Bond duration question

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Bond duration question
Apologize if this was asked before
Lets say I have $100,000 in bond fund with 2 percent yield and 5 year duration
Interest rates go from 2 percent to 5 percent, so my principal goes down $15,000 ( 3 percent times 5 yeR duration)
My $15,000 loss is offset by increase of interest 3 percent of $85,000 or $2550 per year, so it would take me about 6 yrs to regain my loss
Am I correct??
Lets say I have $100,000 in bond fund with 2 percent yield and 5 year duration
Interest rates go from 2 percent to 5 percent, so my principal goes down $15,000 ( 3 percent times 5 yeR duration)
My $15,000 loss is offset by increase of interest 3 percent of $85,000 or $2550 per year, so it would take me about 6 yrs to regain my loss
Am I correct??
Re: Bond duration question
Yes, essentially.
However, the loss is a little smaller and the gain a little quicker, both because of reinvestment effects. These two effects combine to lower the "point of indifference" to about 5 years  that is, after this much time you're better off than you would have been without the yield increase.
I would also note that the higher 3 percent increased "interest" is more accurately called "yield". It's quite likely you'll receive some of it as dividends and some of it as capital gains, as depreciated bonds regain value the more they approach maturity. When you say "interest" people might think only of dividends.
However, the loss is a little smaller and the gain a little quicker, both because of reinvestment effects. These two effects combine to lower the "point of indifference" to about 5 years  that is, after this much time you're better off than you would have been without the yield increase.
I would also note that the higher 3 percent increased "interest" is more accurately called "yield". It's quite likely you'll receive some of it as dividends and some of it as capital gains, as depreciated bonds regain value the more they approach maturity. When you say "interest" people might think only of dividends.

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 Joined: Sat Aug 11, 2012 8:44 am
Re: Bond duration question
Let's make this simple.
Let's assume that you put $100,000 in a zerocoupon bond with a 2% yield, maturing in 5 years.
That's not a bond fund, which would behave somewhat differently. But, as a fund is just a collection of individual bonds that have precise contractual obligations, a fund will remain as well behaved as the bonds it contains. So, understanding how a single zerocoupon bond behaves will give you a pretty good idea of how things are.
1) If you hold this bond to maturity, you will get exactly: $100,000 X 1.02^5 = $110,408.08 in 5 years.
2) Just after you bought the bond (e.g. seconds later), 5year yields jump to 5%. If you wanted to sell your bond at that point, what would you get for it?
You have a zerocoupon bond which will mature in 5 years at a par value of $110,408.08 (that's the bond's promise!). So, as current 5year yields are now 5%, the current value of the bond is: $110,408.08 / 1.05^5 = 86,507.62
So, the immediate loss is $100,000  $86,507.62 = $13,492.38
3) Note that if you hold the zerocoupon bond to maturity, you will always get $110,408.08. A bond is a contract with exact numbers written on it! So, in 5 years, you will have more than $100,000. You'll have the same amount as if yields had not changed.
4) How long until you recover $100,000?
You have to solve N in: $86,507.62 X 1.05^N = $100,000. The solution is 2.97.
You can round this to 3 and verify: $86,507.62 X 1.05^3 = $100,143.38
So, in a little less than 3 years, you'll have recovered the initial invested amount.
Had I used a normal bond that throws coupons, the calculations would have been more complex if I wanted to be really precise, but the results would have been very similar (but not identical due of differences in compounding). The use of a zerocoupon bond greatly simplifies calculations and allows to easily understand the behavior of a bond when yields change.
Let's assume that you put $100,000 in a zerocoupon bond with a 2% yield, maturing in 5 years.
That's not a bond fund, which would behave somewhat differently. But, as a fund is just a collection of individual bonds that have precise contractual obligations, a fund will remain as well behaved as the bonds it contains. So, understanding how a single zerocoupon bond behaves will give you a pretty good idea of how things are.
1) If you hold this bond to maturity, you will get exactly: $100,000 X 1.02^5 = $110,408.08 in 5 years.
2) Just after you bought the bond (e.g. seconds later), 5year yields jump to 5%. If you wanted to sell your bond at that point, what would you get for it?
You have a zerocoupon bond which will mature in 5 years at a par value of $110,408.08 (that's the bond's promise!). So, as current 5year yields are now 5%, the current value of the bond is: $110,408.08 / 1.05^5 = 86,507.62
So, the immediate loss is $100,000  $86,507.62 = $13,492.38
3) Note that if you hold the zerocoupon bond to maturity, you will always get $110,408.08. A bond is a contract with exact numbers written on it! So, in 5 years, you will have more than $100,000. You'll have the same amount as if yields had not changed.
