Bond Duration education

Discuss all general (i.e. non-personal) investing questions and issues, investing news, and theory.
Post Reply
workerbeeengineer
Posts: 35
Joined: Thu Jun 02, 2016 12:22 am

Bond Duration education

Post by workerbeeengineer » Fri Sep 22, 2017 11:28 pm

BHs, requesting a little education on bond duration as follows: I think I get the basics...duration is a measure of a bond or bond fund interest rate sensitivity. Where I need education: does the rate of change of prevailing interest rates higher (or lower) have any impact on much a bond will lose (or gain) in value? In other words, does a long gradual trend up in interest rates versus a sudden spike up in rates make any difference in how much a bond would lose in value. I'm guessing the answer is no...however we got from interest rate point A to higher interest rate point B, a bond will have decreased in value the same amount. Thanks.

VaR
Posts: 386
Joined: Sat Dec 05, 2015 11:27 pm

Re: Bond Duration education

Post by VaR » Sat Sep 23, 2017 6:59 pm

Duration is an instantaneous measure of first order sensitivity to a parallel shift in the yield curve, so there are probably four effects that you will want to learn about:
1. As time passes, the duration of a bond goes down. So a bond with a duration of 5 years will have a duration of 4 years after a year.
2. At the end of a period, it doesn't matter whether rates changed at the begining, at the end, or gradually over the year. The end market value of the bond will be the same.
3. Most of the time, the yield curve is upward sloping so as time passes, a bond's price will tend to increase slightly over time.
4. As a bond approaches maturity, its price will tend to move toward par. Note that you can't "take advantage" of this because this effect is already priced in.

But besides just educating yourself, why do you want to know? Is there some decision you're thinking of making that depends on this knowledge? Unless you're a bond trader, this information is unlikely to be useful. :)
Last edited by VaR on Sat Sep 23, 2017 10:15 pm, edited 1 time in total.

User avatar
grabiner
Advisory Board
Posts: 20879
Joined: Tue Feb 20, 2007 11:58 pm
Location: Columbia, MD

Re: Bond Duration education

Post by grabiner » Sat Sep 23, 2017 9:30 pm

It is simplest to see the effect with a zero-coupon bond. Suppose that a zero-coupon bond is currently 10 years from maturity and has a 3% yield with a par of $10,000; the bond is now worth $7441. In two years, if the yield becomes 5%, the value will be $6768, regardless of what the yield did along the way.

Individual bonds with coupon payments will behave similarly, even though each coupon should really have a different yield. If the yield on a coupon bond is 5% two years from now, that determines its value.

However, bond funds change their holdings over time, and thus the timing of interest-rate moves affects the return. If you hold a bond fund which keeps a constant 5-year duration, with a 3% yield, and the yield rises to 5% tomorrow, the fund will lose 10%, but it will then earn 5% over the two following years to break even (actually a small loss because of compounding). In contrast, if the fund keeps its 3% yield for two years, and then has its yield rise suddenly to 5% two years from now, you will have only earned 6% over the two years, and will then lose 10%. If the yield rises steadily over two years, you will earn 8% over the two years from the yield and lose 10% to the decreased bond price.
David Grabiner

User avatar
#Cruncher
Posts: 2420
Joined: Fri May 14, 2010 2:33 am
Location: New York City
Contact:

Re: Bond Duration education

Post by #Cruncher » Sat Sep 23, 2017 9:33 pm

VaR wrote:
Sat Sep 23, 2017 6:59 pm
3. Most of the time, the yield curve is upward sloping so as time passes, a bond's price will tend to increase slightly over time.
4. As a bond approaches maturity, its price will tend to move toward par.
For a coupon bond these two trends may pull in opposite directions. Which one predominates depends on the size of the coupon, the yield curve, and the time remaining to maturity. For a zero-coupon bond, however, the two trends reinforce each other. Here is a listing of the year-by-year price changes of two hypothetical 10-year bonds each with an initial yield-to-maturity of 2.26%: one has a 2.26% coupon and the other has no coupon. The yield curve is taken from the Daily Treasury Yield Curve for 9/22/2017.

