By: Bob Collie, FIA, Chief Research Strategist, Americas Institutional
JANUARY 2012
The implications for bond prices of
changes in interest rates
Issue: There is a direct relationship between interest rates (the yield curve) and
bond prices. However, because a certain degree of interest rate change is already
priced into the market, the relationship is not as simple as “if yields go up, bond
prices fall, and vice versa”. How should investors take account of this as they make
investment decisions?
Response: If yields change exactly as priced into the forward curve, then all
bonds will deliver the same level of return. Pension plan liabilities (if calculated
using a mark-to-market approach) will likewise change by that same amount.
Therefore, when investors are considering a portfolio position (for example, a longor short-duration position relative to their liabilities), they should be comparing their
yield curve expectations not to the current yield curve, but to the future yield curve
that is implied by forward rates. At present (January 2012), that forward-implied
future yield curve is materially higher than the current yield curve.
Importantly, this means that when an investor’s belief is that interest rates are likely
to increase, it does not necessarily follow that they should take a short position
unless the expected increase is greater than the increase already priced into the
forward curve.
Russell Investments // The implications for bond prices of changes in interest rates
Investors…should
be comparing their
yield curve
expectations not to
the current yield
curve, but to the
future yield curve
that is implied by
forward rates.
Background
The forward-implied yield curve
Exhibit 1 shows the Citigroup Pension Discount Curve1 as of January 1, 2012. It implies
that the yield available over a one-year horizon is 1.15%; over two years, 1.26% per
year; and so on, up to 4.71% a year over a 30-year horizon.
Spot yield
Exhibit 1:
6%
4%
2%
0%
0
5
10
15
20
25
30
Term of bond (years)
What might that yield curve look like in one year, on January 1, 2013? In this note we will
focus mainly on one particular possibility, which we will refer to as the forward-implied
yield curve. It is shown in Exhibit 2.
Spot yield
Exhibit 2
6%
4%
2%
0%
0
5
10
15
20
Term of bond (years)
Today's yield curve
25
30
Forward-implied yield curve
Hypothetical analysis shown for illustrative purposes only and based on the assumptions outlined in this publication.
1
Source: the Society of Actuaries, “Professional Interests: Citigroup Pension Discount Curve,” at
http://www.soa.org/professional-interests/pension/resources/pen-resources-pension.aspx.
Russell Investments // The implications for bond prices of changes in interest rates
/ p2
This curve is derived as follows: Today’s two-year spot rate of 1.26% a year implies that
$1 will grow to $1.0254 over two years. Since today’s one-year spot rate implies that $1
will grow to $1.0115 in one year, the implied one-year spot rate at January 1, 2013, is
1.37%.2 Similar calculations enable forward-implied spot rates to be derived for other
maturities; see Appendix 1 for an illustration of this.
We note that the forward-implied curve is noticeably higher than today’s curve. For
example, the rate at 2 through 9 years is more than 50 basis points higher than the spot
rate at January 1, 2012.
This curve is, in effect, the state of the world one year hence that is implied by current
bond prices. If yields were to move in exactly this way, then every bond (no matter the
maturity) would return the same amount. In this example, that amount is 1.15% (the oneyear spot rate as of January 1, 2012.) Similarly, pension liabilities that are priced from
this yield curve would also increase by 1.15%.3 (These calculations are described in
more detail in Appendix 2).
Implications for choosing portfolio positions
This leads to the observation that yields could increase materially over the next 12
months, without bond portfolios falling in value. This is, perhaps, surprising at first look.
This result shows that a material rise in yields is priced into today’s market. Only if rates
increase faster than this would a loss arise. This means that portfolio positions taken on
a view of future interest rate changes must be based on a comparison of the expected
future yield curve to the forward-implied yield curve, not to the current yield curve; these
two yield curves can be substantially different.
A material rise in
yields is priced into
today’s market.
Only if rates
increase faster than
this would a loss
arise.
Such positions might be based, for example, on a view that there is a tendency for yields
to mean-revert (which would in turn lead to a view that rates are likely to move up from
levels that are low by the standards of recent history). It is also common to base a view
of future interest rate movements on analysis of monetary policy and of how that policy
might develop. Account might also be taken of increased demand among corporate
pension plans for long credit, a trend that was boosted by the Pension Protection Act.