4) How long until you recover $100,000?
You have to solve N in: $86,507.62 X 1.05^N = $100,000. The solution is 2.97.
You can round this to 3 and verify: $86,507.62 X 1.05^3 = $100,143.38
So, in a little less than 3 years, you'll have recovered the initial invested amount.
Had I used a normal bond that throws coupons, the calculations would have been more complex if I wanted to be really precise, but the results would have been very similar (but not identical due of differences in compounding). The use of a zerocoupon bond greatly simplifies calculations and allows to easily understand the behavior of a bond when yields change.
Last edited by longinvest on Tue Jul 14, 2015 2:43 pm, edited 8 times in total.
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Re: Bond duration question
Would this be due because of the effect of convexity?ogd wrote:However, the loss is a little smaller and the gain a little quicker, both because of reinvestment effects.
I'm still learning about bonds.
Re: Bond duration question
The "loss is a little smaller", yes.skepticalobserver wrote:Would this be due because of the effect of convexity?ogd wrote:However, the loss is a little smaller and the gain a little quicker, both because of reinvestment effects.
I'm still learning about bonds.
The value loss is not a linear function of the yield difference. It's pretty close, but the higher the difference the bigger the deviation. In an extreme example, a yield increase of 25% does not lose you 125%.
The "gain a little quicker" is simply because of compounding, which is more pronounced at 5% than it was at 2%.

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Re: Bond duration question
Thank you.
As I understand it convexity "refines" the duration number.
As I understand it convexity "refines" the duration number.
Re: Bond duration question
Not quite, assuming we are talking about modified duration. Duration is the average price movement if the yield goes up or down. However, the up price is not exactly the same as the down price. The default measure is a change of .01% in the yield. However, you could choose something other jump, like 1%. If you did so you would get a different duration number.skepticalobserver wrote:Thank you.
As I understand it convexity "refines" the duration number.
To be even more precise, graph the yield (xaxis) and bond price (yaxis). The duration is the slope of that line. If the curve of the line was constant we would not have this issue.
Applying continuous calculus to a system that is not continuous has many practical problems, like the one we are playing around with.
Re: Bond duration question
If it's a onetime rise in interest rates and then they level off, you are correct per comments above from others. However, a rate rise often occurs incrementally in spurts over months or years. This complicates the picture and makes it difficult to calculate "recovery" time in advance.Jackhenryport wrote:Apologize if this was asked before
Lets say I have $100,000 in bond fund with 2 percent yield and 5 year duration
Interest rates go from 2 percent to 5 percent, so my principal goes down $15,000 ( 3 percent times 5 yeR duration)
My $15,000 loss is offset by increase of interest 3 percent of $85,000 or $2550 per year, so it would take me about 6 yrs to regain my loss
Am I correct??
Re: Bond duration question
AA = 40/55/5. Expected CAGR = 3.8%. GSD (5y) = 6.2%. USD inflation (10 y) = 1.8%. AWR = 4.0%. TER = 0.4%. Port Yield = 2.82%. Term = 33 yr. FI Duration = 6.0 yr. Portfolio survival probability = 95%.
 abuss368
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Re: Bond duration question
Hi Jackhenryport,Jackhenryport wrote:Apologize if this was asked before
Lets say I have $100,000 in bond fund with 2 percent yield and 5 year duration
Interest rates go from 2 percent to 5 percent, so my principal goes down $15,000 ( 3 percent times 5 yeR duration)
My $15,000 loss is offset by increase of interest 3 percent of $85,000 or $2550 per year, so it would take me about 6 yrs to regain my loss
Am I correct??
Essentially yes. The return from a bond fund is essentially the interest component. Any fluctuation in NAV is worthless over time. Jack Bogle has provided excellent examples of this in his books.
Best.
John C. Bogle  Two Fund Portfolio: Total Stock & Total Bond. "Simplicity is the master key to financial success."
Re: Bond duration question
Your answer is wrong. It would take you 3 years to recover from your 9% loss.
Assume: duration = 5.00 yr. SEC Yield = 2.00%. New SEC Yield = 5.00%. Difference in SEC Yield = +3.00%.
NAV loss = 3(5.00  2.00) = 9.00%.
9.00/3.00 = 3.00 yr to recover NAV.
