Market Valuations and Duration - TreasuryDirect by chenshu


									 The need for Market Valuation of your


SFFAS 1 – Accounting for Selected Assets and

   72. Disclosure of market value. For investments in
   Market-based and marketable Treasury securities, the
   market valuation should be disclosed.
From the FedInvest system you can select Prior Days Prices which
takes you to a listing of price files. You can choose the price file
for the particular day you wish to value your portfolio.
Once you choose the price file you use the End of Day
price to calculate your market value.
 • We know:
    – An increase in interest rates causes bond
      prices to fall, and a decrease in interest
      rates causes bond prices to rise.

• We also know that longer maturity debt
  securities tend to be more volatile in price.
   – For a given change in interest rates, the price of a
     longer term bond generally changes more than
     the price of a shorter term bond.
• Two bonds with the same term to
  maturity do not have the same interest-
  rate risk.
 – A 10 year zero coupon bond makes all of
   its payments at the end of the term.

  – A 10 year coupon bond makes payments
    before the maturity date.
• When interest rates rise, the prices of
  low coupon securities tend to fall faster
  than the prices of high coupon securities.

• Similarly, when interest rates decline, the
  prices of low coupon rate securities tend
  to rise faster than the prices of high
  coupon rate securities.
• Knowledge of the impact of varying
  coupon rates on security price volatility
  led to the development of a new index
  of maturity other than straight calendar
• The new measure permits analysts to
  construct a linear relationship between
  term to maturity and security price
  volatility, regardless of differing coupon
is the measure of the price sensitivity of a
fixed-income security to an interest rate
change of 100 basis points. The calculation is
based on the weighted average of the present
values for all cash flows.
Duration is measured in years; however, do not
confuse it with a bond’s maturity. For all bonds,
duration is shorter than maturity except zero
coupon bonds, whose duration is equal to
maturity. This is because all cash flows are
received at maturity.
The term “duration,” having a special meaning in the context of bonds, is a
measurement of how long in years it takes for the price of a bond to be repaid by
its internal cash flows. It is an important measure for investors to consider, as
bonds with higher durations are more risky and have higher price volatility than
bonds with lower durations.

     For each of the two basic types of bonds the duration is the following:
       1. Zero-coupon bond – Duration is equal to its time to maturity.
       2. Straight bond – Duration will always be less than its time to maturity.

Here are some visual models that demonstrate the properties of duration for a zero-
coupon bond and a straight bond.
           Duration of a Zero-Coupon Bond

The red lever above represents the four-year time period it takes for a zero coupon to
mature. The money bag balancing on the far right represents the future value of the
bond, the amount that will be paid to the bondholder at maturity. The fulcrum, or the
point holding the lever, represents duration, which must be positioned where the red
lever is balanced. The fulcrum balances the red lever at the point on the time line
when the amount paid for the bond and the cash flow received from the bond are
equal. Since the entire cash flow of a zero-coupon bond occurs at maturity, the
fulcrum is located directly below this one payment.
             Duration of a Straight Bond
Consider a straight bond that pays coupons annually and matures in five years. Its cash
flows consist of five annual coupon payments and the last payment includes the face value
of the bond.

The moneybags represent the cash flows you will receive over the five-year period. To
balance the red lever (at the point where total cash flows equal the amount paid for the
bond), the fulcrum must be further to the left, at a point before maturity. Unlike the zero-
coupon bond, the straight bond pays coupon payments throughout its life and therefore
repays the full amount paid for the bond sooner.
  Factors Affecting Duration
It is important to note, however, that duration changes as the coupons are paid to the
bondholder. As the bondholder receives a coupon payment, the amount of the cash
flow is no longer on the timeline, which means it is no longer counted as a future cash
flow that goes towards repaying the bondholder. Our model of the fulcrum
demonstrates this: as the first coupon payment is removed from the red lever (paid to
the bondholder), the lever is no longer in balance (because the coupon payment is no
longer counted as a future cash flow).
The fulcrum must now move to the right in order to balance the lever again:

Duration increases immediately on the day a coupon is paid, but throughout the life of
the bond, the duration is continually decreasing as time to the bond’s maturity decreases.
The movement of time is represented above as the shortening of the red lever: notice
how the first duration had five payment periods and the above diagram has only four.
This shortening of the timeline, however, occurs gradually, and as it does, duration
continually decreases. So, in summary, duration is decreasing as time moves closer to
maturity, but duration also increases momentarily on the day a coupon is paid and
removed from the series of future cash flows – all this occurs until duration, as it does for
a zero-coupon bond, eventually converges with the bond’s maturity.
Duartion – Other factors:
Coupon rate and Yield also affect the bond’s duration. Bonds with high coupon
rates and in turn high yields will tend to have a lower duration than bonds that pay
low coupon rates, or offer a low yield. This makes sense, since when a bond pays a
higher coupon rate the holder of the security received repayment for the security at
a faster rate. The diagram below summarizes how duration changes with coupon
rate and yield.
                          Macaulay Duration
The formula usually used to calculate a bond’s basic duration is the Macaulay
duration, which was created by Frederick Macaulay in 1938 but not commonly used
until the 1970s.
Macaulay duration is calculated by adding the results of multiplying the present value of
each cash flow by the time it is received, and dividing by the total price of the security. The
formula for Macaulay duration is as follows:

                                                            n = number of cash flows
                                                            t = time to maturity
                                                            C = cash flow
                                                            i = yield to maturity
                                                            M = maturity par value

     Let’s go through an example:
If you hold a five-year bond with a par value of $1,000 and a coupon rate of 5%. For
simplicity, assume that the bond is paid annually and that interest rates are 3% (yield).

                                               n = number of cash flows
                                               t = time to maturity
                                               C = cash flow
                                               i = yield to maturity
                                               M = maturity par value

Fortunately if you are seeking the Macaulay duration of a zero-coupon bond, the duration
would be equal to the bond’s maturity, so there is no calculation required.
Therefore…the lower the coupon rate, the
     higher the duration of the bond.
Coupon Bonds: duration is shorter than maturity

 Discount bonds (yield is greater than coupon):
duration increases at a decreasing rate up to a point,
after which it declines

 Par value bonds: duration increases with

 Premium bonds (yield is less than coupon):
duration increases throughout but at a lesser rate
than with a par value bond.
 Duration depends on yield-to-maturity.

 The higher the yield the shorter the duration,
other things being equal.
 In Treasury bonds, the only source of risk
  stems from interest rate changes.
 Duration is a measure of this source of

 Duration allows bonds of different
  maturities and coupon rates to be directly
calculates the annual duration for a security that pays periodic

A security has a July 1, 1993, settlement date and a December 1,
1998, maturity date. The semiannual coupon rate is 5.50% and the
annual yield is 5.61%. The bond has a 30/360 day-count basis.
To determine the security's annual duration:
@DURATION(@DATE(93;7;1);@DATE(98;12;1);0.055;0.0561;2;0) =
DURATION(settlement,maturity,coupon yld,frequency,basis)

A bond has the following terms:
January 1, 1998, settlement date
January 1, 2006, maturity date
8 percent coupon
9.0 percent yield
Frequency is semiannual
Actual/actual basis
The duration (in the 1900 date system) is:
DURATION("1/1/1998","1/1/2006",0.08,0.09,2,1) equals
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