Interest rates ; Government securities ; Risk

Brian Dvorak

Joseph Bisignano

English

Research Papers in Economics

April 3, 1981 
Risk and Duration 
During the last year and a half, individuals 
have lived with considerably more interest-
rate variability than they had experienced in 
the previous twenty years. The three-month 
Treasury bill rate in 1980 fluctuated between 
a high of 16.7 percent and a low of 6.5 per-
cent-about a 1  O-percentage-point spread, 
compared with a high-low spread of less than 
4 percentage points in 1979. Moreover, long-
term interest rates were over fou r ti mes more 
variable in 1980 than they were (on average) 
during the 1960-79 period, on the basis of 
rate movements in the representative 20-year 
Treasury bond (Chart 1). 
The reasons for the increased interest-rate 
variability-especially for long-term rates-
still remain somewhat of a puzzle. Rather 
than hold a postmortem on why interest rates 
were so variable in 1980, let us consider the 
relationship between interest-rate variability 
and "risk," and the relationship between risk 
and the maturity of u.s. government debt. 
The recent change in these relationships sug-
gests that  the average matu rity of  government 
debt may be a misleading indicator of  the 
availability of long-term debt to the public. 
Secondly, the change suggests that the risk 
of holding government bonds in terms of 
price variability depends on an assessment 
of  a bond's "duration," and not simply its 
maturity. 
Risk factors 
The first question facing an investor in a debt 
security is whether to purchase a short- or 
long-term asset. In answering this question, 
the individual must balance two types of 
risk -price risk and coupon risk. Price risk is 
simply the risk that the market value of  the 
security will change while the individual 
holds the asset, given the chance that he may 
want/have to sell the security before maturity. 
Consider, for example, the individual who 
purchased a three-year Treasury security in 
July 1980 at 9.3 percent, and sold it in De-
cember when its yield rose to 13.7 percent. 
Since bond yield and price are inversely re-
lated, the investor in that case would have 
suffered a substantial capital loss. 
Coupon risk, on the other hand, is the risk a 
short-term investor faces when reinvesting 
his money at some uncertain interest rate in 
the future. It would be to the individual's 
advantage to know what the trade~off is be-
tween coupon risk and price risk, since the 
two move inversely to one another (a rise in 
interest rates reduces current mar.ket price but 
increases the return on reinvested funds). This 
requires, however, that the investor estimate 
what interest rates might be in the future. 
Yield curve 
To develop such an estimate, economists 
often focus on the "term structure" of  interest 
rates-the relationship between the market 
yield on a security and the number of  years to 
maturity. Typically economists focus on Trea-
sury debt, since its yield is not affected by the 
risk of  default. An efficient market generally 
wi  II provide approximately the same yield for 
a long-term bond as for a sequence of short-
term securities held over the same time per-
iod, assuming that investors do not demand 
any premium for holding a less liquid long-
term security rather than a more liquid short-
term asset. 
Some economists argue that the current long-
term bond rate is an average of current and 
expected future short-term rates. Hence, a 
rising yield curve implies that the market 
expects short-term rates to rise in the future. 
That is, if long-term rates are higher than 
short-term rates, expected future short-term 
rates have to be higher than current short-
term rates to preserve the equality of returns 
between the two investments. This so-called 
"expectation theory" of the term structure is 
widely employed by financial analysts and 
economists who compute "forward rates"-
implied future short-term interest rates-from 
the relationship between current short-term IF~cl~lf([\ll ~~~~1fW~ 
rEb ~\  1l~  Jk  ~  TI 
&  r"ii.\  17\\  TIP  '):4 ~\ D"0 (Ie' ~ ~  ((V ((\\  ~~ww  1J,  C9-0.u~,-jl..l~~~ 
0.30 
0.10 
Chart 1 
Twenty·  year Bond  Rates 
Standard Deviation' 
OpirlioflS expreSS2ci  ir~  this iie\,\'s!etter clo  not 
necessari:y refiect the vie"vv5  of the management 
of the Federal Reserve Bank of San  Francisco, 
nor of the Board of Governors of the Federai 
Reserve Systern. 
1968  1972  1976 
*Standard deviation in weekly average change in U.s. 
Government 20-year bond yield, calculated for each 
annual period 1960-1980. 
and long-term interest rates. The yield curve 
thus contains information on the expected 
movement of short-term interest rates. 
In evaluating the potential price risk of hold-
ing long- versus short-term securities, many 
economists argue that for a given yield 
change a bond price change generally wi  II be 
