## Bond duration question

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Topic Author
Jackhenryport
Posts: 26
Joined: Sat Jul 02, 2011 5:48 pm

### 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??

ogd
Posts: 4875
Joined: Thu Jun 14, 2012 11:43 pm

### 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.

longinvest
Posts: 4085
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 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.
Bogleheads investment philosophy | One-ETF global balanced index portfolio | VPW

skepticalobserver
Posts: 1024
Joined: Tue Jul 29, 2014 11:29 am

### Re: Bond duration question

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?

ogd
Posts: 4875
Joined: Thu Jun 14, 2012 11:43 pm

### Re: Bond duration question

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?

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
Posts: 1024
Joined: Tue Jul 29, 2014 11:29 am

### Re: Bond duration question

Thank you.

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

alex_686
Posts: 5151
Joined: Mon Feb 09, 2015 2:39 pm

### Re: Bond duration question

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.

Munir
Posts: 2566
Joined: Mon Feb 26, 2007 4:39 pm
Location: Oregon

### Re: Bond duration question

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.

galeno
Posts: 1576
Joined: Fri Dec 21, 2007 12:06 pm

### Re: Bond duration question

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%.

abuss368
Posts: 17043
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

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."

galeno
Posts: 1576
Joined: Fri Dec 21, 2007 12:06 pm

### Re: Bond duration question

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: 17043
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?
John C. Bogle - Two Fund Portfolio: Total Stock & Total Bond. "Simplicity is the master key to financial success."

ogd
Posts: 4875
Joined: Thu Jun 14, 2012 11:43 pm

### Re: Bond duration question

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.

galeno
Posts: 1576
Joined: Fri Dec 21, 2007 12:06 pm

### 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(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
Posts: 5151
Joined: Mon Feb 09, 2015 2:39 pm

### Re: Bond duration question

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.

galeno
Posts: 1576
Joined: Fri Dec 21, 2007 12:06 pm

### 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%.

ogd
Posts: 4875
Joined: Thu Jun 14, 2012 11:43 pm

### Re: Bond duration question

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.

NightFall
Posts: 269
Joined: Wed Mar 12, 2014 4:38 pm

### Re: Bond duration question

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.

galeno
Posts: 1576
Joined: Fri Dec 21, 2007 12:06 pm

### Re: Bond duration question

"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."

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%.

ogd
Posts: 4875
Joined: Thu Jun 14, 2012 11:43 pm

### Re: Bond duration question

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."