Code: Select all

Years to          ---- Price ----     - Cum Return -
 Mature   Yield    2.26%     Zero     2.26%     Zero
     10   2.26%   100.00    79.97
      9   2.20%   100.49    82.21     2.75%    2.80% [*]
      8   2.15%   100.80    84.35     2.66%    2.70%
      7   2.10%   101.03    86.46     2.60%    2.63%
      6   1.99%   101.51    88.85     2.62%    2.67%
      5   1.88%   101.80    91.11     2.60%    2.64%
      4   1.73%   102.03    93.37     2.58%    2.61%
      3   1.58%   101.98    95.41     2.52%    2.55%
      2   1.46%   101.57    97.14     2.44%    2.46%
      1   1.30%   100.95    98.72     2.36%    2.37% [*]
      0           100.00   100.00     2.26%    2.26%
For the 2.26% coupon bond's price, point # 3 predominates until three years before maturity when point # 4 takes over and the price starts to fall. For the zero-coupon bond, however, the price rises the entire ten years. Note how the rising yield curve improves the cumulative return for both bonds every year until maturity -- assuming it does not change during the ten years.

* Example price and cumulative return computation using the Excel PV and RATE functions:

Code: Select all

Price
   9: 100.49 = -PV(2.2%, 9, 2.26, 100, 0)
       82.21 = -PV(2.2%, 9, 0,    100, 0)
   1: 100.95 = -PV(1.3%, 1, 2.26, 100, 0) = 102.26 / 1.013
       98.72 = -PV(1.3%, 1, 0,    100, 0) = 100.00 / 1.013
Cumulative Return
   9:  2.75% = RATE(1, 2.26, -100.00, 100.49, 0)
       2.80% = RATE(1, 0,     -79.97,  82.21, 0)
   1:  2.36% = RATE(9, 2.26, -100.00, 100.95, 0)
       2.37% = RATE(9, 0,     -79.97,  98.72, 0)

workerbeeengineer
Posts: 35
Joined: Thu Jun 02, 2016 12:22 am

Re: Bond Duration education

Post by workerbeeengineer » Sun Sep 24, 2017 9:30 am

Thanks VaR, Grabiner, and #Cruncher for the education. To answer VaR's question - just education...no big bond related decisions looming. Mostly thinking about duration as it applies to bond funds. Grabiner's example scenarios were helpful - thanks again.

VaR
Posts: 386
Joined: Sat Dec 05, 2015 11:27 pm

Re: Bond Duration education

Post by VaR » Sun Sep 24, 2017 1:10 pm

grabiner wrote:
Sat Sep 23, 2017 9:30 pm
However, bond funds change their holdings over time, and thus the timing of interest-rate moves affects the return. If you hold a bond fund which keeps a constant 5-year duration, with a 3% yield, and the yield rises to 5% tomorrow, the fund will lose 10%, but it will then earn 5% over the two following years to break even (actually a small loss because of compounding). In contrast, if the fund keeps its 3% yield for two years, and then has its yield rise suddenly to 5% two years from now, you will have only earned 6% over the two years, and will then lose 10%. If the yield rises steadily over two years, you will earn 8% over the two years from the yield and lose 10% to the decreased bond price.
In scenario 1 with the immediate rise in rates to 5%, I think the increase in yield to 5% over the two years is off of a base that is 10% lower, so you're actually receiving $4.50 a year based on an original investment of $100. But the concept is still correct in that your total return over two years is $9 - $10 = loss of $1 vs scenario 2 where $6 - $10 = loss of $4 per $100 where rates rise by 2% at the end of the period.

Note that in scenario 1 for an individual bond held for year 1, assuming that you bought a 5.5 year par bond (duration=5) at the start and held them for a year, that the price of the bond would immediately drop to 90 (cents on the dollar) but rise 1/5.5 or 1.8 (cents on the dollar) after the first year due to the pull-to-par effect. So for a $100 investment you get $3 in interest, $10 mtm loss in price at the beginning of the period, but a $2 mtm gain in price by the end of the period. Note that at the end of the period you are holding a 4.5 year bond (duration = 4.1) with a market price of 92, having received $3 in interest from coupons.

At the end of year 2 you expect to be holding a 3.5 year bond (duration = 3.2) with a market price of 94, having received a total of $6 in interest from coupons. Your total return should be $0.

Post Reply