Day Count Conventions: Actual/Actual

57 Pages·2010·0.15 MB·English

Day Count Conventions: Actual/Actual
(cid:15)
The first “actual” refers to the actual number of days in
a month.
(cid:15)
The second refers to the actual number of days in a
coupon period.
(cid:15)
The number of days between June 17, 1992, and
October 1, 1992, is 106.
{ 13 days in June, 31 days in July, 31 days in August,
30 days in September, and 1 day in October.
⃝c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 66
Day Count Conventions: 30/360
(cid:15)
Each month has 30 days and each year 360 days.
(cid:15)
The number of days between June 17, 1992, and
October 1, 1992, is 104.
{ 13 days in June, 30 days in July, 30 days in August,
30 days in September, and 1 day in October.
(cid:15)
In general, the number of days from date
(cid:17) (cid:17)
D (y ; m ; d ) to date D (y ; m ; d ) is
1 1 1 1 2 2 2 2
(cid:2) (cid:0) (cid:2) (cid:0) (cid:0)
360 (y y ) + 30 (m m ) + (d d ):
2 1 2 1 2 1
(cid:15)
Complications: 31, Feb 28, and Feb 29.
⃝c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 67
Full Price (Dirty Price, Invoice Price)
(cid:15)
In reality, the settlement date may fall on any day
between two coupon payment dates.
(cid:15)
Let
number of days between the settlement
and the next coupon payment date
(cid:17)
! : (6)
number of days in the coupon period
(cid:15)
The price is now calculated by
n∑(cid:0)1
C F
PV = ( ) + ( ) : (7)
!+i !+n(cid:0)1
1 + r 1 + r
i=0 m m
⃝c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 68
Accrued Interest
(cid:15)
The buyer pays the quoted price plus the accrued
interest — the invoice price:
number of days from the last
coupon payment to the settlement date
(cid:2) (cid:2) (cid:0)
C = C (1 !):
number of days in the coupon period
(cid:15)
The yield to maturity is the r satisfying Eq. (7) when
P is the invoice price.
(cid:15)
The quoted price in the U.S./U.K. does not include the
accrued interest; it is called the clean price or flat price.
⃝c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 69
(cid:0)
C(1 !)
6
coupon payment date coupon payment date
(cid:27) -(cid:27) -
(cid:0)
(1 !)% !% -
⃝c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 70
Example (“30/360”)
(cid:15)
A bond with a 10% coupon rate and paying interest
semiannually, with clean price 111.2891.
(cid:15)
The maturity date is March 1, 1995, and the settlement
date is July 1, 1993.
(cid:15)
There are 60 days between July 1, 1993, and the next
coupon date, September 1, 1993.
⃝c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 71
Example (“30/360”) (concluded)
(cid:15) (cid:2) 180(cid:0)60
The accrued interest is (10=2) = 3:3333 per
180
$100 of par value.
(cid:15)
The yield to maturity is 3%.
(cid:15)
This can be verified by Eq. (7) on p. 68 with
{ ! = 60=180,
{ m = 2,
{ C = 5,
{ PV= 111:2891 + 3:3333,
{ r = 0:03.
⃝c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 72
Price Behavior (2) Revisited
(cid:15)
Before: A bond selling at par if the yield to maturity
equals the coupon rate.
(cid:15)
But it assumed that the settlement date is on a coupon
payment date.
(cid:15)
Now suppose the settlement date for a bond selling at
par (i.e., the quoted price is equal to the par value) falls
between two coupon payment dates.
(cid:15)
Then its yield to maturity is less than the coupon rate.
{ The short reason: Exponential growth is replaced by
linear growth, hence “overpaying” the coupon.
⃝c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 73
Bond Price Volatility
⃝c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 74
“Well, Beethoven, what is this?”
— Attributed to Prince Anton Esterh´azy
⃝c 2010 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 75