Measures of yield (expected return)
US – Treasury Bond 11 5/8 coupon, maturity 11/15/2004, issued 10/30/84
coupon
11.625
maturity
11/15/04
SD
8/9/1999 124.08
124.14
ask
NCD
LCD
w
v
yield
6.12% 11/15/99 5/15/99 0.4674 0.5326
9/04/2003
111.31
1.47% 11/15/03 5/15/03 0.6087 0.3913
8/31/04
bid
ask
102.03 1.422% 11/15/04 5/15/04 0.5870
0.4130
accrued
ask
2.717
124.4375 127.15421
3.538
111.96875 115.50675
3.412
102.09375 105.50575
On any date except a coupon payment date bond transactions will involve accrued interest. The buyer of
the bond will pay the seller of the bond the “clean” price plus accrued interest. The sum of the “clean”
price and the accrued interest is often called the invoice price; it is the cash outlay necessary to purchase the
bond.
The quote information from Barron’s 8/9/99 reflects secondary market dealer quotes for the 115/8% of
2004 Treasury bond.
The “clean” bid price is 124 8/32 = $124.25 per $100 face value. The dealer is willing to pay $124.25 to
purchase this bond.
The “clean” ask price is 124 14/32 = $124.4375 per $100 face value. The dealer is willing to sell this bond
for $124.4375.
The accrued interest for a transaction on the settlement date, 8/9/99, will depend on the days since the last
coupon payment and the day count conventions in the Treasury market. In the Treasury market the
actual/actual day count convention is used.
w
days between LCD and SD
 (1  v)
days between LCD and NCD
The fraction (w) is the portion of a six-month period since the last coupon payment date.
# days between 5/15/99 and 8/9/99 = 86
# days between 5/15/99 and 11/15/99 = 184
w = 86/184 = .467391
The calculation of accrued interest depends on; w, face value, and coupon rate. For this example the face
value is $100.
accrued interest = (1/2)*(.11625)*(0.46739)*(100) = 2.7167
To purchase this bond the invoice price = “clean” ask price + accrued interest.
invoice price = $124.4375 + 2.7167 = 127.1542.
invoice
Potential sources of a bond’s dollar return
1. The coupon payments
2. Capital gain/loss at maturity/call date/sale date
3. Reinvestment income
Current yield: Considers only promised coupon payments ignores capital gain/loss and
reinvestment income.
Current yield (8/9/1999) = 100*($11.625 / 127.1542) = 9.14%
“The yield (to maturity, to call, to put) on any investment is the interest rate that will
make the present value of the cash flows from the investment equal to the full price
(invoice) of the investment.”
Yield-to-maturity (internal rate of return)
Yield-to-call
Yield-to-put
Yield-to-worst
Yield-to-maturity, Yield-to-call, Yield-to-put are all measures of the internal rate of
return from investment in a fixed income security. The invoice price (clean + accrued
interest) is analogous to the initial investment in a capital budgeting decision. The
promised cashflows (coupon payments, maturity value, call price, put price) and invoice
price are used to find the discount rate (y) that equates the present value of the promised
payments with the invoice price.
n
P 
C
t 1 (1  0.5 * y )
t
 M * (1  0.5 * y)  n
P = invoice price
y = yield
M = maturity value or call price or put price
Bond description:
Semi-annual pay
Current date = settlement date = coupon payment date
Coupon rate = c = 5%
M = $1,000
B0 = invoice amount = $1,050
Time to maturity = 10 years
Time to first call = 4 years, first call price = 102
Time to first put = 5 years, first put price = 100
Scheduled cash flows to maturity
1
2
($1,050)
$25
$25
........
20
$1025
Scheduled cash flows to first call
1
2
($1,050)
$25
$25
........
8
$1045
Scheduled cash flows to first put
1
2
($1,050)
$25
$25
........
10
$1025
Semi-annual pay, Settlement date = coupon payment date,
Coupon rate = c = 5%, M = $1,000, B0 = $1,050
ytm
4.3772%
ytc
4.0973%
ytp
3.8899%
ytw
3.8899%
Yield-to-maturity, yield-to-call, yield-to-put are measures of the expected return from
owning a fixed income investment. These measures indicate the return per dollar
invested for a specific scenario.
Yield-to-maturity scenario –


All promised cash flows through the bond’s maturity date inclusive of coupons
and maturity value are received as promised.
All cash flows received prior to maturity date are reinvested at the yield-tomaturity till the maturity date.
Total return – A measure of expected return that incorporates a specific user defined
scenario specifying the reinvestment rate(s), and terminal security value for the
investment horizon.
Bond-equivalent TR semi-annual compounding
 accumulation 1 / h 

TR  
 1  2
B0





accumulation – Sum of coupons, reinvestment income, and horizon value of security on
the horizon date.
Bond and scenario description –
Semi-annual pay
Current date = settlement date = coupon payment date
Time-to-maturity = 7.5 years
Coupon rate = c = 4.5%
M = $5,000
ytm = 6%
B0 = invoice amount = $4,552.33
h = investment horizon = 5 years
g = reinvestment rate = 3%
yh = yield-to-maturity on horizon date = 6%
Using future value of annuity to find the accumulation of coupons and reinvestment
income on the horizon date.
 1  0.01510  1
$112.50 * 
  $1,204.06
0.015


Using bond valuation rule to find horizon value of bond
1  1  0.03 5 
5
Bh  $112.50 * 
  $5,000 * (1.03)  $4,828.26
0
.
03


accumulation = $1,204.06 + $4,828.26 = $6,032.32
 $6,032.17 1 / 10 
TR  2 * 
 1  5.71%

 $4,552.33 
