Value of Floating Rate Bond

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Chapter 2
Pricing of Bonds

Learning Objectives
After reading this chapter, you will understand
 the time value of money
 how to calculate the price of a bond
 that to price a bond it is necessary to estimate the expected cash flows and determine
the appropriate yield at which to discount the expected cash flows
 why the price of a bond changes in the direction opposite to the change in required
yield
 that the relationship between price and yield of an option-free bond is convex
 the relationship between coupon rate, required yield, and price
 how the price of a bond changes as it approaches maturity
 the reasons why the price of a bond changes
 the complications of pricing bonds
 the pricing of floating-rate and inverse-floating-rate securities
 what accrued interest is and how bond prices are quoted
1
Time value of money (TVM)
(1) Future Value
How much is received at the end is based on repayment of principal,
expected cash flows (ordinary annuity), and reinvestment of those cash
flows over the investment horion!





 − +
=
y
y
C FV
n
1 ) 1 (
" #ar $alue
% y & period rate (annual reinvestment rate ad'usted for number of payments per
year)
(2) Present Value
The price of any security equals the PV of the security’s expected cash flows.
(o, to price a bond we need to )now!
• *he sie and timing of the bond+s expected cash flows, and
• *he required return (commensurate with how uncertain , ris)y the cash flows are)
#$ of a lump sum!
n
y
C
PV
) 1 ( +
=
 -n the case of a bond, ./0 is the principal
#$ of an ordinary annuity!












+

=
y
y
C PV
n
) 1 (
1
1
Pricing a on!
1
2e begin with a simple bullet bond!
• 3oncallable (maturity )nown with certainty)
• /oupons are paid every six months
• *he next coupon is received exactly six months from now
• *he first interest rate at which the coupons can be invested is fixed for the life of
the bond
• #rincipal is paid at maturity (no amortiing)
• /oupon is fixed for the life of the bond
-n general, for a fixed rate, ordinary payment, semi-annual coupon, the bullet bond price
(P) can be computed using the following formula!
n n
y
M
y
C
y
C
y
C
y
C
P
) 1 ( ) 1 (
444
) 1 ( ) 1 ( ) 1 (
5 1
+
+
+
+ +
+
+
+
+
+
=
6r,
n
n
n
n
t
n
y
M
y
y
C
y
M
y
C
P
) 1 (
) 1 (
1
1
) 1 ( ) 1 (
1
+
+












+

=
+
+
+
=

=
Exaple
/onsider a 17-year 178 coupon bond with a par value of 91,777, semi-annual payments and
a required yield (.mar)et interest rate0) of 1184
:iven C & 741(91,777) , 1 & 9;7, n & 1(17) & <7 and y & 7411 , 1 & 747;;, the present value of
the coupon payments (P) is!

/alculate
6r,
2hat if we change the yield higher or lower= >8= 118= -nterpretation=
A few complications!
#ro'ecting cash flows for fixed income securities is relatively straightforward, but
sometimes it can be less clear!
5
• e4g4, if the issuer or the investor has the option to change the contractual due date
for the payment of the principal (callable or putable bonds)
• if the coupon payment is rest periodically by a formula based on some value or
values of reference rates (floating rate securities)
• if the investor has the choice to convert or exchange the security into common
stoc) (convertible bond)
Pricin! "ero#coupon $onds
?or "ero#coupon $onds, the investor realies interest as the difference between the
maturity value and the purchase price4 *he equation is!
n n
y
M
y y y y
P
) 1 ( ) 1 (
7
444
) 1 (
7
) 1 (
7
) 1 (
7
5 1
+
+
+
+ +
+
+
+
+
+
=

6r,
n
y
M
P
) 1 ( +
=
%ero#&oupon 'ond Exaple
#rice a ero that expires 1; years from today if its maturity value is 91777 and the
required return is @4<8
/alculate
Price#(ield )elationship

