Why are short term bond funds less interest rate sensitive and long term bond funds more sensitive to increasing interest rates?

Thank you Bogleheads.

## Quick question about bond funds

### Re: Quick question about bond funds

I started to look for a quick and easy reference to this and found plenty of stuff, including the Wiki, in which duration is defined as

"The formal definition of duration is the average value of the time to each future payment, . . ."

and the statement is made that

" The duration of a bond, or a bond fund, is a measure of its sensitivity to interest rates."

But I didn't find anything in that article, or in other articles, that shows the mathematics for why the second statement follows from the definition.

Maybe someone can find an exposition of this theorem.

"The formal definition of duration is the average value of the time to each future payment, . . ."

and the statement is made that

" The duration of a bond, or a bond fund, is a measure of its sensitivity to interest rates."

But I didn't find anything in that article, or in other articles, that shows the mathematics for why the second statement follows from the definition.

Maybe someone can find an exposition of this theorem.

### Re: Quick question about bond funds

Because the have a much shorter term.

The short term bonds, say 1 - 3 years, offer much lower interest because they end or turn over so quickly and new ones can adjust to interest rate changes very quickly.

Long term bonds offer higher interest rates, but they're locked in for 15 - 20 years... a lot can change in in that amount of time.

Just a simple explanation and ignoring potential selling of long term bonds and all. Basically, more reward equals more risk.

The short term bonds, say 1 - 3 years, offer much lower interest because they end or turn over so quickly and new ones can adjust to interest rate changes very quickly.

Long term bonds offer higher interest rates, but they're locked in for 15 - 20 years... a lot can change in in that amount of time.

Just a simple explanation and ignoring potential selling of long term bonds and all. Basically, more reward equals more risk.

### Re: Quick question about bond funds

It's bonds in general, not just bond funds.

Here's a simplified example: suppose I have $100 of a bond paying 1% / year coupons, and market yields (not just "interest rates", which are generally understood to be zero-term Fed rates -- but yield for that specific type of bond) increase from that 1% to 2%.

If the bond was 3 year, the total payments from my bond will be $103, whereas a fresh bond will pay $106. So the price of my bond will drop to about $97 to compensate, to make my bond competitive -- a buyer can get the same $6 profit by a buying a new bond ($106 - $100) or buying 1.03 units of the old bond ($103 * 1.03 - $97 * 1.03). Any higher and my bond would be a bad deal; any lower and my bond will be too good of a deal.

If the bond was 10 year, the total payments from my bond will be $110 and a fresh bond will pay $120 over its lifetime. The price of my bond drops to roughly $91.6 to compensate -- a buyer can get the same $20 profit by buying a new bond ($120 - $100), or by buying 1.09 units of the old bond ($110 * 1.09 - $91.6 * 1.09).

So the 10 year bond declined 3x more than the 3 year bond, in response to the same move. Basically, being stuck for 10 years at a lower yield is worse than being stuck for only 3 years, and it costs you more to get out of that contract.

This is simplified math and ignores dividend reinvestment -- you can look up articles like http://www.investopedia.com/articles/bo ... _yield.asp for the full formulas. But this is the gist of it.

Funds are simply a large collection of such bonds, some shorter some longer, but the average bond for a long fund is substantially longer than for a short fund, so the price drop is steeper.

Here's a simplified example: suppose I have $100 of a bond paying 1% / year coupons, and market yields (not just "interest rates", which are generally understood to be zero-term Fed rates -- but yield for that specific type of bond) increase from that 1% to 2%.

If the bond was 3 year, the total payments from my bond will be $103, whereas a fresh bond will pay $106. So the price of my bond will drop to about $97 to compensate, to make my bond competitive -- a buyer can get the same $6 profit by a buying a new bond ($106 - $100) or buying 1.03 units of the old bond ($103 * 1.03 - $97 * 1.03). Any higher and my bond would be a bad deal; any lower and my bond will be too good of a deal.

If the bond was 10 year, the total payments from my bond will be $110 and a fresh bond will pay $120 over its lifetime. The price of my bond drops to roughly $91.6 to compensate -- a buyer can get the same $20 profit by buying a new bond ($120 - $100), or by buying 1.09 units of the old bond ($110 * 1.09 - $91.6 * 1.09).

So the 10 year bond declined 3x more than the 3 year bond, in response to the same move. Basically, being stuck for 10 years at a lower yield is worse than being stuck for only 3 years, and it costs you more to get out of that contract.

This is simplified math and ignores dividend reinvestment -- you can look up articles like http://www.investopedia.com/articles/bo ... _yield.asp for the full formulas. But this is the gist of it.

Funds are simply a large collection of such bonds, some shorter some longer, but the average bond for a long fund is substantially longer than for a short fund, so the price drop is steeper.

### Re: Quick question about bond funds

Someone else may comment, although this is usually true, but this is not always the case. Interest rates (from very short term to very long term) do not always move together and, in fact, at times may move in opposite directions. So, for example, if short term rates spike up a short term bond will lose "value" and if, at the same time, long term rates drop, a long term bond will gain value.

- Phineas J. Whoopee
**Posts:**6656**Joined:**Sun Dec 18, 2011 6:18 pm

### Re: Quick question about bond funds

dm200 wrote:Someone else may comment, although this is usually true, but this is not always the case. Interest rates (from very short term to very long term) do not always move together and, in fact, at times may move in opposite directions. So, for example, if short term rates spike up a short term bond will lose "value" and if, at the same time, long term rates drop, a long term bond will gain value.

The observed facts of the world agree with you, and I'll go even further if I may substitute the more tightly-defined term yield. It is seldom the case that yields of similar-quality bonds at all maturities move the same amount, or even proportionally per ogd's very illustrative though as-stated simplified math.

Yes, they can move in opposite directions. Furthermore, and I'm not in the habit of changing my asset allocation because of economic forecasts, sometimes there's even an inverted yield curve, in which short-term bonds yield more than similar longer-term ones. Investopedia, in its noncommittal passive voice, says it "is considered to be a predictor of economic recession," without identifying anybody in particular who thinks of it that way.

I can't disagree with Investopedia's direct claim, that there exist people out there who consider it to be that. One of the problems is it has a track record of predicting more recessions than occur.

PJW