Assume: duration = 5.00 yr. SEC Yield = 2.00%. New SEC Yield = 5.00%. Difference in SEC Yield = +3.00%.
NAV loss = 3(5.00  2.00) = 9.00%.
9.00/3.00 = 3.00 yr to recover NAV.
Jackhenryport wrote:Apologize if this was asked before
Lets say I have $100,000 in bond fund with 2 percent yield and 5 year duration
Interest rates go from 2 percent to 5 percent, so my principal goes down $15,000 ( 3 percent times 5 yeR duration)
My $15,000 loss is offset by increase of interest 3 percent of $85,000 or $2550 per year, so it would take me about 6 yrs to regain my loss
Am I correct??
AA = 40/55/5. Expected CAGR = 3.8%. GSD (5y) = 6.2%. USD inflation (10 y) = 1.8%. AWR = 4.0%. TER = 0.4%. Port Yield = 2.82%. Term = 33 yr. FI Duration = 6.0 yr. Portfolio survival probability = 95%.
 abuss368
 Posts: 17205
 Joined: Mon Aug 03, 2009 2:33 pm
 Location: Where the water is warm, the drinks are cold, and I don't know the names of the players!
 Contact:
Re: Bond duration question
I did not check the math.
5 duration. 3% increase. Result is 15% drop.
New yield is 5%. 3 years needed to recover.
Is this formula correct?
5 duration. 3% increase. Result is 15% drop.
New yield is 5%. 3 years needed to recover.
Is this formula correct?
John C. Bogle  Two Fund Portfolio: Total Stock & Total Bond. "Simplicity is the master key to financial success."
Re: Bond duration question
You got your numbers mixed and ended up multiplying two percentages. The OP is approximately correct.galeno wrote: NAV loss = 3(5.00  2.00) = 9.00%.
Recovery to zero is a low standard. Recovery to the original 2% / year standard takes 5 years.abuss368 wrote:New yield is 5%. 3 years needed to recover.
Re: Bond duration question
The math is correct. Assuming investment grade bonds. Every 1% increase in interest rates will cause a loss in NAV = duration  SEC Yield. The increase in interest rates = 3% so 3(52) = 9% loss in NAV.
Since the net increase in SEC Yield = 3%. It will take 3 years for an $91 bond (Bond A)increasing by 5% per year to catch up to a $100 bond (Bond B) increasing by 2%.
After 3 years, Bond B runs away from Bond A. I'm actually looking forward to increases in interest rates.
Since the net increase in SEC Yield = 3%. It will take 3 years for an $91 bond (Bond A)increasing by 5% per year to catch up to a $100 bond (Bond B) increasing by 2%.
After 3 years, Bond B runs away from Bond A. I'm actually looking forward to increases in interest rates.
Last edited by galeno on Tue Jul 14, 2015 6:26 pm, edited 1 time in total.
AA = 40/55/5. Expected CAGR = 3.8%. GSD (5y) = 6.2%. USD inflation (10 y) = 1.8%. AWR = 4.0%. TER = 0.4%. Port Yield = 2.82%. Term = 33 yr. FI Duration = 6.0 yr. Portfolio survival probability = 95%.
Re: Bond duration question
I do not. In nominal terms bonds are at historically low levels. In real terms they are low but not that low  say at the 33% for the past 80 years. If bond yields were increasing because real yields were increasing I would agree with you. However, I fear that bond yields are going to go up because of inflation. If that is true, in real terms Bond B never catches up with Bond A in real terms.galeno wrote:After 3 years, Bond B runs away from Bond A. I'm actually looking forward to increases in interest rates.
Re: Bond duration question
In Bogleheadland math trumps feelings and emotions. Every time.
AA = 40/55/5. Expected CAGR = 3.8%. GSD (5y) = 6.2%. USD inflation (10 y) = 1.8%. AWR = 4.0%. TER = 0.4%. Port Yield = 2.82%. Term = 33 yr. FI Duration = 6.0 yr. Portfolio survival probability = 95%.
Re: Bond duration question
Nope. Every 1% increase in interest rates will cause approximately duration decrease in NAV.galeno wrote:The math is correct. Assuming investment grade bonds. Every 1% increase in interest rates will cause a loss in NAV = duration  SEC Yield. The increase in interest rates = 3% so 3(52) = 9% loss in NAV.
The SEC yields were already subtracted from each other to get that 3%. You never subtract yields from duration, i.e. percentage from years.
Re: Bond duration question
This is a great explanation.longinvest wrote:Let's make this simple.
