Unformatted text preview: Fixed Income Securities
and Markets
Chapter 3
YieldtoMaturity
Yield to Each cash flow of a fixed income security
Each
must be discounted at the factor or rate
appropriate for the term of that cash flow. YieldtoMaturity
Yield toYieldtomaturity, or yield, is the single
Yield torate that when used to discount a
bond's cash flows, produces the
bond's market price.
Yield is not a good measure of relative
Yield
value or of realized return to maturity.
If a bond's YTM remains unchanged over a
If
short time period, that bond's realized total
rate of return equals its yield. 1 102 + 18.125 3.125
3.125
3.125
103.125
=
+
+
+
2
3
4
y⎛
32
y⎞ ⎛
y⎞ ⎛
y⎞
1+
2 ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎜1 + 2 ⎟
⎝
⎠⎝
⎠⎝
⎠ YTM=4.8875%
YTM=4.8875% The price of a Tyear security making
Tsemiannual payments of c/2 and a final
principal payment of F is:
2T P (T ) = ∑
t =1 c/2 (1 + y / 2 ) c⎡ ⎛ 1 ⎞
P(T ) = ⎢1 − ⎜
⎟
y ⎢ ⎝ 1+ y / 2 ⎠
⎣ t 2T + F (1 + y / 2 ) 2T 2T
⎤
⎛1⎞
⎥+F⎜
⎟
⎥
⎝ 1+ y / 2 ⎠
⎦ From the formula
When Coupon Rate = YTM. Par bonds.
When
When Coupon Rate > YTM, Premium
When
bonds. The bonds are selling for
more than face value.
When Coupon Rate < YTM, Discount
When
bonds, or the bond sells at a discount
to par. 2 Bond prices approach par as they
Bond
approach maturity: pull to par. If a bond's YTM over a sixmonth period
sixremains unchanged, the annual total return
of the bond over that period equals its YTM. P0 = c/2
c/2
1+ c / 2
+
+L +
2
2T
1 + y / 2 (1 + y / 2 )
(1 + y / 2 ) P2=
1/ c
c/2
1+ c / 2
+
+L +
2T −1
2 1+ y / 2
(1 + y / 2 ) P 2 = (1 + y / 2) P0
1/ 6.25s of Feb. 15, 2003 ⎛P
⎞
y = 2 ⎜ 1/ 2 − 1⎟
P0
⎝
⎠ YTM=4.8875% Spot Rates: 5.008%,4.929%,4.864%,4.886% 102 + 18.125 3.125
3.125
3.125
103.125
=
+
+
+
2
3
4
y⎛
32
y⎞ ⎛
y⎞ ⎛
y⎞
1+
1 + ⎟ ⎜1 + ⎟ ⎜1 + ⎟
2 ⎜ 2⎠ ⎝ 2⎠ ⎝ 2⎠
⎝
3.125
3.125
3.125
103.125
=
+
+
+
ˆ
r.5 ⎛ r ⎞ 2 ⎛ r ⎞3 ⎛ r ⎞ 4
ˆ
ˆ
ˆ2
1
1.5
1+
2 ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎜1 + 2 ⎟
⎝
⎠⎝
⎠
⎠⎝
Flat term structure of spot rates, YTM=Spot
Flat
Upward sloping, YTM is below the last spot
Upward
Downward sloping, YTM above last spot
Downward 3 Yield, cash flow, and spot rates Yield of a zero coupon bond of a
Yield
particular maturity equals the spot
rate of that maturity.
Par coupon bonds: coupon bonds
Par
selling at par, the yield equals the
coupon rate c. Par Coupon Bonds 100c 2T
∑ d (t / 2) + 100d (T ) = 100
2 t =1 c= 2 [1 − d (T ) ]
2T ∑ d (t / 2)
t =1 4 Par Nonprepayable Mortgage: present value=amount borrowed=par
yield discounts future cash flows into PV
2T X ∑ d (t / 2) = 100
t =1 ⎛ 2T
⎞
X = 100 / ⎜ ∑ d (t / 2) ⎟
⎝ t =1
⎠
2T 100 = X ∑
t =1 1 (1 + yT / 2 ) t Discussion of Figure 3.2
All equal at .5 years
All
Shortend, downward sloping, par yields
Shortexceed zero yields, negligible
Mediumterm, upward sloping, zero yields
Mediumexceed par yields, spread increasing
Longend, down sloping, spread narrows
LongQualitatively, relative to zero yield,
Qualitatively,
mortgage yield is just like par yield, except
more pronounced YTM and relative value:
the coupon effect
The impact of coupon level on the
The
YTM of coupon bonds with the same
maturity is called the coupon effect.
The size of the coupon effect depends
The
very much on the shape of the term
structure of interest rates and the cash
flow structure of the securities. 5 YTM and realized return
102 + 18.125 3.125
3.125
3.125
103.125
=
+
+
+
2
3
4
y⎛
32
y⎞ ⎛
y⎞ ⎛
y⎞
1+
2 ⎜1 + 2 ⎟ ⎜1 + 2 ⎟ ⎜1 + 2 ⎟
⎝
⎠⎝
⎠⎝
⎠ 4
3
2
⎡⎛
y⎞
y⎞ ⎛
y⎞ ⎛
y ⎞⎤
⎛
102.5664 ⎜1 + ⎟ = 3.125 ⎢⎜1 + ⎟ + ⎜ 1 + ⎟ + ⎜ 1 + ⎟ ⎥ + 103.125
⎝ 2⎠
⎢⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎥
⎣
⎦ YTM is not the bond's return if held to maturity
YTM
Coupons are invested at uncertain future rates
Coupons 6 ...
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 Fall '08
 ANDERSON

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