Download Lecture 2 The Valuation of Debt Securities

The Yield Curve and Valuation
1
Yield Curve
• The graphical depiction of the relationship between the yield on bonds of
the same credit quality but different maturities is known as the yield curve.
• The yield curve is an unsatisfactory measure of the relation between
required yield and maturity since securities with the same maturity may
actually carry different yields.
2
Shape of the Yield Curve
3
Shape of the Yield Curve (Continued)
• Four shapes: normal or positively sloped yield curves, flat yield curves,
inverted or negatively sloped yield curves, and humped yield curves.
• Market participants pay attention to the difference between long-term
Treasury yields and short-term Treasury yields.
• The spreads between these yields for two maturities is referred to as the
steepness or slope of the yield curve.
• There is no industry-wide accepted definition of the maturity used for the
long-end and the maturity used for the short-end of the yield curve.
• 1). the 30-year yield vs. the 3-month yield, 2). the 30-year yield vs. the 2year yield.
4
Yield Curve Shifts
•
•
•
•
A parallel shift in the yield curve vs. a nonparallel shift in the yield curve
Parallel shifts: upward parallel shift and downward parallel shift.
Nonparallel yield curve shifts: 1) twists and 2) Butterfly shifts
Twists:
1) flattening of curve, 2) steepening of curve
• Butterfly shifts:
1) A positive butterfly means that the yield curve becomes less humped.
利率上升時，短期與長期利率上升幅度大於中期上升幅度；反之，
利率下降時，短期與長期利率下降幅度小於中期下降幅度。
2) A negatively butterfly means the yield curve becomes more humped.
5
Spot Rate and Spot Rate Curve
• The spot rate is the yield on a zero-coupon Treasury.
• The spot rate curve is the graphical depiction of the relationship between
the spot rate and maturity.
• However, it is not possible to construct such a curve solely from
observations of market activity on Treasury securities.
• It is necessary to derive this curve from theoretical considerations as
applied to the yields of the actually traded Treasury debt securities.
• Such a curve is called a theoretical spot rate curve or term structure of
interest rate.
6
Treasury Yield Curve and Treasury Spot Rate Curve
• The on-the-run Treasury yield curve shows the relationship between the
yield for on-the-run Treasury issues and maturity.
• The term structure of interest rates is the relationship between the
theoretical yield on zero-coupon Treasury securities and maturity.
• An analysis of Treasury spot rates is critical because they are used to value
fixed-income securities.
• Also, they are benchmarks used to establish the minimum yields that
investors want when investing in a non-Treasury security.
7
Principal Component of Analysis (PCA) of Yield Curve
• Rober Litterman and Jose Scheinkmanm “Common Factors Affecting Bond
Returns,” Journal of Fixed Income (June 1991), pp. 54-61.
• Litterman and Scheinkman found that three factors explained historical
returns for zero-coupon Treasury securities for all maturities.
•
•
•
•
Factor 1: changes in the level of rates
Factor 2: changes in the slope of the yield curve
Factor 3: changes in the curvature of the yield curve
They determined the importance of each factor by its coefficient of
determination.
• Explanatory power: about 90% for factor 1 and about 8.5% for factor 2.
8
Constructing the Theoretical Spot Rate Curve for
Treasuries
•
The curve can be constructed from the yields on Treasury securities.
1.
2.
3.
4.
Treasury coupon strips
On-the-run Treasury issues
On-the-the-run Treasury issues and selected off-the-run Treasury issues
All Treasury coupon securities and bills
9
Treasury Coupon Strips
•
Use the observed yield on Treasury coupon strips to construct an actual
spot rate curve.
•
1.
Three problems:
The liquidity of the strips market is not as great as that of the Treasury
coupon market.
The tax treatment of strips is different from that of Treasury coupon
securities.
There are maturity sectors where non-U.S. investors find it advantageous
to trade off yi8eld for tax advantages associated with a strip.
2.
3.
10
On-the-Run Treasury Issues
• The on-the-run Treasury issues are the most recently auctioned issues of a
given maturity.
• In the U.S., these issues include the 1-month, 3-month, and 6-month
Treasury bills, and the 2-year, 5-year, 10-year Treasury notes.
• Only those on-the-run coupon issues trading at par are used.
• Reason: to eliminate the effect of the tax treatment for securities selling at a
discount or premium.
11
On-the-Run Treasury Issues and Selected Off-the-Run
Treasury Issues
• One of the problems with using just the on-the-run issues is the large gap
between maturities.
• To mitigate this problem, selected off-the-run Treasury issues are included,
and a linear interpolation method is used to fill in the gaps for the other