Any of these perspectives might lead investors to decide to take active positions. The
key point of this note is that those active positions should be based on the investor’s
expectations relative to the forward-implied curve – i.e., relative to what is priced into the
market today – not relative to the current yield curve.
The asset portfolio will earn a return above (below) the one-year spot rate if the curve
moves to a level below (above) the forward-implied curve. Similarly, a short-duration
position causes a pension plan’s assets to earn a return that is higher than the increase
in the liabilities only if the curve moves above forward-implied rates. If rates rise less
than is priced into the market, then a short position would result in assets failing to keep
pace with liabilities.
The question is not whether rates rise or fall, but whether they rise or fall by more than
what is priced in.
It follows that any
prediction for the
future course of
interest rates must
include an expected
timeline.
It follows that any prediction for the future course of interest rates must include an
expected timeline. For example, the opinion that the five-year spot rate will rise by 1%
over one year (which is 0.40% more than implied by the forward curve) might lead to a
decision to reduce exposure. If the expected rate increase does come to pass, but over
two years rather than one, then that short position will result in a loss (because the
2
1.0115 x 1.0137 = 1.0254, so 1.37% is the rate required to turn $1.0115 into $1.0254.
3
Liabilities calculated using other than a mark-to-market methodology would behave differently.
Russell Investments // The implications for bond prices of changes in interest rates
/ p3
increase is less than that implied by the forward-implied curve at a two-year horizon).
The hoped-for profit is not merely delayed by the wrong timing, it is turned into a loss.
Decomposing the effect of yield curve changes
One way to look at this effect is to consider the various components of the return to a
bond portfolio. For example, when we compare a long bond to a short bond, the returns
generally differ, because of:
a)
a difference in yield (a higher yield to the long bond when the yield curve is
upward-sloping);
b) greater sensitivity of the long bond to a change in yields; and
c)
different degrees of change in the long and short yields.
It is when yields evolve according to the forward curve that these differences exactly
balance out.
For example, the initial yield-to-maturity of a five-year bond is 2.44%, and the forwardimplied scenario implies that this changes to 2.81% over the course of twelve months.
The return of 1.15% can therefore be approximately decomposed into:
a)
a yield of 2.81%4; and
b) a loss of roughly 1.6% due to the rise in yields.
i.e. Using the approximation5 that
Return ≈ yield
+ duration x change in yield, then:
1.15% ≈ 2.81% +
4.39 x (2.44%-2.81%).
In contrast, the initial yield-to-maturity of the thirty-year bond is 4.33%, rising to 4.55%
twelve months out in the forward-implied scenario. The return of 1.15% can therefore be
approximately decomposed into:
a)
a 4.55% yield; and
b) a loss of roughly 3.3% due to the rise in yields.
Or: 1.15%
≈ 4.55% +
14.91 x (4.33%-4.55%).
4
The ending yield is used in this version of the formula in order to be consistent with the use of the initial
duration of the bond. This implies that, for the purposes of applying this formula, the change in rates occurs at
the start of the year.
5
This relationship is an approximation because the true impact of a change in yield is non-linear and because
the effects are not additive, but rather compound each other.
Russell Investments // The implications for bond prices of changes in interest rates
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The impact of other potential changes in the yield curve
We show below the returns over the next twelve months on bonds with maturities of one
through 30 years, in various scenarios.
(A) YIELD CURVE MOVES IN LINE WITH THE FORWARD-IMPLIED CURVE
We start by showing the effect – described above – of rates riding the forward curve. At
left we compare what this implies about the yield curve on January 1, 2013, and at right
the impact on bond returns and on the increase in a typical pension plan’s liabilities. As
described above, all bonds would give the same return.6
Exhibit 3a and 3b:
6%
12-month return
Spot yield
6%
4%
2%
4%
2%
0%
0%
0
5
10
15
20
25
1
30
3
5
7
9 11 13 15 17 19 21 23 25 27 29
Term of bond (years)
Term of bond (years)
Hypothetical analysis shown for illustrative purposes only and based on the assumptions outlined in this publication.
