The bond/gilt market

Mikeeee_2

OldScientist said:

zagfles said:

OldScientist said:

Mikeeee_2 said:

Bobziz said:

Very useful, thank you @Mikeeee_2 Could you explain the 14.96% calculation please.

It's quite a complex formula but the Modified Duration (which is the % above) is derived from the Macaulay Duration:

Usefully, the duration function (in most/all spreadsheet packages) can be used to calculate the Macualay duration - the modified duration is then a (relatively) trivial calculation. I aldo note that for back of the envelope calculations, the modified duration is close enough to the Macualay duration except when the YTM is high.

While I agree that care must be taken in chasing coupons (particularly since, when held outside of a tax sheltered environment, they attract tax), they do have the effect of reducing the Macaulay duration (and hence modified duration). In other words, a zero coupon gilt will have a higher volatility with changes in interest rates than one with a non-zero coupon. For example, the following graph shows the modified duration as a function of YTM for a bond with a maturity of 20 years and with semi-annual coupons of 0%, 1%, 2%, 3%, 4%, and 5%.

 

For a YTM of 4%, the modified duration ranges from about 19% for a 0% coupon to about 13% for a 5% coupon.

Of course, if you are holding the gilt to maturity then the volatility is not a significant concern, but might be disturbing for anyone holding a gilt ladder.

In passing., I note that this graph also illustrates why bond prices reacted with such volatility to changes in yields over the last couple of years. Firstly, the starting YTM were low (~1.5%) and secondly, since gilts are usually issued with positive coupons close to the prevailing yield many gilts issued in the last decade or so, have had relatively small coupons.

It's interesting that the implication of this, if I've understood correctly, is that interest rate changes now are assumed to affect the expected interest rate decades down the line. It's clear why gilts will go up/down with interest rates, what I don't really get is why long dated gilts are so much more volatile than mid dated gilts. Say a 15 year gilt compared to a 50 year one. This seems to assume that any interest change now is going to affect interest rates in 15+ years time.

Probably easiest to illustrate this an example. Imagine we have two zero coupon bonds (this just makes the maths a bit easier) one with 15 years to maturity and the other with 50 years. The price, P is then given by 100/(1+y)^n where y is the yield expressed as a fraction and n is the number of years to maturity. Assuming that both bonds have the same yield of 3% (i.e., the yield curve is flat) then

For the 15 year bond, the price is 100/(1+0.03)^15=64.186 (the modified duration is 14.6)

For the 50 year bond, the price would be 100/(1+0.03)^50=22.811 (the modified duration is 48.5)

You can start with the current price and check whether the final price is 100. So, for the 15 year bond final price=64.186*(1+0.03)^15=100.

Now let us assume that the yield increases to 5% (again, for the purposes of this example, the yield is the same for both bonds).

For the 15 year bond the price is now P=100/(1+0.05)^15=48.102

While for the 50 year bond the price falls to 100/(1+0.05)^50=8.720

The price of the 15 year bond has fallen by about 25%, while that of the 50 year bond has fallen by 60% - i.e. the volatility of the longer bond is greater. This happens because of the power term in the equation - the effect of the yield change is played out over a much longer time for the longer maturity bond. I also note that this example illustrates the approximate nature of the rule of thumb associated with modified duration - it is only accurate for small changes in yields (but is otherwise is good enough).

OldScientist said:

zagfles said:

OldScientist said:

Mikeeee_2 said:

Bobziz said:

Very useful, thank you @Mikeeee_2 Could you explain the 14.96% calculation please.

It's quite a complex formula but the Modified Duration (which is the % above) is derived from the Macaulay Duration:

Usefully, the duration function (in most/all spreadsheet packages) can be used to calculate the Macualay duration - the modified duration is then a (relatively) trivial calculation. I aldo note that for back of the envelope calculations, the modified duration is close enough to the Macualay duration except when the YTM is high.

While I agree that care must be taken in chasing coupons (particularly since, when held outside of a tax sheltered environment, they attract tax), they do have the effect of reducing the Macaulay duration (and hence modified duration). In other words, a zero coupon gilt will have a higher volatility with changes in interest rates than one with a non-zero coupon. For example, the following graph shows the modified duration as a function of YTM for a bond with a maturity of 20 years and with semi-annual coupons of 0%, 1%, 2%, 3%, 4%, and 5%.