maturities.
• The bootstrapping method is then used to construct the theoretical spot rate
curve.
12
All Treasury Coupon Securities and Bills
• Using on on-the-run issues and a few off-the-run issues fails to recognize
the information embodied in Treasury prices that are not included in the
analysis.
• Some argue that it is more appropriate to use all outstanding Treasury
coupon securities and bills.
• When all coupon securities and bills are used, statistical methodologies
must be employed rather than bootstrapping.
• Several statistical methodologies have been proposed, including quadratic
spline and cubic spline by McCulloch(1971,1975), exponential splines by
Vasicek and Fong(1982), Nelson-Siegel function by bNelson and
Siegel(1987), and smoothing spline by Fisher, Nychka and Zervos(1995)
etc.。
13
Expectations Theories of the Term Structure of Interest
Rates
• The three forms of the expectations theory (the pure expectations theory,
the liquidity preference theory, and the preferred habitat theory) assume
that the forward rates in current long-term bonds are closely related to the
market’s expectations about future short-term rates.
• The three forms of the expectations theory differ on whether or not other
factors also affect forward rates, and how.
• The pure expectations theory postulates that no systematic factors other
than expected future short-term rates affect forward rates.
14
The Pure Expectation Theory
There are several interpretations of the pure expectations theory:
• Broadest Interpretation
The broadest interpretation of the pure expectations theory suggest that
investors expect the return for any investment horizon to be the same,
regardless of the maturity strategy selected.
• Local Expectations
Local expectations suggest that the return will be the same over a shortterm investment horizon starting today.
• Return-to-maturity Expectations Interpretation
It suggests that the return that an investor will realize by rolling over shortterm bonds to some investment horizon will be the same as holding a zerocoupon bond with a maturity that is the same as that investment horizon.
15
Liquidity Preference Theory
• This theory states that investors will hold longer-term maturities if they are
offered a long-term rate higher than the average of expected future rates by
a risk premium that is positively related to the term to maturity.
• The forward rates should reflect both interest rate expectations and a
“liquidity” premium and the premium should be higher for longer
maturities.
• Therefore, forward rates will not be an unbiased estimate of the market’s
expectations of future interest rates
• Thus, an upward-sloping yield curve may reflect expectations that future
interest rates either (1) will rise, or (2) will be unchanged or even fall.
16
The Preferred Habitat Theory
• The preferred habitat theory asserts that if there is an imbalance between
the supply and demand for funds within a given maturity range, investors
and borrowers will not be reluctant to shift their investing and financing
activities out of their preferred maturity sector to take advantage of any
imbalance.
• This theory also adopts the view that the term structure reflects the
expectation of the future path of interest rates as well as a risk premium.
• The theory rejects the assertion that the risk premium must rise uniformly
with maturity.
17
The Swap Curve
• The swap rate is the rate at which fixed cash flows can be exchanged for
floating cash flows.
• In a LIBOR-based swap, the swap curve provides a yield curve for LIBOR.
• The swap spread is primarily a gauge of the credit risk associated with a
country’s banking sector.
• Reasons for increased used of swap curve:
1) No government regulation,
2)The supply depends only on the number of counterparties.
3) Comparison across countries is meaningful.
4) There are more maturity points available to construct a swap curve.
• Similarly, LIBOR spot rate curve can be derived from the swap curve, and
the LIBOR forward rate curve can be further derived from the spot rate
curve.
18
Valuation Approaches
• Traditional approach
To discount every cash flow of a fixed income security by the same
discount rate.
• The Arbitrage-free approach
Any financial asset is viewed as a package of zero-coupon bonds.
• Valuation models: binomial model and Monte Carlo simulation model.
19
General Principles of Valuation
• The fair value is equal to the present value of its expected cash flows.
• Step 1: Estimate the expected cash flows.
• Step 2: Determine the appropriate interest rate or interest rates.
• Step 3: Calculate the present value of the expected cash flows found in step
1 using the interest rate or interest rates determined in step 2.
20
Present Value
• A lump sum payment
PV 
Pn
(1  r ) n
PV 
Pn
(1  r / m) nm
• An ordinary annuity
1  (1  r )  n 
 1
1
1 
PV  A  