Let's assume that you put $100,000 in a zerocoupon bond with a 2% yield, maturing in 5 years.
That's not a bond fund, which would behave somewhat differently. But, as a fund is just a collection of individual bonds that have precise contractual obligations, a fund will remain as well behaved as the bonds it contains. So, understanding how a single zerocoupon bond behaves will give you a pretty good idea of how things are.
1) If you hold this bond to maturity, you will get exactly: $100,000 X 1.02^5 = $110,408.08 in 5 years.
2) Just after you bought the bond (e.g. seconds later), 5year yields jump to 5%. If you wanted to sell your bond at that point, what would you get for it?
You have a zerocoupon bond which will mature in 5 years at a par value of $110,408.08 (that's the bond's promise!). So, as current 5year yields are now 5%, the current value of the bond is: $110,408.08 / 1.05^5 = 86,507.62
So, the immediate loss is $100,000  $86,507.62 = $13,492.38
3) Note that if you hold the zerocoupon bond to maturity, you will always get $110,408.08. A bond is a contract with exact numbers written on it! So, in 5 years, you will have more than $100,000. You'll have the same amount as if yields had not changed.
4) How long until you recover $100,000?
You have to solve N in: $86,507.62 X 1.05^N = $100,000. The solution is 2.97.
You can round this to 3 and verify: $86,507.62 X 1.05^3 = $100,143.38
So, in a little less than 3 years, you'll have recovered the initial invested amount.
Had I used a normal bond that throws coupons, the calculations would have been more complex if I wanted to be really precise, but the results would have been very similar (but not identical due of differences in compounding). The use of a zerocoupon bond greatly simplifies calculations and allows to easily understand the behavior of a bond when yields change.
Re: Bond duration question
"Benz: Right. Earlier this summer, we did see a fair amount of volatility in more interestratesensitive securities. Here, I would urge retirees to do that duration stress test that we've talked about before with their fixedincome holdings. I've written an article on this topic, but the basic idea is that you find duration and you find the SEC yield for each of your bond funds. This will really only yield a meaningful result with highquality bond funds. You subtract that SEC yield from the duration. The amount that you're left over with is the rough amount you would see that investment lose in a oneyear period in which rates rose by one percentage point."
Full article: http://www.morningstar.com/Cover/videoC ... ?id=705749
Full article: http://www.morningstar.com/Cover/videoC ... ?id=705749
ogd wrote:Nope. Every 1% increase in interest rates will cause approximately duration decrease in NAV.galeno wrote:The math is correct. Assuming investment grade bonds. Every 1% increase in interest rates will cause a loss in NAV = duration  SEC Yield. The increase in interest rates = 3% so 3(52) = 9% loss in NAV.
The SEC yields were already subtracted from each other to get that 3%. You never subtract yields from duration, i.e. percentage from years.
AA = 40/55/5. Expected CAGR = 3.8%. GSD (5y) = 6.2%. USD inflation (10 y) = 1.8%. AWR = 4.0%. TER = 0.4%. Port Yield = 2.82%. Term = 33 yr. FI Duration = 6.0 yr. Portfolio survival probability = 95%.
Re: Bond duration question
Ah, I see what she's saying. At first I was like, there's no way Christine Benz could make that kind of error.galeno wrote:"Benz: Right. Earlier this summer, we did see a fair amount of volatility in more interestratesensitive securities. Here, I would urge retirees to do that duration stress test that we've talked about before with their fixedincome holdings. I've written an article on this topic, but the basic idea is that you find duration and you find the SEC yield for each of your bond funds. This will really only yield a meaningful result with highquality bond funds. You subtract that SEC yield from the duration. The amount that you're left over with is the rough amount you would see that investment lose in a oneyear period in which rates rose by one percentage point."
Full article: http://www.morningstar.com/Cover/videoC ... ?id=705749
She's talking specifically about a 1% increase in rates. In this case, the NAV loses duration but you also get back SEC yield as usual, leaving you with, say, a 3% loss for a 5 year fund going from 2>3% that year.
But the calculation only works this way for 1% increase, precisely. This is because you get the SEC yield only once, no matter how much the interest rates increased. The difference cannot be multiplied with the increase.
So the formula for total return for the first year is more like: (y1  y0) x duration  sec_yield. The formula for the NAV loss is only the first part. And, as I mentioned, this is an approximation (although one that works reasonably for small increases).
More on this in our wiki: http://www.bogleheads.org/wiki/Bonds:_a ... s#Duration