 

For a YTM of 4%, the modified duration ranges from about 19% for a 0% coupon to about 13% for a 5% coupon.

Of course, if you are holding the gilt to maturity then the volatility is not a significant concern, but might be disturbing for anyone holding a gilt ladder.

In passing., I note that this graph also illustrates why bond prices reacted with such volatility to changes in yields over the last couple of years. Firstly, the starting YTM were low (~1.5%) and secondly, since gilts are usually issued with positive coupons close to the prevailing yield many gilts issued in the last decade or so, have had relatively small coupons.

It's interesting that the implication of this, if I've understood correctly, is that interest rate changes now are assumed to affect the expected interest rate decades down the line. It's clear why gilts will go up/down with interest rates, what I don't really get is why long dated gilts are so much more volatile than mid dated gilts. Say a 15 year gilt compared to a 50 year one. This seems to assume that any interest change now is going to affect interest rates in 15+ years time.

Probably easiest to illustrate this an example. Imagine we have two zero coupon bonds (this just makes the maths a bit easier) one with 15 years to maturity and the other with 50 years. The price, P is then given by 100/(1+y)^n where y is the yield expressed as a fraction and n is the number of years to maturity. Assuming that both bonds have the same yield of 3% (i.e., the yield curve is flat) then

For the 15 year bond, the price is 100/(1+0.03)^15=64.186 (the modified duration is 14.6)

For the 50 year bond, the price would be 100/(1+0.03)^50=22.811 (the modified duration is 48.5)

You can start with the current price and check whether the final price is 100. So, for the 15 year bond final price=64.186*(1+0.03)^15=100.

Now let us assume that the yield increases to 5% (again, for the purposes of this example, the yield is the same for both bonds).

For the 15 year bond the price is now P=100/(1+0.05)^15=48.102

While for the 50 year bond the price falls to 100/(1+0.05)^50=8.720

The price of the 15 year bond has fallen by about 25%, while that of the 50 year bond has fallen by 60% - i.e. the volatility of the longer bond is greater. This happens because of the power term in the equation - the effect of the yield change is played out over a much longer time for the longer maturity bond. I also note that this example illustrates the approximate nature of the rule of thumb associated with modified duration - it is only accurate for small changes in yields (but is otherwise is good enough).

Couldn't have explained it any better. I've been in work so thanks for this 👏

JohnWinder

So a low yielding longer dated gilt will always have to move more than anything else.

And isn't there a lot hiding in 'low yielding'.....https://portfoliocharts.com/2019/05/27/high-profits-at-low-rates-the-benefits-of-bond-convexity/

zagfles

OldScientist said:

zagfles said:

OldScientist said:

Mikeeee_2 said:

Bobziz said:

Very useful, thank you @Mikeeee_2 Could you explain the 14.96% calculation please.

It's quite a complex formula but the Modified Duration (which is the % above) is derived from the Macaulay Duration:

Usefully, the duration function (in most/all spreadsheet packages) can be used to calculate the Macualay duration - the modified duration is then a (relatively) trivial calculation. I aldo note that for back of the envelope calculations, the modified duration is close enough to the Macualay duration except when the YTM is high.

While I agree that care must be taken in chasing coupons (particularly since, when held outside of a tax sheltered environment, they attract tax), they do have the effect of reducing the Macaulay duration (and hence modified duration). In other words, a zero coupon gilt will have a higher volatility with changes in interest rates than one with a non-zero coupon. For example, the following graph shows the modified duration as a function of YTM for a bond with a maturity of 20 years and with semi-annual coupons of 0%, 1%, 2%, 3%, 4%, and 5%.

 

For a YTM of 4%, the modified duration ranges from about 19% for a 0% coupon to about 13% for a 5% coupon.

Of course, if you are holding the gilt to maturity then the volatility is not a significant concern, but might be disturbing for anyone holding a gilt ladder.

In passing., I note that this graph also illustrates why bond prices reacted with such volatility to changes in yields over the last couple of years. Firstly, the starting YTM were low (~1.5%) and secondly, since gilts are usually issued with positive coupons close to the prevailing yield many gilts issued in the last decade or so, have had relatively small coupons.