 A 

2
n
(
1

r
)
r
(
1

r
)
(
1

r
)




• An annuity due
PV= (1+r) × ordinary annuity PV
21
Future Value
• A lump sum payment
FV  P0 (1  r ) n
FV  P0 (1  r / m) nm
• An ordinary annuity

FV  A  (1  r )  (1  r ) 2   (1  r ) n

 (1  r ) n  1
 A 

r


• An annuity due
FV=(1+r)×ordinary annuity FV
22
Pricing a Bond
• 對固定利率債券而言，其評價過程相當於an ordinary annuity現值與a
lump sum現值的合計數或表示為
 1
1  (1  r )  n 
1
1 
1
1
P  A 




M


A


M




2
r
(1  r ) n 
(1  r ) n
(1  r ) n


 (1  r ) (1  r )
P為債券價格，r為貼現率，A為每期利息以及M為面額。
• 若r為required rate，則等號右邊現值合計數為投資人主觀評估債券的
合理價格
• 若r為ytm，則現值合計數等於市價。(ytm是為期間的平均報酬率)
23
Pricing Zero-Coupon Bonds
• For zero-coupon bonds, the investor realizes interest as the difference
between the maturity value and the purchase price. The equation is:.
P
M
(1  r )n
24
Next Coupon Payment Due in Less than Six Months
• When 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
C
M
+
V
t 1
(1 + r )V (1 + r )n1
t =1 (1 + r ) (1 + r )
P =
where v = (days between settlement and next coupon) / (days in six-month period).
25
Price-Yield Relationship
•
債券價格(p)與收益率(r)呈反向關係且為非線性
•
債券價格與收益率呈convex，且為positive convexity，不過這是
針對純債券而言，若為callable bonds則可能出現部分曲線呈
negative convexity的現象。
26
Relationship between Coupon Rate, Discount Rate, and
Price Relative to Par Value
• Coupon rate = ytm, therefore price = par value
• Coupon rate < ytm, therefore price < par value (discount)
• Coupon rate > ytm, therefore price > par value (premium)
27
債券價格變動因素
•
即使coupon rate維持不變，債券價格可能因下列因素而變動:
1. 發行人的信用評等改變引起必要報酬率改變
2. 市場因素使得必要報酬率改變
3. 時間流逝會產生pull-to-par的現象。
28
Change in a Bond’s Value Toward Maturity
1.
Decreases over time if the bond is selling at a premium
2.
Increases over time if the bond is selling at a discount
3.
Is unchanged if the bond is selling at par value
29
Movement of Bonds Towards Maturity
Years
Premium bond
Discount bond
Par bond
20
119.63
80.36
100.00
18
118.74
81.26
100.00
16
117.70
82.30
100.00
14
116.49
83.51
100.00
12
115.07
84.93
100.00
10
113.42
86.58
100.00
8
111.49
88.51
100.00
6
109.25
90.75
100.00
4
106.62
93.38
100.00
2
103.57
96.43
100.00
0
100.00
100.00
100.00
30
Bond Equivalent Yield and Effective Yield
Bond-equivalent yield: for a semiannual pay bond, doubling the periodic
interest rate
effective annual yield = (1 + periodic interest rate)^m – 1
31
當期收益率(current yield, cy)
coupon
cy 
price
• 一簡單、方便的報酬指標
• 不考慮債券的資本損利、再投資利息以及到期期限的長短
等因素，因此有時會是一個失真的指標
• 當到期期限很長時，此時當期收益率趨近於到期收益率，
可以當期收益率替代到期收益率。
32
到期收益率(yield to maturity, ytm)
T
c
M
p0  

t
T
(
1

ytm
)
(
1

ytm
)
t 1
 c  PVIFAytm,T  M  PVIFytm,T
1  (1  ytm) T
or  c  
ytm


M

T
(
1

ytm
)