(B) NO CHANGE IN THE YIELD CURVE
Next, we show the effect of yields being unchanged over the 12-month period. Shorterduration bonds would deliver lower returns than longer-duration, while pension plan
liabilities would move up by roughly 4.4%.
Exhibit 4a and 4b:
6%
12-month return
Spot yield
6%
4%
2%
4%
2%
0%
0%
0
5
10
15
20
25
30
Term of bond (years)
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29
Term of bond (years)
Hypothetical analysis shown for illustrative purposes only and based on the assumptions outlined in this publication.
6
Note that while the forward-implied curve captures the degree of change that is priced into the market today, this is not exactly the
same as saying that it reflects the market’s expectations. A slightly upward-sloping curve, for example, might result from investors
demanding a risk premium or an illiquidity premium for investing in long bonds, rather than necessarily from an expectation of future
rises.
Russell Investments // The implications for bond prices of changes in interest rates
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(C) CURVE FLATTENS TO BE 4% AT ALL DURATIONS
If the yield curve were to flatten to be 4% at all durations, then it would be above the
forward-implied curve for terms of less than 11 years, while for 12-year and longer terms
it would be below. Bonds with a duration less than 12 years (which implies a term of
some 23 years) would fail to keep pace with the increase of 6.6% in the typical plan’s
liability.
Exhibit 5a and 5b:
12%
12-month return
Spot yield
6%
4%
2%
8%
4%
0%
-4%
0%
0
5
10
15
20
25
1
30
3
5
7
9 11 13 15 17 19 21 23 25 27 29
Term of bond (years)
Term of bond (years)
Hypothetical analysis shown for illustrative purposes only and based on the assumptions outlined in this publication.
(D) CURVE MOVES TO A LEVEL HALF A PERCENT BELOW THE FORWARD-IMPLIED
CURVE AT ALL POINTS
Next, we show the pattern that would result from the yield curve moving to half a percent
below the forward-implied curve. This would equate to little change from current levels at
shorter durations, but to a drop at longer durations. The pension plan liability would
move up by 7.4%. Once again, only bonds with maturities of 23 years or longer (i.e.,
durations of greater than 12 years) would outpace this growth.
Exhibit 6a and 6b:
10%
12-month return
Spot yield
6%
4%
2%
8%
6%
4%
2%
0%
0%
0
5
10
15
20
25
30
Term of bond (years)
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29
Term of bond (years)
Hypothetical analysis shown for illustrative purposes only and based on the assumptions outlined in this publication.
Russell Investments // The implications for bond prices of changes in interest rates
/ p6
(E) CURVE MOVES TO A LEVEL HALF A PERCENT ABOVE THE FORWARD-IMPLIED
CURVE AT ALL POINTS
Finally, if the yield curve were to move to a level half a percent above the forwardimplied curve, then short positions would deliver gains relative to the liability, which
would fall in value by 4.5%.
Exhibit 7a and 7b:
2%
12-month return
Spot yield
6%
4%
2%
0%
-2%
-4%
-6%
0%
0
5
10
15
20
25
1
30
3
5
7
9 11 13 15 17 19 21 23 25 27 29
Term of bond (years)
Term of bond (years)
Hypothetical analysis shown for illustrative purposes only and based on the assumptions outlined in this publication.
Summarizing, the table below shows – for each of the five scenarios – the change in our
typical plan’s liabilities; the return on a five-year bond; the return on a 30-year bond; and
the change in funded status (relative to a 100% starting point) that would result from
investing entirely in either of those instruments.
Exhibit 8:
Scenario
Liability change
Follow forward-implied curve
+1.2%
No change
+4.4%
Flatten to 4% all durations
+6.6%
Move to ½% below forward-implied
+7.4%
Move to ½% above forward-implied
–4.5%
5-year bond return
(funded status change)
+1.2%
(0.0%)
+4.0%
(–0.4%)
–2.8%
(–8.8%)
+2.9%
(–4.2%)
–0.5%
(+4.1%)
Russell Investments // The implications for bond prices of changes in interest rates
30-year bond return
(funded status change)
+1.2%
(0.0%)
+4.5%
(0.0%)
+9.6%
(+2.9%)
+8.6%
(+1.1%)
–5.5%
(–1.1%)
/ p7
Appendix 1: Anatomy of the forward curve
The relationship between a forward rate and the current spot rate is illustrated below.
