Download It is not appropriate to discount the cash flows of a bond by the yield

Test 2 FIN 4550 11/8/04 100 points
It is not appropriate to discount the cash flows of a bond by the yield to maturity of a
Treasury security with corresponding time to maturity, because of differences in the
timing and size of cash flows. Differences in the timing and size of cash flows will
produce differences in duration, convexity and tax liability. Each of these has an impact
on bond value.
To appropriately price a bond it is necessary to change the perspective. Through Chapter
4 in the text a bond was treated as a package of cash flows. All cash flows were
discounted with the same rate regardless of the timing of the cash flows.
In Chapter 5 we will take the perspective that a bond is a package of individual zero
coupon bonds, each zero coupon bond to be treated as a unique entity, requiring discount
by the appropriate zero coupon rate. The appropriate zero coupon rate for each cash
flow is the spot rate for a zero coupon bond with time to maturity equal to the timing of
the cash flow.
The process for deriving the spot rate curve.
The spot rate curve (term structure) is extracted from prices of US Treasury securities (Tbills, notes, bonds or STRIPS).
Steps to construct the spot rate curve from market information.
1. Identify set of Treasury securities to be utilized. On-the-run and Off-the-run
Treasury securities
Yield Book
http://www.publicdebt.treas.gov/
2. Assemble the par coupon curve. - Linear interpolation
ni
nj
nk
ni < nj < nk
time to maturity in semi annual periods
ytmi
of on-the-run Treasury
time to maturity in semi annual periods
ytmj
of par/coupon curve (Exhibit 5-7)
time to maturity in semi annual periods
ytmk
of on-the-run Treasury
annualized yield to maturity
of on-the-run Treasury
annualized yield to maturity
or par/coupon rate
annualized yield to maturity
of on-the-run Treasury
( ytmk  ytmi )
(nk  ni )
3. Bootstrap the spot rate curve from the par rate curve.
par/coupon rate = ytm j  ytmi  (n j  ni ) 
1
1/ n















1 c
z n  2  
 1






n 1
1





1  c  


z
i 1 (1  i ) i 




2 



4. From the spot rate curve interpolate spot rates appropriate for the bond’s cash
flow dates if necessary.
5. Use the spot rate curve and risk premium if necessary to find the value of a bond.
Forward rates and the term structure of interest rates:
A forward rate is the rate earned during the forward period that equates the realized
return from a sequential investment to the realized return of a single investment for the
entire investment horizon.
0---------- j ----------- k
Calculating a forward rate from spot rates:
(1  0.5  z j ) j  (1  0.5 j f (k  j ) ) (k  j )  (1  0.5  z k ) k

 1  0.5  z k k
f

2


j (k  j )
j
 1  0.5  z j



1

 (k  j ) 

 1




It is possible for a security market participant such as a bond dealer to either borrow or
lend at the implied forward rates.
Transacting at the long term rate “locks-in” the forward rate for the forward period.
The term structure of interest rates is affected by the interaction between implied forward
rates, f, market participants expectations of spot rates in forward periods, E(z), and
market participants interest rate risk tolerance.
Only a zero coupon security that matches a decision maker’s time horizon is free of both
price risk and reinvestment risk.
If E(z)s equal forward rates then total return is equal for all investment strategies
spanning an investment horizon.
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If E(z)s are not equal forward rates then total returns are not equal for all investment
strategies spanning an investment horizon.
Conditioned on E(z)s and decision maker’s risk tolerance, a decision maker may select a
strategy that produces an interest rate risk exposure if the expected return from the
investment strategy or expected cost from borrowing strategy is sufficient to compensate
for the additional interest rate risk exposure.
The term structure theories discussed in the text (pages 111-116) are predicated on
different assumptions concerning market participant’s interest rate risk tolerance.
Pure expectations theory;
Liquidity theory;
Preferred Habitat theory;
Segmented Markets theory;
Treasury bills are pure discount securities issued by US Treasury. Treasury bills are sold
with an initial maturity of one-month, 3-months, 6-months and one-year. Cash
management bills are also sold very short original maturity.
In the secondary market, Treasury bills are quoted on a bank discount basis using a 360
day year. A quote of 3% indicates the price of the bill is calculated as a 3% discount
from face value.
Given, M=$1,000,000 and t = 82 a quote of Yd = 3% can be used to find the purchase
price for the bill:
D = dollar discount = $1,000,000 * 0.03 * 82/360 = $6,833.33
P = M – D = $1,000,000 – 6,833.33 = 993,166.67
To state the yield on a bill in a comparable format to the yield on Treasury notes and
bonds it is necessary to account for the discount interest of the bill and the 360 day
money-market convention.
For bills with less than 182 days till maturity
BEY =
D 365
$6,833.33 365

=
= 3.063%

P
t
$993,166.67 82
To calculate bond equivalent yield, for bills with maturity greater than 182 days it is also
necessary to account for the fact that bills pay no coupons but notes and bonds do.
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Given, M=$1,000,000 and t = 200 a quote of Yd = 3% can be used to find the purchase
price for the bill:
D = dollar discount = $1,000,000 * 0.03 * 200/360 = $16,666.67
P = M – D = $1,000,000 – $16,666.67 = 983,333.33
BEY =
 2 t
2
365
2 t
365t 2  365
 11  M 
P
2 t
1
365
= 3.089%
To compare the yield of a Treasury bill with the yield offered on non-discount money
market instruments such as CDs the return on a Treasury bill must be adjusted to account
for the discount interest of the bill. The CD equivalent (money market equivalent) yield
on a Treasury bill can be found simply from the discount rate (all bill maturities)
CDEY =
360  Yd
360  t  Yd
For the 82 day bill examined above: CDEY =
360  0.03
 3.021%
360  82  0.03
Treasury auctions – Treasury bills notes and bonds are sold originally by the Treasury
through regularly scheduled auctions.
Single price auction
Multiple price auction
Tail
Bid-to-cover ratio
Secondary market quotations
T-bills
T-notes
T-bonds
Clean price, Full price and Accrued interest
STRIPS
TIPS
Tax treatment of STRIPS
Tax treatment of TIPS
Accrued interest TIPS:
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0.50  c  M 
(SD  LCD) CPI  U SD
where CPI-U0 = Index level on auction date.

( NCD  LCD) CPI  U 0
If settlement date is 11/15/04
CPI-USD = CPI-U11/15/04 = CPI-U8/01/04 + 15/31*(CPI-U9/01/04 – CPI-U8/01/04)
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