Price"#iel! $elationship for a 2%"#ear &%' Coupon on!
<
Aield #rice (9) Aield #rice (9) Aield #rice (9)
747;; 1,;<14BC 747@7 1,7@1471 7411; >1B4B77
747C7 1,<C1457 747@; 1,7<<4<1 74157 B>B4>17
747C; 1,5>>4C; *.1** 1+***.** 7415; B;@4B;7
747B7 1,517455 7417; @;>4>5 741<7 B5545B7
747B; 1,1;C4>@ 74117 9@1@4BB 741<; B7>4;57
747>7 1,1@B4@5 7411; >>54; 741;7 C>;41<7
747>; 1,1<547> 74117 ><@4;< 741;; CC547>7
)elationship 'etween &oupon )ate+ )equired (ield+ and Price
 2hen yields in the mar)etplace rise above the coupon rate at a given point in
time, the price of the bond falls so that an investor buying the bond can realie
capital appreciation4
;
 *he appreciation represents a form of interest to a new investor to compensate
for a coupon rate that is lower than the required yield4
 2hen a bond sells below its par value, it is said to be selling at a discount4
 A bond whose price is above its par value is said to be selling at a premium4
-n other words,
Aield D /oupon  Eond price F par value  .Eond sells at a discount0
Aield F /oupon  Eond price D par value  .Eond sells at a premium0
)elationship 'etween 'ond Price and Tie if ,nterest )ates -re .nchan!ed
 ?or a bond selling at par value, the coupon rate equals the required yield4
 As the bond moves closer to maturity, the bond continues to sell at par4
 -ts price will remain constant as the bond moves toward the maturity date4
 *he price of a bond will not remain constant for a bond selling at a premium or a
discount4
*ime path of a 17-year 178 coupon bond selling at a discount and the same bond
selling at a premium as it approaches maturity4
C
(ear
Price of /iscount 'ond
0ellin! to (ield 121
Price of Preiu 'ond
0ellin! to (ield 2.31
1747 9 ><@4;< 91,111477
1C47 >;@41C 1,1@@41<
1147 >B<4;7 1,1C@4<;
1747 >>;457 1,1;74>5
>47 >@>4@< 1,11@415
<47 @5B4@7 1,7B<45B
747 1,777477 1,777477
 *he discount bond increases in price as it approaches maturity, assuming that the
required yield does not change4
 ?or a premium bond, the opposite occurs4
 ?or both bonds, the price will equal par value at the maturity date4
Three 4ain Factors that -ffect 'ond Prices5
*he price of a bond can change for three reasons!
B
• Gar)et interest rate (A*G)! there is a change in the required yield owing to changes
in the credit quality of the issuer
• *ime to Gaturity! there is a change in the price of the bond selling at a premium or a
discount, without any change in the required yield, simply because the bond is
moving toward maturity
• /redit Huality (default ris))! there is a change in the required yield owing to a
change in the yield on comparable bonds (i4e4, a change in the yield required by the
mar)et)
Complications
*he framewor) above for pricing a bond assumes the following!
• the next coupon payment is exactly six months away
• the cash flows are )nown
• the appropriate required yield can be determined
• one rate is used to discount all cash flows
(1) The next coupon payent is less than six onths away
2hen an investor purchases a bond whose next coupon payment is due in less than
six months, the accepted method for computing the price of the bond is as
follows!