greater, the greater is the time to maturity. 
This, however, is not always the case. More-
over, the change in price due to a change in 
yield, for a given maturity, will not always be 
the same. This means that the prices of long-
er-term securities are not always more sensi-
tive than prices of shorter-term securities to a 
given change in interest rates. The relation-
ship between price change and the move-
ment in interest rates can be misunderstood, 
because the actual maturity 6f a bond only 
tells the investor when the final payment is 
due, and conveys little information on the 
time pattern of payments to be received over 
the life of  the bond. We need to recall thatthe 
higher the market yield, the more important 
to the bond's present value are the coupon 
payments relative to the final payment at 
maturity. This information is ignored if  we 
focus only on the yield-maturity relationship. 
Macaulay's duration 
In his monumental study of interest rates, 
Frederick R. Macaulay (1938) attempted to 
distill a bond time structure from a com-
parison of bonds with different maturities, 
coupon payments and market yields. To ac-
complish this task, Macaulay introduced the 
concept of "duration"  -the  average life of a 
debt i  nstru ment where the ti me between the 
present and each future payment is weighted 
by the present value of  the respective coupon 
or principal payments, relative to the bond's 
current price. Duration thus measures the 
weighted average of  the future periods during 
which the bond generates coupon and prin-
cipal payments, where the weights are the 
marginal present-value contributions of all 
payments received. 
Consider, as an example, the case where 
interest rates rise.  In that case, the present 
2 
value of a future payment falls, and the pres-
ent value of a payment far in the future falls 
relatively more than a near-term payment. 
Hence the bond's duration falls. Similarly, 
when interest rates fall, the bond's duration 
lengthens. Analysts miss this information on a 
bond's ti me profi Ie if  they consider simply the 
bond's maturity or term structure of rates. 
The advantage of focusing on duration rather 
than maturity can be summarized with one 
simplepoint. For a given change in interest 
rates, the price of a bond will vary propor-
tionately (and inversely) with its duration but 
not proportionately with its maturity. For ex-
ample, if  a bond has a duration of  five years, a 
one-percentage point increase in market 
yield from 10 to 11  percent wi  II resu It ina 
decline'in market price equal to the duration 
divided by one plus the initial market yield, or 
about 4.5 percent.* Thus, price risk and dur-
ation are intimately linked-as economists 
and financial analysts are only now begin-
ning to appreciate. 
Since duration lengthens when interest rates 
fall dnd shortens when interest rates rise, we 
may paraphrase Gertrude Stein's comment 
on roses to say that a one-percentage point 
change in interest rates is not a one-
percentage point change in interest rates. 
That is, the same change in interest rates 
generates a larger change in market price 
when interest rates are low than when th~y~ 
are high, because a given bond will have a 
longer duration at low than at high interest 
rates. Thus to understand the riskiness of a 
bond, an investor must know its duration ~nd 
not simply its maturity. 
Macaulay's duration concept is valuable be-
cause it tells an investor how to convert a 
risky asset into essentially a much less risky 
*Algebraically this may be written as: 
Change in bond price (%)  = 
Duration 
- --- x  (Change'in market yield) 
(1  + r) 
where r is the initial market yield; change in market 
yield in basis points divided by 100. 
198( r  r  ( 
Yield to  Yield vs.  Maturity 
maturity(%) 
16 
14 
12 
)2/31180 
10  r9128179 
8 
O'  I  I  I  I  I  I  I  I  I  I 
0  4  8  12  16  20  24  28 
Maturity  (years) 
asset, by balancing price risk against coupon 
reinvestment risk. Abstracting from default 
risk and taxes, an investor can "immunize" 
his rate of  total return from effects of unex-
pected interest-rate movements if he chooses 
a portfolio whose duration (and not maturity) 
exactly equals his desired holding period. 
The investor with an immunized portfolio 
balances the cost of  suffering a decline in the 
prices of his assets with the benefit of being 
able to reinvest in higher earning assets-or 
inversely, if interest rates decline. By acting in 
this way, the investor guarantees that the 
yield expected at the time of  the initial in-
vestment is actually obtained when the port-
folio is liquidated. 
The duration concept is readily applicable to 
a financial institution's risk sensitivity. The 
institution whose assets have a longer dura-
tion than its liabilities will lose when interest 
rates rise and gain when interest rates fall. The 
institution's risk thus depends on the time 
profile-duration-of its balance sheet. An 