It's interesting that the implication of this, if I've understood correctly, is that interest rate changes now are assumed to affect the expected interest rate decades down the line. It's clear why gilts will go up/down with interest rates, what I don't really get is why long dated gilts are so much more volatile than mid dated gilts. Say a 15 year gilt compared to a 50 year one. This seems to assume that any interest change now is going to affect interest rates in 15+ years time.

Probably easiest to illustrate this an example. Imagine we have two zero coupon bonds (this just makes the maths a bit easier) one with 15 years to maturity and the other with 50 years. The price, P is then given by 100/(1+y)^n where y is the yield expressed as a fraction and n is the number of years to maturity. Assuming that both bonds have the same yield of 3% (i.e., the yield curve is flat) then

For the 15 year bond, the price is 100/(1+0.03)^15=64.186 (the modified duration is 14.6)

For the 50 year bond, the price would be 100/(1+0.03)^50=22.811 (the modified duration is 48.5)

You can start with the current price and check whether the final price is 100. So, for the 15 year bond final price=64.186*(1+0.03)^15=100.

Now let us assume that the yield increases to 5% (again, for the purposes of this example, the yield is the same for both bonds).

For the 15 year bond the price is now P=100/(1+0.05)^15=48.102

While for the 50 year bond the price falls to 100/(1+0.05)^50=8.720

The price of the 15 year bond has fallen by about 25%, while that of the 50 year bond has fallen by 60% - i.e. the volatility of the longer bond is greater. This happens because of the power term in the equation - the effect of the yield change is played out over a much longer time for the longer maturity bond. I also note that this example illustrates the approximate nature of the rule of thumb associated with modified duration - it is only accurate for small changes in yields (but is otherwise is good enough).

I understand that, but it misses the point I was making. Clearly, if the YTM changes from 3% to 5% on both the 15 and 50 year gilt, that would mean the price drop will be much greater on the longer dated gilt, as you illustrate above.

The point is I don't understand why the YTM should change the same for both gilts, unless the assumption is that events now which eg cause interest to rise, will result in higher rates in 15+ years time than previously expected.

I've already given an example, but it'll be the same with yours. 

Say "risk free return" expected by the market is 3%, and that's considered the long term norm. So the YTM on both short and long gilts will be 3%. 

Something changes in the economy and the BoE raises interest rates. The market now wants 5% "risk free return". If the market believes the change in the economy will result in rates staying at 5% for the next 50 years, then fine, your logic is flawless and the gilts should move as you describe.

But if the market believes the interest rate rise is temporary and that rates should return to the 3% "norm" at some stage in the next 15 years, then it doesn't make sense that the 50 year gilt moves more than the 15 year gilt. Because if they did return to "normal", then the YTM on the 50 year gilt in 15 years' time will be 3%, and if you pretend that's maturity (it's really the expected price based on that assumption), then the maths will show both gilts moving the same %.

So basically - event occurs - is the event going to cause a permanent interest rate shift - YTMs on all gilts should change the same and longer dated gilts will move more. No argument there.

Event occurs - the event is likely to cause a temporary interest change lasting n years but not a permanent change, any gilts with a maturity more than n years away the price change % should be the same, the YTM on longer dated gilts will change less than on shorter gilts because the expectation of interest rates after n years isn't affected.

MK62

Surely if the risk free rate was 5% today, and the market expected that risk free rate to return to 3% at some point in the next 15 years, then the YTMs on the two bonds would not both be 5% today........

OldScientist

zagfles said:

I understand that, but it misses the point I was making. Clearly, if the YTM changes from 3% to 5% on both the 15 and 50 year gilt, that would mean the price drop will be much greater on the longer dated gilt, as you illustrate above.

The point is I don't understand why the YTM should change the same for both gilts, unless the assumption is that events now which eg cause interest to rise, will result in higher rates in 15+ years time than previously expected.

The simple answer is that, outside simple illustrative examples like the one I chose, the YTM will not necessarily change by the same amount for gilts of different maturities for several reasons:
a) The yield curve is unlikely to be flat since it tends to rise as maturity increases. If I understand correctly, this is because longer maturity gilts are less 'risk free' since there is always the possibility of 'unknown unknowns' leading to non payment of capital. For example, the probability of the UK defaulting on its debt in the next year is much lower than over the next 50 years and therefore the market usually expects to be paid more to hold 50 year gilts than one year gilts.
b) Outside of simple examples, gilts of different maturities do not start with the same yield and therefore changes in yield across the yield curve are also likely to be different.