Exhibit 9:
1. The n-year spot rate is the
rate used by the market to
put a price on future
cashflows
2. The (1 year) forward rate is the
n-1 year spot rate that is implied
1 year from now, to produce the
same price today
3. It follows that if a return equal to
the 1-year spot rate is earned, then
the spot rate one year from now will
be the forward rate
1-year
spot rate
4. Equivalently, if the spot rate one year
from now is the forward rate, then the
return earned in the next year must equal
the 1-year spot rate
Bonds and pension liabilities are simply a series of future cash flows, and their price is
calculated by discounting those cash flows at the discount rate applying to the
appropriate time horizon.
So a cash flow, say C, due to occur n years in the future is priced as:
C/(1+yn)n
In one year, that same cashflow will be one year closer to being due and will be priced at
C/(1+zn-1)n-1, where zn-1 is the spot rate at that time.
But note how the forward rate is calculated. It is based on the relationship:
(1+fn-1)n-1 = (1+yn)n / (1+y1)
From this it follows that if zn-1 (the spot rate in one year) is equal to fn-1 (the rate implied
by the forward curve) then the price in one year is equal to the price today multiplied by
1+y1.
In other words, the forward-implied rate is the rate which makes all bonds return the one
year spot rate or, equivalently, the rate that results if every bond earns that same return.
Russell Investments // The implications for bond prices of changes in interest rates
/ p8
Appendix 2: Details of the calculations
The bond returns shown in this paper assume a 6% coupon, paid annually at the end of
each year. It is assumed that there are no defaults. Bonds are bought and held
throughout the period of analysis.
Thus, if the initial one-year spot rate is y1, the initial two-year spot rate y2, and so on,
then the price of an n-year bond today is calculated as:
6/(1+y1) + 6/(1+y2)2 + … + 106/(1+yn)n.
If, in twelve months, the one-year spot rate is z1, the two-year spot rate z2, and so on,
then the price at that one-year horizon (immediately prior to the payment of the first
coupon) is calculated as:
6 + 6/(1+z1) + … + 106/(1+zn-1)n-1.
Calculation of liabilities is based on a representative set of assumed future cash flows c1,
c2, … (the first 30 years of which are shown below), with an initial liability duration of 12
years. Benefit payments, accruals of new benefits and contributions are all assumed to
be zero.
Exhibit 10:
70
60
50
40
30
20
10
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
Hypothetical analysis shown for illustrative purposes only and based on the assumptions outlined in this publication.
The initial liability value is calculated as:
 ci / (1+yi)i.
The value after one year (immediately prior to the first benefit payment) is calculated as:
 ci /(1+zi-1)i-1.
Russell Investments // The implications for bond prices of changes in interest rates
/ p9
A note on assumptions regarding buy-and-hold and accrual of new benefits: if we were
to change the assumption of a buy-and-hold strategy, and assume instead that bonds
are rebalanced in order to maintain duration,7 results would be similar but not identical,
since there would be frequent trading of positions – and the outcome would therefore
depend not only on the yield curve at the end of the year, but also on how it changed
over the course of the 12 months. Under the assumption of a smooth change in yields
over the course of the year, the return for each maturity of bond would lie between the
return shown in the main body of this paper for that maturity and the return shown for a
bond with a maturity of one year longer.
Likewise, if we were to change our assumption from a frozen plan to an open plan (i.e.,
assume that benefits continue to accrue – with contributions being made equal to the
service cost), results would be similar to those shown in the main body of this paper. The
liability changes in each of the five scenarios would be roughly: (a) 1.15%; (b) 4.6%; (c)
7.2%; (d) 7.9%; and (e) –4.9%.
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Call Russell at 800-426-8506 or
visit www.russell.com/institutional
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appropriateness of any investment, nor a solicitation of any type. The general information contained in this publication should not be
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First used: January 2012
USI-12116-01-14
7
This is roughly equivalent to assuming that short holdings roll out of the portfolio and new bonds are purchased
at longer durations to maintain a constant duration.
Russell Investments // The implications for bond prices of changes in interest rates
/ p 10