=
− −
+ +
+
+ +
=
n
t
t v t v
y y
M
y y
C
P
1
1 1
) 1 ( ) 1 ( ) 1 ( ) 1 (
2here v & (I days between settlement and next coupon) , (I days in a C-month period)
(2) &ash Flows 4ay 6ot 'e 7nown5
>
• ?or a noncallable bond cash flows are )nown with certainty (assuming that
issuer does not default)
• However, most bonds are callable
• -nterest rates then determine the cash flow!
 -f interest rates drop low enough below the coupon rate, the issuer will call the
bond
• Also, /?+s on floaters and inverse floaters change over time and are not
)nown (more below)
(8) /eterinin! the appropriate required yield5
*he required yield for a bond is
RP r R
f
+ =
• f
r
is obtained from an appropriate maturity *reasury security
• RP should be obtained from RP+s of bonds of similar ris) (requires
'udgmentJ)
(9) :ne /iscount )ate -pplica$le to -ll &ash Flows
2e have assumed that all bond cash flows should be discounted at one discount rate
• However, usually we are facing an upward sloping yield curve!
 Kach /? should be discounted at a rate consistent with the timing of its
occurrence4
• -n other words, we can view a bond as a pac)age of ero-coupon bonds!
 Kach cash coupon (and principal payment) is a separate ero-coupon bond and
should be discounted at a rate appropriate for the .maturity0 of that cash flow
@
Pricing (loating"$ate an! )nverse"(loating"$ate
*ecurities
 *he cash flow is not )nown for either a floating-rate or an inverse-floating-
rate securityL it depends on the reference rate in the future4
Price of a Floater
 *he coupon rate of a floating-rate security (or floater) is equal to a reference
rate plus some spread or margin4
Kxample! coupon rate on a floater & 5-month *-bill rate " B; bps
 *he price of a floater depends on
• the spread over the reference rate
- the price of this floater decreases below par value when mar)et spread increases above
the stated spread (i4e4 rate required by the mar)et)
- price of a floater increases above par value when mar)et spread decreases below the
stated spread
• any restrictions that may be imposed on the resetting of the coupon rate
17
Price of an ,n;erse#Floater
 -n general, an inverse floater is created from a fixed-rate security4
-nterest rate on an inverse floater & fixed rate M reference rate
 *he security from which the inverse floater is created is called the collateral4
 ?rom the collateral two bonds are created! a floater and an inverse floater4
Nelationships!
-/oupon on inverse floater " coupon on floater & coupon on the fixed-rate security
-par value on inverse floater " par value on floater & par value on the fixed-rate security
Kxample!
/ollateral is a ;-year, @8 coupon, semiannual coupon, 917 million par value bond4
-?loater is created to have a 917 million par value, coupon rate of (5-month *-bill rate "
177 bps)
--nverse floater is created to have a 917 million par value, coupon rate of (1C8-5-month
*-Eill rate)
6verall, (74;)%(5-month *-bill rate"177 bps)
" (74;)%(1C8-5-month *-bill rate)
& @8 (coupon on the fixed-rate bond)
 *he price of a floater depends on (i) the spread over the reference rate and (ii) any
restrictions that may be imposed on the resetting of the coupon rate4
 ?or example, a floater may have a maximum coupon rate called a cap or a minimum
coupon rate called a floor4
 Here, set a floor on the inverse floater at 784 *his )eeps the coupon rate on the
inverse floater from becoming negative
 (et a cap on the floater at 1>8, so that the sum of coupons paid on the floater and
inverse floater do not exceed the coupon on the fixed rate bond (collateral)
11
Price +uotes an! ,ccrue! )nterest
• Eond prices are usually different than their face valuesL those with prices that
trade above (below) the par amount trade at a premium (discount)
• 6ne basis point is 1,177
th
of a percent, i4e4 1 basis point & 7477714 ?or example,
the difference between bond yields of ;4;7 and ;4;1 is 1 bp4
(ome Eloomberg *reasury 3ote and Eond prices on Oec 57, 177@!
/6P#63
GA*PN-*A
OA*K
/PNNK3*
#N-/K,A-KQO
#N-/K,A-KQO
/HA3:K
*-GK
;-Aear 14C1; 11,51,171< @@-1< , 14C> -7-17 , 47CB 1<!7@
B-Aear 541;7 11,51,171C @@-7;R , 545> -7-17" , 47;5 1<!1@
17-Aear 545B; 11,1;,171@ @C-7> , 54>5 -7-11" , 47<@ 1<!1@
57-Aear <45B; 11,1;,175@ @;-1B , <4C5 -7-11 , 471< 1<!1@
• *he number after the hyphen is a .tic)0 and is in 51
nd
+s4 *he overall price is a
percentage of the par amount!
e4g4 the ;-year price & @@-1< & @@"1<,51 & @@4B;8
the B-year price & @@-7;R & @@";4;,51 & @@41B1>B;8
• -n general! quotes are in 8 of the par value
K4g4 @@4B5; means @@4B5;8 of the par value
11
Accrued -nterest
2hen an investor purchases a bond between coupon payments, the investor
must compensate the seller of the bond for the coupon interest earned from the
time of the last coupon payment to the settlement date of the bond4
 *his amount is called accrued interest4
 ?or corporate and municipal bonds, accrued interest is based on a 5C7-day year,
with each month having 57 days4
 *he amount that the buyer pays the seller is the agreed-upon
price plus accrued interest4
 *his is often referred to as the full price or dirty price4
 *he price of a bond without accrued interest is called the flat or clean price4
 *he exceptions are bonds that are in default4
 (uch bonds are said to be quoted flat, that is, without accrued interest4
Accrued -nterest

/oupon 1 paid Eond sold /oupon 1 paid
?ull (dirty) price & bond (clean or flat) price " accrued interest
Payments Between Days of Number
Payment ast !ince Days of Number
"nterest #ccrued =
15
&ountin! up days in Excel5
3umber of Oays since Qast #ayment &/6P#OAAE((settlement,maturity,frequency,basis)
3umber of Oays Eetween #ayments &/6P#OAA((settlement,maturity,frequency,basis) Eond
#rice & #N-/K(settlement,maturity,rate,ytm,redemption,frequency,basis)
P),&E+ P),&E/,0&+ P),&E4-T+ and /,0& Functions in 4icrosoft :ffice
Excel for &alculatin! 'ond Prices and :ther 0ecurities Payin! ,nterest
Gicrosoft Kxcel has several formulas for calculating bond prices and other securities
paying interest, such as *reasuries or certificates of deposit (/Os)4 *hese prices ta)e into
account the accrued interest, if any4
Eond #rice (8 of ?$) & #N-/K(settlement,maturity,rate,yield,redemption,frequency,basis)
Sero /oupon Eond #rice & #N-/KO-(/(settlement,maturity,discount,redemption,basis)
Sero /oupon Eond Aield & O-(/(settlement,maturity,price,redemption,basis)
#rice of (ecurity that pays interest only at maturity &
#N-/KGA*(settlement,maturity,issue,rate,yield,basis)
• (ettlement & Oate in quotes of settlement4
• Gaturity & Oate in quotes when bond matures4
• Nate & 3ominal annual coupon interest rate in decimal form4
• Aield & Annual yield to maturity in decimal form4
• -ssue & -ssue date of the security4
• #rice & #rice of security as a percent of par value4
• Nedemption & $alue of security at redemption per 9177 of face value4 Gost often,
redemption will equal 1774
• ?requency & 3umber of coupon payments per year4
o 1 & Annual
o 1 & (emiannual
o < & Huarterly
• Easis & Oay count basis4
o 7 & 57,5C7 (P4(4 3A(O basis)4 *his is the default if the basis is omitted4
o 1 & actual,actual (actual number of days in month,year)4
o 1 & actual,5C7
o 5 & actual,5C;
o < & Kuropean 57,5C7
1<
Exaples<.sin! 4icrosoft :ffice Excel for &alculatin! 'ond Prices and /iscounts
*he following basic factsTwhere they applyTwill be used for each of the example
calculations for a 17-year bond originally issued in 1,1,177> with a par value of 91,777!
• (ettlement date & 5,51,177>
• Gaturity date & 11,51,171B
• -ssue date & 1,1,177>
• /oupon rate & C8
• Aield to maturity & >8
• #rice (per 9177 of face value) & 114@@
• Nedemption & 177
• ?requency & 1 for most coupon bonds4
• Easis & 1 (actual,actual)
2hat is the price of a bond selling for a yield to maturity of >8=
'ond Price &#N-/K(U5,51,177>U,U11,51,171BU,747C,747>,177,1,1) & >C4C17@1 & =3>>.21
2hat is the discount price of a ero coupon bond with a par value of 91,777 yielding >8=
Price /iscount &#N-/KO-(/(U5,51,177>U,U11,51,171BU,747C,747>,177,1) & 114@@1>> &
=21?.?8
2hat is the interest rate of a discounted ero coupon bond selling for 911@4@7 that pays
91,777 at maturity=
,nterest )ate of 'ond /iscount & O-(/(U5,51,177>U,U11,51,171BU,114@@,177,1) &
747>7775 & 31
• 3ote that the price found with the #N-/KO-(/ function has been rounded and
plugged into the O-(/ function as a way to verify values4 (114@@ & 911@4@7 for a
bond with a 91,777 par value)4
1;
*he last function, #N-/KGA*, calculates the price of a security that pays all of its
interest at maturity, which includes negotiable money mar)et certificates of deposit (/O)4
2hat is the price of a negotiable, @7-day /O originally issued for 9177,777 on 5,1,177>
paying a rate of >8 with a current yield of C8 and a settlement date of <,1,177>=
Here we use the Gicrosoft Kxcel /ate function, which ta)es the format
/-TE(year+onth+day) to do some calendar arithmetic4 2e also use the ban)erVs year of
5C7 days, so we choose a basis of 7, which we could have omitted, since it is the default4
4ar@et Price of &/ &
#N-/KGA*(OA*K(177>,<,1),OA*K(177>,5,1)"@7,OA*K(177>,5,1),747>,747C,7)&
1**.8131 per 9177 of face value & =1**+813.1*
6ote5 *he above calculations were made using Gicrosoft 6ffice Kxcel 177B4 *hese
functions are also available in earlier versions of Kxcel4
1C

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