Again, that probably addresses 'how' more than 'why'. I'm not sure the latter has a definitive answer since the price (and hence yield) of gilts is somehow tied up with perceptions of how UKplc is doing and is likely to perform in the future (i.e., an assessment of 'default risk'). The current S&P AA rating presumably reflects their analysts view of those prospects both now and in the future. A downgrade in credit rating or other reasons for a lack of confidence in the UK would see a sell off of gilts and rising yields. My understanding is that the base rate has a greater effect at the short end of maturity (and that can be seen in the current yield curves which are far from flat).

zagfles

OldScientist said:

zagfles said:

I understand that, but it misses the point I was making. Clearly, if the YTM changes from 3% to 5% on both the 15 and 50 year gilt, that would mean the price drop will be much greater on the longer dated gilt, as you illustrate above.

The point is I don't understand why the YTM should change the same for both gilts, unless the assumption is that events now which eg cause interest to rise, will result in higher rates in 15+ years time than previously expected.

The simple answer is that, outside simple illustrative examples like the one I chose, the YTM will not necessarily change by the same amount for gilts of different maturities for several reasons:

Yet the formula you quoted assumes it will, if I've understood it correctly

a) The yield curve is unlikely to be flat since it tends to rise as maturity increases. If I understand correctly, this is because longer maturity gilts are less 'risk free' since there is always the possibility of 'unknown unknowns' leading to non payment of capital. For example, the probability of the UK defaulting on its debt in the next year is much lower than over the next 50 years and therefore the market usually expects to be paid more to hold 50 year gilts than one year gilts.

Which makes sense, but we're talking about a change in the YTM/"risk free rate"/price. Fair enough, if an event occurs which is likely to change the prospect of the govt's ability to repay debt in 50 years time. But my point was an event occuring which is likely to cause a temporary rise in interest rates.

zagfles

MK62 said:

Surely if the risk free rate was 5% today, and the market expected that risk free rate to return to 3% at some point in the next 15 years, then the YTMs on the two bonds would not both be 5% today........

Exactly my point summarised nicely! But the forumale quoted above seem to assume the YTMs on all gilts will change the same regardless of duration (resulting in the price on long dated gilts changing far more than short dated)

OldScientist

zagfles said:

OldScientist said:

zagfles said:

I understand that, but it misses the point I was making. Clearly, if the YTM changes from 3% to 5% on both the 15 and 50 year gilt, that would mean the price drop will be much greater on the longer dated gilt, as you illustrate above.

The point is I don't understand why the YTM should change the same for both gilts, unless the assumption is that events now which eg cause interest to rise, will result in higher rates in 15+ years time than previously expected.

The simple answer is that, outside simple illustrative examples like the one I chose, the YTM will not necessarily change by the same amount for gilts of different maturities for several reasons:

Yet the formula you quoted assumes it will, if I've understood it correctly

a) The yield curve is unlikely to be flat since it tends to rise as maturity increases. If I understand correctly, this is because longer maturity gilts are less 'risk free' since there is always the possibility of 'unknown unknowns' leading to non payment of capital. For example, the probability of the UK defaulting on its debt in the next year is much lower than over the next 50 years and therefore the market usually expects to be paid more to hold 50 year gilts than one year gilts.

Which makes sense, but we're talking about a change in the YTM/"risk free rate"/price. Fair enough, if an event occurs which is likely to change the prospect of the govt's ability to repay debt in 50 years time. But my point was an event occuring which is likely to cause a temporary rise in interest rates.

While the formula I quoted makes an assumption about the coupons (zero coupon) in order to make the maths more straightforward), it makes no assumption about the yields, so you have not understood it correctly. However, in the example I gave where I used the formula I assumed that the yield curve was flat (as I stated).

If you are interested in the effect of news on gilt prices, there is an interesting paper by Elmendorf et al., "The effect of news on bond prices: Evidence from the United Kingdom, 1900-1920" (a google search will find it - it is freely available).

If you're looking to use such 'events' to predict what will happen to gilt prices you're probably going to be disappointed!

masonic

zagfles said:

masonic said:

zagfles said:

Johnjdc said:

Mikeee is correct. The risk free rate is the risk free rate, that's what it's called and it just means the nominal return you can get without risking your nominal capital.

