall as the yields rise, investors pile into government bonds which carry no risk of default so prices rise and yields fall, the difference between the yields on the two investments is the spread between corporate and government bonds, during recessions the spread is wide, as economic conditions improve the opposite happens, investors become more confident that corporations will pay their debt so investors are keen to buy the bonds, those who previously held the safer government bonds realize they can get better returns from corporate bonds and so the yield spread between government and corporate debt narrows. While it is true that in better times the yields on corporate debt has fallen this alone does not make them an inferior investment relative to government bonds because the reason that corporate debt yields have fallen is that corporations have become more likely to actually pay what they owe while during periods of wide credit spread there is an expectation that defaults will increase and while buying a higher yield may at first seem like a no-brainer it's actually a sign that going forward the market suspects that not all corporations will be able to pay what they owe.
In other words yield spreads widen as the expectation of corporate defaults in the near future rises, and yield spreads narrow as expectations of corporate defaults fall.

[TheMotorcycleBoy]

Yes that's about it, the cycle works like this......
....
In other words yield spreads widen as the expectation of corporate defaults in the near future rises, and yield spreads narrow as expectations of corporate defaults fall.

Thanks for this Colin, I see what you mean.

[TheMotorcycleBoy]

Thanks for the replies regarding "duration". The penny is starting to drop. I think I need to get my head around the concept of "the time/present value of money".

In additional to the wiki link GS sited I'm going to peruse these a bit this weekend.

https://www.investopedia.com/articles/03/082703.asp
https://www.investopedia.com/calculator ... cdate.aspx

I think what I failed to gather was that each "cash flow" can itself be used to gain interest. The example I posted earlier i.e. https://www.thestreet.com/topic/46361/duration.html doesn't state this explicitly, however the investopedia linked above does. And of course one is to use the current "yield to maturity" of the bond in question as that interest rate. Or at least that's where my current thoughts lie.

thanks again everyone!

[TheMotorcycleBoy]

Hi all, apologies in advance for my longest post so far:

We are still trying to wrap our heads around the concept of bond duration. I'm much more au fait with the concept: i.e. yield sensitivity to price change, and a measure of how long it will take for the cash flows (present values of) to return one's investment.

However being slightly mathematical and wanting to know how a potential investment of ours works, I've soldiered on with the example that I posted on friday i.e. https://www.thestreet.com/topic/46361/duration.html

Duration measures the time it takes to recover half the present value of all future cash flows from the bond. The discount rate for calculating the present value of the cash flows is the bond's yield. So as a bond's price and yield change, so does its duration.

For example, a bond with 10 years till maturity and a 7% coupon trading at par to yield 7% has a duration of 7.355 years. At a yield of 6% (price 107 14/32), its duration is 7.461 years. At a yield of 8% (price 93 7/32), its duration is 7.246 years.

So based on the replies I received on friday, in particular:

It says present value in the definition. So you have to discount.

and

If you just add up the payments, you are using an interest rate of zero.

I revisited my earlier thoughts: and my initial point of confusion is that Alaric mentioned the term interest rate but the sited thestreet article has no explicit mention of interest rate; but instead it mentions "yield" (presumably the bond's YTM or running yield?). Am I to understand that I should equate the term "yield" in the web article's for Alarics "interest rate"?

Anyway I proceeded with my thoughts, assuming that we are to take the term yield==interest-rate, and went ahead with thestreet's example. After quickly getting irritated with carrying the figures around on paper and being a programmer by profession, I wrote a configurable program, in which I can feed in the numbers of interest (maturity, coupon, rate/yield/? etc.). So below is a screen dump of the program assuming the above bond bought at issue for 100 units maturity=10 years, coupon=7%, discount rate=7%.

The leftmost figure is the number of years since purchase, the next figure is the present value of that year's cash flow (interest payment discounted at rate compounded (1 + r)^n), and the last figure is the accumulated total of the cash flows:

Years:1 PV of cash_flow 6.542056 running total of cash flows PVs 6.542056 
Years:2 PV of cash_flow 6.114071 running total of cash flows PVs 12.656127 
Years:3 PV of cash_flow 5.714085 running total of cash flows PVs 18.370212 
Years:4 PV of cash_flow 5.340266 running total of cash flows PVs 23.710479 
Years:5 PV of cash_flow 4.990903 running total of cash flows PVs 28.701382 
Years:6 PV of cash_flow 4.664396 running total of cash flows PVs 33.365778 
Years:7 PV of cash_flow 4.359248 running total of cash flows PVs 37.725026 
Years:8 PV of cash_flow 4.074064 running total of cash flows PVs 41.799090 
Years:9 PV of cash_flow 3.807536 running total of cash flows PVs 45.606626 
Years:10 PV of cash_flow 54.393374 running total of cash flows PVs 100.000000

So what I still don't get, is what thestreet's article means when it associates half the present value of all future cash flows from the bond and the time value of 7.355 years, since in my above calculations between years 7 and 8, I've accumulated approximately 39 units. What's that half of?

Apologies for my obsessiveness in all this, just eager to make our investments with more confidence, without reliance on a "bond fund" manager. (Love it if there was a decent book for all this....Mark Glowreys book useful though it is, doesn't spend much time on this stuff)

thank Matt

[genou]

So what I still don't get, is what thestreet's article means when it associates half the present value of all future cash flows from the bond and the time value of 7.355 years, since in my above calculations between years 7 and 8, I've accumulated approximately 39 units. What's that half of?
thank Matt

I don't know where they are getting that half from. It is worth pointing out that duration can carry different emphasis - see the explanation here
https://en.wikipedia.org/wiki/Bond_duration

[TheMotorcycleBoy]

I don't know where they are getting that half from.

Fair enough - thanks anyway. Regardless, do you agree that my sums are basically sound i.e. compounding "a discount rate" against those PVs?

Also, was my assumption where I equated yield==interest_rate, for the purposes of interpreting thestreet's artice correct in your opinion.

https://en.wikipedia.org/wiki/Bond_duration

Sure. GS linked that to me earlier. I'll read it in more depth in a bit.

I also found this doc. if anyone else needs help, it seemed to carry some nicer examples.

http://www.treasurer.ca.gov/cdiac/publi ... ration.pdf

Matt

[genou]

I don't know where they are getting that half from.

Fair enough - thanks anyway. Regardless, do you agree that my sums are basically sound i.e. compounding "a discount rate" against those PVs?

Also, was my assumption where I equated yield==interest_rate, for the purposes of interpreting thestreet's artice correct in your opinion.

https://en.wikipedia.org/wiki/Bond_duration

Sure. GS linked that to me earlier. I'll read it in more depth in a bit.

I also found this doc. if anyone else needs help, it seemed to carry some nicer examples.

http://www.treasurer.ca.gov/cdiac/publi ... ration.pdf

Matt

Sorry about the duplicate link, I'm following this thread with one eye, as it were. Alaric should speak for himself, but as I read his "rate of interest" he is talking about the coupon. The Street uses yield to mean running yield in its article, so they are different beasts.

I come at bonds on a more simplistic basis. Leaving aside credit risk, I'm looking at the YTM, as the way I use bonds in my portfolio means that I am highly likely to hold to maturity. Therefore movements in the running yield don't bother me , so I tend not to think about duration / volatility that much.

[Alaric]

I also found this doc. if anyone else needs help, it seemed to carry some nicer examples.

Be careful that it describes American practice.

If you want more insight into how bonds are valued, there's a formula short cut that works when the coupon payments are the same throughout.

Notation
C = coupon per 100 nominal
i = interest rate used for the valuation
assume repayment at par
v = 1/(1+i)
n = term

So value of bond
A = C*v + C*v^2 + ..... C*v^n +100 * v^n

Just look at the terms involving C

V = C*v + C*v^2+ .... C*v^n
then using the trick for valuing geometric progressions which is or was part of school mathematics, multiply both sides by v

V*v = C*v^2 + C*v^3+ ..... C*v^(n+1)

Subtract the second expression from the first

V*(1-v) = C*v * (1-v^n)

or V= C*v *( 1-v^n) * v / (1-v) by dividing both sides by (1-v)

given that v= 1/ (1+i), v/(1-v) simplifies to 1/( (1+i)* (1- 1/(1+i) which is 1/i

Thus the formula for the whole value of the bond is

C * (1- v^n) /i + 100 * v^n

When I talk about the rate of interest I mean the i in the above, not the Coupon. The two are only the same if the Bond has a price of 100.

[TheMotorcycleBoy]

Notation
C = coupon per 100 nominal
i = interest rate used for the valuation
assume repayment at par
v = 1/(1+i)
n = term

So value of bond
A = C*v + C*v^2 + ..... C*v^n +100 * v^n

Yes, thanks, I'd come to that realisation over my ponderings this weekend.

V = C*v + C*v^2+ .... C*v^n
then using the trick for valuing geometric progressions which is or was part of school mathematics, multiply both sides by v

V*v = C*v^2 + C*v^3+ ..... C*v^(n+1)

Subtract the second expression from the first

A ha! Thanks yes I remember it now!

V*(1-v) = C*v * (1-v^n)

or V= C*v *( 1-v^n) * v / (1-v) by dividing both sides by (1-v)

I reckon you grew an extra v term, but otherwise, I'm still following.

Thus the formula for the whole value of the bond is

C * (1- v^n) /i + 100 * v^n

Agreed. Thanks for spending the time to elaborate!

Ok thanks.......in conclusion....I'm forgetting about the thestreet's example! None of you lot know or care why the word half (well certainly not as far as my or Alarics example calculations) so neither will I!

[Alaric]

..I'm forgetting about the thestreet's example!

I think they rather scrambled the explanation. What their definition of duration should be doing is to weight each payment by the number of future periods. So basically they are summing C*t*v^t + 100*n* v^n and then dividing by the bond price.

I did try it and got close to their example but not exactly. It did occur to me that perhaps they had quoted a value from a table of bond results that may have assumed half yearly payments.

There are several variations on the theme of calculating weighted values.

A point to be aware of in practice is that the assumption of a single rate of interest is itself a simplification. In practice the price of money as in "how much to pay today for a payment in the future" isn't a straight line when you plot the implied interest rate. It's usually upward sloping so that you might pay 99 for a payment of 100 due next year, but 90 for a payment of 100 due in 5 years time.

Somewhere there are computers trying to make money out of anomalies this can create, but they usually close rapidly.

More "light" reading, this time on the London market in UK Government Bonds.

http://www.londonstockexchange.com/trad ... -gilts.pdf

It's some years old, so may have been overtaken by events.

[TheMotorcycleBoy]

I think they rather scrambled the explanation. What their definition of duration should be doing is to weight each payment by the number of future periods.

Yes, I agree. That's what I appreciated in this site http://www.treasurer.ca.gov/cdiac/publi ... ration.pdf, that is, on page they clearly exemplify the notion of the weighting of each cash flow. And with that in my mind, I figured out how to modify the computer script I'd written earlier to output Macaulay Durations of various examples, and hey presto, they showed a close match to those figures on the HL site.

[TheMotorcycleBoy]

Sorry to keep going on about this

i = interest rate used for the valuation

and

When I talk about the rate of interest I mean the i in the above, not the Coupon. The two are only the same if the Bond has a price of 100.

So when you (and all the online bond duration calculators) refer to "i" or interest rate in the above, what do you mean exactly? Do you mean the base rate of interest, if so, then all the online bond duration calculators must predict this (which seems improbable), or do you actually mean the bonds YTM? Given the in most simple online duration calcuator I've found so far:

https://www.money-education.com/resourc ... alculation

(the only inputs are coupon rate, YTM and time to maturity)

I suspect as I alluded to earlier, in these duration calculations, we use the bond's YTM as the "rate". Correct?

[TheMotorcycleBoy]

Yes, after finally feeling confident enough to work through https://en.wikipedia.org/wiki/Bond_duration I read what I needed to know:

For most practical calculations, the Macaulay duration is calculated using the yield to maturity to calculate the PVi ...

[Alaric]

the Macaulay duration

I'm not sure it's terribly useful to the private investor. These measures were devised to help institutional investors who hold bond portfolios to match insurance and other liabilities at defined future dates. If in their financial reporting, they are required to mark to market, a term meaning that they have to recalculate at market prices, they don't want to see the value placed on their liabilities going one way, whilst their asset values went the other.

[TheMotorcycleBoy]

I don't know where they are getting that half from. It is worth pointing out that duration can carry different emphasis - see the explanation here https://en.wikipedia.org/wiki/Bond_duration

I figured out the half bit now. If you look at the linked wiki page, just near the end of the "Macaulay duration" derivation there's a (too!) small image. It depicts a see-saw type balance. Moving left to right along the balance is increments of time (measured in years toward maturity of the bond), and resting on the balance are the weighted discounted cash flows that you'd receive each year(or whatever), the time value known as "Macaulay duration" is the point where the fulcrum must be for the whole thing to balance. Thus at either side of fulcrum (i.e. where t=duration) we equal weights, hence on either side there is half of the total weighted DCFs. Bingo.

By the way, you said this earlier on:

Leaving aside credit risk, I'm looking at the YTM, as the way I use bonds in my portfolio means that I am highly likely to hold to maturity.

Yes, that's exactly what we are aiming to do. The last thing we want to do consume any earnings on excessive trading - we don't quite have that much to burn!

[colin]

Out of interest (er forgive the pun) Mat/Melanie , the bonds you are thinking of buying how many years to maturity and what level of yield to maturity would you be looking for, and are you intending to create a continuous running bond ladder or just buy a few individual bonds maturing around the same date?