Bond duration question

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Jackhenryport
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Bond duration question

Post by Jackhenryport » Tue Jul 14, 2015 1:44 pm

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??

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ogd
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Re: Bond duration question

Post by ogd » Tue Jul 14, 2015 1:54 pm

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.

longinvest
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Re: Bond duration question

Post by longinvest » Tue Jul 14, 2015 2:15 pm

Let's make this simple.

Let's assume that you put $100,000 in a zero-coupon 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 zero-coupon 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), 5-year yields jump to 5%. If you wanted to sell your bond at that point, what would you get for it?

You have a zero-coupon bond which will mature in 5 years at a par value of $110,408.08 (that's the bond's promise!). So, as current 5-year 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 zero-coupon 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 zero-coupon 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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skepticalobserver
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Re: Bond duration question

Post by skepticalobserver » Tue Jul 14, 2015 2:28 pm

ogd wrote:However, the loss is a little smaller and the gain a little quicker, both because of reinvestment effects.
Would this be due because of the effect of convexity?

I'm still learning about bonds.

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ogd
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Re: Bond duration question

Post by ogd » Tue Jul 14, 2015 2:34 pm

skepticalobserver wrote:
ogd wrote:However, the loss is a little smaller and the gain a little quicker, both because of reinvestment effects.
Would this be due because of the effect of convexity?

I'm still learning about bonds.
The "loss is a little smaller", yes.

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%.

skepticalobserver
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Re: Bond duration question

Post by skepticalobserver » Tue Jul 14, 2015 2:39 pm

Thank you.

As I understand it convexity "refines" the duration number.

alex_686
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Re: Bond duration question

Post by alex_686 » Tue Jul 14, 2015 2:52 pm

skepticalobserver wrote:Thank you.

As I understand it convexity "refines" the duration number.
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.

To be even more precise, graph the yield (x-axis) and bond price (y-axis). 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.

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Munir
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Re: Bond duration question

Post by Munir » Tue Jul 14, 2015 2:57 pm

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??
If it's a one-time 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.

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galeno
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Re: Bond duration question

Post by galeno » Tue Jul 14, 2015 3:01 pm

This is how to do the simple bond math:

http://socialize.morningstar.com/NewSoc ... px#3661030
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%.

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Re: Bond duration question

Post by abuss368 » Tue Jul 14, 2015 3:06 pm

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??
Hi Jackhenryport,

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."

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galeno
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Re: Bond duration question

Post by galeno » Tue Jul 14, 2015 3:51 pm

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.
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%.

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Re: Bond duration question

Post by abuss368 » Tue Jul 14, 2015 4:13 pm

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?
John C. Bogle - Two Fund Portfolio: Total Stock & Total Bond. "Simplicity is the master key to financial success."

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ogd
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Re: Bond duration question

Post by ogd » Tue Jul 14, 2015 4:35 pm

galeno wrote: NAV loss = 3(5.00 - 2.00) = 9.00%.
You got your numbers mixed and ended up multiplying two percentages. The OP is approximately correct.
abuss368 wrote:New yield is 5%. 3 years needed to recover.
Recovery to zero is a low standard. Recovery to the original 2% / year standard takes 5 years.

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galeno
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Re: Bond duration question

Post by galeno » Tue Jul 14, 2015 6:16 pm

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(5-2) = 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.
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%.

alex_686
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Re: Bond duration question

Post by alex_686 » Tue Jul 14, 2015 6:25 pm

galeno wrote:After 3 years, Bond B runs away from Bond A. I'm actually looking forward to increases in interest rates.
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.

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galeno
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Re: Bond duration question

Post by galeno » Tue Jul 14, 2015 6:32 pm

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%.

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ogd
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Re: Bond duration question

Post by ogd » Tue Jul 14, 2015 7:00 pm

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(5-2) = 9% loss in NAV.
Nope. Every 1% increase in interest rates will cause approximately duration decrease 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.

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Re: Bond duration question

Post by NightFall » Tue Jul 14, 2015 7:42 pm

longinvest wrote:Let's make this simple.

Let's assume that you put $100,000 in a zero-coupon 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 zero-coupon 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), 5-year yields jump to 5%. If you wanted to sell your bond at that point, what would you get for it?

You have a zero-coupon bond which will mature in 5 years at a par value of $110,408.08 (that's the bond's promise!). So, as current 5-year 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 zero-coupon 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 zero-coupon bond greatly simplifies calculations and allows to easily understand the behavior of a bond when yields change.
This is a great explanation.

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galeno
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Re: Bond duration question

Post by galeno » Tue Jul 14, 2015 8:37 pm

"Benz: Right. Earlier this summer, we did see a fair amount of volatility in more interest-rate-sensitive securities. Here, I would urge retirees to do that duration stress test that we've talked about before with their fixed-income 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 high-quality 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 one-year period in which rates rose by one percentage point."

Full article: http://www.morningstar.com/Cover/videoC ... ?id=705749
ogd wrote:
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(5-2) = 9% loss in NAV.
Nope. Every 1% increase in interest rates will cause approximately duration decrease 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%.

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ogd
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Re: Bond duration question

Post by ogd » Tue Jul 14, 2015 9:07 pm

galeno wrote:"Benz: Right. Earlier this summer, we did see a fair amount of volatility in more interest-rate-sensitive securities. Here, I would urge retirees to do that duration stress test that we've talked about before with their fixed-income 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 high-quality 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 one-year period in which rates rose by one percentage point."

Full article: http://www.morningstar.com/Cover/videoC ... ?id=705749
Ah, I see what she's saying. At first I was like, there's no way Christine Benz could make that kind of error.

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

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