The fact that the real terms return might be lower, or even negative, is something to consider when investing, but doesn't affect the definition of the terms.

Now it's just semantics. On that definition you can get "risk free" 27% pa return on Turkish govt bonds

Turkish govt bonds should probably not be considered risk free. That is why there is a significant spread over the lowest risk bonds available.

Why not? Setting aside default risk and currency risk, so eg for a Turk living in Turkey, why aren't Turkish govt bonds "risk free"? It's blatently obvious to any investor in Turkish govt bonds that inflation risk is the biggest risk in buying govt bonds. Whatever definition of "risk free" is commonly used.

It's also obvious to buyers of flat gilts in any country that inflation is a risk, albeit a far lower risk than in Turkey.

Setting aside default risk and currency risk, all countries would have the same credit rating and citizens would universally adopt their local currency in all countries, as they do in places like the UK and the USA. In that world you would be correct, inflation risk would remain and that would be equal whatever currency you held. We would also have no need for forex markets and currency speculators, so it would certainly have its upsides. However, countries have different credit ratings, symbolising different default risks, and more importantly they have currency risk: When a currency devalues, things get relatively more expensive in that currency than they are in other currencies. As a result, sometimes citizens of a country utilise bonds of different countries because they are closer to the "risk free rate". They may even use the currency of a different country in the more extreme cases because they cannot predict what they will have to pay for things in their local currency from week to week or month to month, even if prices in other currencies are stable. This is all currency risk, and a risk premium above the "risk free rate" is priced into the yield of bonds issued in that currency. As defined by the financial sector, the "risk free rate" is currency independent in an efficient market (and I am putting the term in quotes to be clear that I am using this this accepted definition of it being free of capital risk, rather than asserting there are no risks at all - nothing provides this guarantee). If you see a higher rate of return, then it will be due to the inclusion of an additional risk premium.

I agree with you that it is obvious to buyers of fiat bonds in any country that inflation is a risk (hopefully it is or they are in for disappointment). But the "risk free rate" is proxied by short duration treasuries (conventionally the 3 month Treasury is adopted as the proxy for the "risk free rate"), so inflation is not usually a concern over such a short term. Indeed, with the lag of index linking, it is unlikely you'd be able to replace your 3 month Treasury with an alternative index linked bond that delivers the desired result. Consumers with no appetite for risk generally shouldn't invest at the "risk free rate" because they have access to FSCS-backed savings products that usually pay higher rates than can be obtained by all and sundry including institutional investors on bond markets. These are also labelled risk free, despite having no inflation protection.

Considering longer duration bonds, which normally pay more than the "risk free rate" due to a duration risk premium, this is where inflation can be more material. But while buyers of the inflation linked variety avoid the risk of their investment devaluing as compared with some official measure of inflation, their outcome in nominal terms could be above or below the "risk free rate" and/or the implied rate of inflation determined by the market price of the bond when they invest. The return in real terms can be either positive or negative due to the same factors, but it will be known from the outset. I'm a big advocate of securing inflation linked returns where possible, but there are times when it makes no financial sense to do so. But that's another discussion.

Linton

MK62 said:

Surely if the risk free rate was 5% today, and the market expected that risk free rate to return to 3% at some point in the next 15 years, then the YTMs on the two bonds would not both be 5% today........

People seem to be making heavy weather of this , gilts reallty are  pretty straightforward though the maths can be a little messy.  You can work out the behaviour of the markets purely on the basis of the coupon and capital repayment value at naturity being fixed at the point you buy the bond and that market pricing operates on the basis of bonds being held to maturity.  This means that expectations of future prices are pretty irrelevent. 

WHat does matter is the current interest rate.  The price of the shortest duration gilts match the interest rates set by the government.  Longer duration ones are set by the markets but are strongly nfluenced by the prices of shorter dated ones - Clearly the market will quickly remove any step change at one particular naturity date.

On the question of risk free....

Bonds are just a loan like any other with interest paid and capital repayment being calculated purely on £ terms.   Take your mortgage for example. Your repayment is not index linked.  Have the mortgage companies being doing it wrong all these years? Clearly they dont see future inflation as a risk to their business of lending money..

Any risk from individual gilts comes from using them inappropriately, as is the case for any investment or any tool used for any activity whatsoever.  What they dont have is the additional risk of market uncertainty.

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