institution can insulate itselffrom movements 
in interest rates if its asset- and liability-
payment streams have the same weighted-
average duration. 
Duration and term structure 
Consider now the effect of interest-rate 
movements on the yield-maturity (Chart 2) 
and yield-duration (Chart 3) relationships for 
Treasu ry secu rities. (These charts plot  data for 
all Treasury securities outstanding except 
bills and "flower bonds," special Treasury 
securities with estate-tax benefits.) Chart 2 
shows a similar-shaped yield-maturity struc-
ture,both downward sloping, for the dates 
September 28, 1979, and December 31, 
1980. The term structure between those 
two dates appears to have been "lifted" by 
about 3 % percentage points. Also, the term-
structure relationship for December 1980 
appears sharply downward sloping, possibly 
, suggesti ng an expectation of a rapid down-
turn in short-term interest rates. In contrast, 
3 
, 
Yteld to  Yield vs. ~ 
maturity  (%) 
12/31/80 
r 
9/28179 
0;  ,  ,  !  I  , 
0  2  4  6  8  10 
Duration (years) 
the yield-duration relationship (Chart 3) 
shows a much shorter and slightly steeper 
yield curve, particularly in late 1980. Be-
tween September 1979 and December 1980, 
the longest duration bond went from about 
10.4 to 9.0 years. This reduction in duration 
implies that, for long-term bonds, a one-
percentage point change in interest rates 
would have resulted in about 16 percent less 
bond-price variability in December 1980 
than in September 1979. 
Higher interest rates in late 1980 thus essen-
tially shrunk the yield curve in the yield-
duration relationship. On the other hand, the 
yield-maturity relationship could be mislead-
ing, since'by late 1980 it displayed an exten-
sion of  the term-to-maturity structure, as well 
as a flatter yield curve than that shown in the 
yield-duration relationship. Government se-
curities with longer maturities were available 
in late 1980, but the true availability of 
"long-term" government securities was less 
then than in the fall of 1979. Moreover, for-
ward rates-expected future short-term 
rates-could be significantly different in the 
two different relationships. 
The weighted average duration of privately-
held Treasury debt (excluding Treasury bills) 
was about three years in late 1980, compared 
with a weighted average maturity of about 
five years. (The weights were volumes out-
standing measured at face value.) This fact 
reinforces the argument about the short-term 
nature of  the Federal government's debt. 
Thus, previous stud ies of  the effects of Trea-
sury debt management on interest rates may 
be in serious error, because of  their focus on' 
the "average maturity rather than average dur-
ation of Treasury debt. Duration and not ma-
turity gives a much truer picture of the time 
profile of  Treasury debt obligations-and 
of  the riskiness of potential interest-rate 
changes. 
Joseph Bisignano and Brian Dvorak BANKING DATA-TWELFTH fEDERAL RESERVE DISTRICT 
(Dollar amounts in millions) 
Selected Assets and liabilities 
Large Commercial Banks 
Loans (gross, adjusted) and investments* 
Loans (gross, adjusted) - total# 
Commercial and industrial 
Real estate 
Loans to individuals 
Securities loans 
U.s. Treasury securities* 
Other securities* 
Demand deposits - total # 
Demand deposits - adjusted 
Savings deposits -'-total 
Time deposits - total# 
Individuals, part. & corp. 
(Large negotiable CD's) 
Amount 
Outstanding 
3/18/81 
146,834 
124,400 
36,641 
51,318 
23,399 
1,409 
6,768 
15,666 
40,969 
28,813 
30,008 
77,252 
68,219-
29,904 
Change 
from 
3/11/81 
416 
473 
351 
2 
- 25 
37 
- 53 
- 4 
- 439 
945 
100 
217 
290 
137 
uOll3U!lISeM. lIeln • uol3aJO  •  epei\aN  • 0ljepi 
!!eMeH  •  e!uJoJ!leJ  euoz!JV.  e>jselV 
CG)~<§~~mJ~JrdI mJ~~ 
~  CG)  4ImJ~<tJ( 
@&Jr@<§@.CQ[  TI~Jr@1P@dI 
:JJ \ill~m:JJJJ'W<W~cm ~~JJ'W~~~CQI 
Change from 
year ago 
Dollar  Percent 
8,294  6.0 
7,986  6.9 
2,123  6.2 
6,086  13.5 
- 1,076  - 4.4 
381  37.1 
43  0.6 
265  1.7 
- 2,506  - 5.8 
1,885  - 6.1 
2,687  9.8 
16,172  26.5 
15,757  30.0 
8,303  38.4 
Weekly Averages  Weekended  Weekended  Comparable 
of Daily Figures 
Member Bank Reserve Position 
Excess Reserves (  + )/Deficiency (  -) 
Borrowings 
Net free reserves (  + )/Net borrowed{ - ) 
* Excludes trading account securities. 
#  Includes items not shown separately. 
3/18/81 
n.a. 
29 
n.a. 
3/11/81  year -ago period 
n.a.  20 
40  263 
n.a.  243 
Editorial comments may be addressed to the editor (William Burke) or to the author ....  Free copies of this 
and other Federal Reserve publications can be obtained by calling or writing the Public Infonnation Section, 
Federal Reserve Bank of San Francisco, P.O. Box 7702, San Francisco 94120. Phone (415) 544-2184. 

Risk and duration

http://www.frbsf.org/publications/economics/letter/1981/el81-14.pdf

Risk and duration

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics