Measuring Yield
Chapter 3
Computing Yield
yield = interest rate that solves the following
CF
CF
P = CF + CF
+
+ . . .+

1
1 

2
y
1
1 
y
3
1 
2
y
3
1 
internal rate of return
Promised Annual Payments
(Cash Flow to Investor)
Years from Now
1
$100
2
100
3
100
4
1,000
N
y
N
IRR
Years from Now
1
2
3
4
Years from Now
Promised Annual Payments (Cash Flow to Investor)
$100
100
100
1,000
Promised Annual Payments Present Value of Cash
(Cash Flow to Investor)
Flow at 10%
1
$100
$90.91
2
100
82.64
3
100
75.13
4
1,000
683.01
Present value = $931.69
IRR
Years from Now Promised Annual PaymentsPresent Value of Cash
(Cash Flow to Investor)
Flow at 12%
1
$100
$89.29
2
100
79.72
3
100
71.18
4
1,000
635.52
Present value = $875.71
Years from Now Promised Annual Payments )Present Value of Cash
(Cash Flow to Investor)
Flow at 11%
1
$100
$90.09
2
100
81.16
3
100
73.12
4
1,000
658.73
Present value = $903.10
Yields





simple annual interest rate
effective annual yield
EAY = (1 + periodic interest rate)m – 1
examples
current yield = (annual $ coupon int.) / price

only considers coupon interest not capital gain/loss
if selling at discount/premium
Yields

YTM

bond-equivalent yield

convention in bond market to move from semiannual yield to an annual yield
by doubling the semiannual yield




Why is practice of doubling a semiannual yield followed?
Wouldn’t it be more appropriate to compute EAY by compounding?
YTM considers current coupon and capital gain/loss
factors affecting reinvestment risk


for a given YTM and a given non-zero coupon rate, the longer the
maturity, the more the bond’s total dollar return depends on reinvestment
income to realize the YTM at time of purchase (ie, the larger the
reinvestment risk)
for a coupon paying bond, for a given maturity and a given YTM, the
higher the coupon rate, the more dependent the bond’s total dollar return
will be on the reinvestment of the coupon payments in order to produce
the YTM at time of purchase
Yields

Yield to Call
in practice, YTM and YTC calculated for callable
bonds
 to calculate, find PV of all coupons until bond is
called and then use call price as final value
 convention is to calculate yield to first call (or yield
to next call), yield to first par call, and yield to
refunding

Yield to Call
Annual Interest Semiannual Rate Present Value of 6 PV of $103
PV of CFs
Rate (%)
y (%)
Payments of $3.5
6 Periods from Now
5
2.5
$19.28
$88.82
108.10
5.2
2.6
19.21
88.30
107.51
5.4
2.7
19.15
87.78
106.93
5.6
2.8
19.09
87.27
106.36



8 year 7% coupon bond with maturity value of $100
first call date is end of year 3
call price of $103

note that YTC assumes that all CFs can be reinvested at YTC until
assumed call date – may not be true
Yields

Yield to Put
rate that makes PV of CFs to first put date equal to
price plus accrued interest
 example


Yield to Worst
calculate yield to call/put for all possible dates and
YTM and then pick minimum of all of these
 does not mean much since problem with all yield
measures are they do not identify potential return
over investment period

Yields

Yield (IRR) for a Portfolio

not simply weighted average of YTMs for all bonds in portfolio
Bond
A
B
C
Coupon Rate (%)
Maturity (years) Par Value
Price
YTM (%)
7
5 $10,000,000 $9,209,000
9
10.5
7
20,000,000 20,000,000
10.5
6
3
30,000,000 28,050,000
8.5
Period CF Received Bond A
Bond B
Bond C
Portfolio
1
$350,000 $1,050,000
$900,000
$2,300,000
2
350,000
1,050,000
900,000
2,300,000
3
350,000
1,050,000
900,000
2,300,000
4
350,000
1,050,000
900,000
2,300,000
5
350,000
1,050,000
900,000
2,300,000
6
350,000
1,050,000
30,900,000
32,300,000
7
350,000
1,050,000
—
1,400,000
8
350,000
1,050,000
—
1,400,000
9
350,000
1,050,000
—
1,400,000
10
10,350,000
1,050,000
—
11,400,000
11
—
1,050,000
—
1,050,000
12
—
1,050,000
—
1,050,000
13
—
1,050,000
—
1,050,000
14
—
21,050,000
—
21,050,000
Yields

Cash Flow Yield

MBS and ABS have CFs that include interest and principal –
amortizing securities




prepayment speed must be assumed to project CFs needed to
calculate yield
yield calculated using assumed prepayment rate is cash flow yield **
example
limitations


projected CFs assumed to be reinvested at CF yield
MBS or ABS is assumed to be held until final payoff of all loans
based on a prepayment assumption
Spread/Margin Measures for
Floating Rate Securities


coupon rate for floater changes periodically
“margin” measures

spread for life (simple margin)
SpreadForLife  [

100 (100  Pr ice)
100
 QuotedM arg in] * (
)
Maturity
Pr ice
discount margin




determine CFs assuming reference rate does not change over life
select a margin
discount CFs in step 1 by current value of reference rate plus margin
compare PV of CFs in step 3 to price plus accrued interest



if PV = security’s price + acc. int., discount margin is margin assumed in step 2
if PV does not equal, go back to step 2 and try another margin
for bond selling at par, discount margin is quoted margin in coupon reset
formula
Discount Margin
Exhibit 3-1. Calculation of the Discount Margin for a Floating-Rate Security
Floating-rate security:
Maturity: six years
Coupon rate: reference rate + 80 basis points
Reset every six months
PV of CF at Assumed Annual Margin (bp)
Reference
Cash
Period
Rate
Flow
80
84
88
96
100
1
10%
5.4
5.1233
5.1224
5.1214
5.1195
5.1185
2
10
5.4
4.8609
4.859
4.8572
4.8535
4.8516
3
10
5.4
4.6118
4.6092
4.6066
4.6013
4.5987
4
10
5.4
4.3755
4.3722
4.3689
4.3623
4.359
5
10
5.4
4.1514
4.1474
4.1435
4.1356
4.1317
6
10
5.4
3.9387
3.9342
3.9297
3.9208
3.9163
7
10
5.4
3.7369
3.7319
3.727
3.7171
3.7122
8
10
5.4
3.5454
3.5401
3.5347
3.524
3.5186
9
10
5.4
3.3638
3.358
3.3523
3.3409
3.3352
10
10
5.4
3.1914
3.1854
3.1794
3.1673
3.1613
11
10
5.4
3.0279
3.0216
3.0153
3.0028
2.9965
12
10
105.4 56.0729 55.9454 55.8182 55.5647 55.4385
Present Value =
100 99.8269 99.6541 99.3098 99.1381
Sources of Bond Return




coupon payments
capital gain/loss on sale of bond (or when
called)
reinvestment of coupon payments – interest on
interest
yields
current
 YTM
 CF Yield

 1  r n  1
C
  nC

r

Dollar Return

coupon interest + interest on interest =
 1  r n  1
C


r


interest on interest =
 1  r n  1
C
  nC

r



example
Total Dollar Return
Total Return

measure of yield that makes an assumption
about the reinvestment rate
bond
A
B
C
D

coupon
5
6
11
8
maturity
3
20
15
5
YTM
9
8.6
9.2
8
If all 4 have the same credit quality, which is the
most attractive?
Total Return
1.
2.
3.
4.
5.
Compute the total coupon payments plus the interest
on interest based on the assumed reinvestment rate.
Determine the projected sale price at the end of the
planned investment horizon.
Sum the values computed in steps 1 and 2.
To obtain the semiannual total return, use {total
future $ / purchase price}1/h -1
Double the interest rate in step 4 (as interest is
assumed to be paid semiannually.)
Yield Changes

absolute yield change
measured in basis points
 absolute value of difference between two yields


percentage yield change
ln of the ratio of the change in yield
 % yield change = 100 x ln(new yield / initial yield)