33
ytm (continued)
•
•
•
•
•
到期收益率又稱為殖利率
到期收益率是衡量持有至到期的報酬率
修正當期收益率的諸多缺點
一內部收益率(internal rate of return)的概念
假設，就是每一筆現金流量均以ytm進行再投資 (缺點)
34
Yield to Call
• the yield to call can be expressed as follows:
P=
C
C
C
C
M*
+
+
+
.
.
.
+
+
(1 + y) (1 + y )2 (1 + y )3
(1 + y )n* (1 + y )n*
• where M* = call price (in dollars) and n* = number of periods until the
assumed call date (number of years times 2)
• It assumes that the coupon interest payments will be reinvested at the yield
to call..
35
Yield to Put
• When an issue is putable, a yield to put is calculated.
• The yield to put is the interest rate that makes the present value of the cash
flows to the assumed put date plus the put price on that date as set forth in
the put schedule equal to the bond’s price.
• The formula is the same as for the yield to call, but M* is now defined as
the put price and n* is the number of periods until the assumed put date.
The procedure is the same as calculating yield to maturity and yield to call.
36
Yield to Worst
• A practice in the industry is for an investor to calculate the yield to maturity,
the yield to every possible call date, and the yield to every possible put date.
•
The minimum of all of these yields is called the yield to worst.
37
Cash Flow Yield
• For amortizing securities, the cash flow each period consists of three
components: (i) coupon interest, (ii) scheduled principal repayment, and (iii)
prepayments.
• For amortizing securities, market participants calculate a cash flow yield.
• It is the interest rate that will make the present value of the projected cash
flows equal to the market price.
38
Yield for a Portfolio
• The yield for a portfolio of bonds is not simply the average or weighted
average of the yield to maturity of the individual bond issues in the
portfolio.
• It is computed by determining the cash flows for the portfolio and
determining the interest rate that will make the present value of the cash
flows equal to the market value of the portfolio.
39
永續債券的評價、當期收益率與到期收益率


c
c
1
p0  

 (1  ytm)t
t
(
1

ytm
)
(
1

ytm
)
t 1
t 0
c
1
c
(1  ytm)




1
(1  ytm)
(1  ytm)
ytm
1
1  ytm
 c
ytm
c
p0 
ytm
c
 ytm 
p0
因此永續債券的當期收益率等於當期收益率。
40
ytm for a zero-coupon bond
1/ n
 M
y= 
 P
1
41
Sources of Return
• An investor who purchases a bond can expect to receive a dollar return
from one or more of these sources:
1. the coupon interest payments made by the issuer
2. any capital gain (or capital loss) when the security matures, is called, or
is sold
3. income from reinvestment of interim cash flows
42
Interest-on-Interest Dollar Return
• The coupon interest plus interest on interest :
 (1 + r )n  1
C

r


• The total coupon interest: nC
• The interest-on-interest:
 (1 + r )n  1
C
  nC
r


43
An Example on Sources of Return
Suppose that an investor purchases a bond with a coupon rate of 7% that
matures in 8 years. The price and the yield to maturity for this bond are
$94.17 and 8%, respectively.
An investment of $94.17 will generate $176.38. (future value)
Coupon interest = $3.5 * 16 = $56
Capital gain = $100 - $94.17 = $5.83
Interest-on-interest = $76.38 - $56 = $20.38
44
Yield to Maturity and Reinvestment Risk
• The investor will realize the yield to maturity at the time of purchase only if
the bond is held to maturity and the coupon payments can be reinvested at
the computed yield to maturity.
• The risk that the investor faces is that future reinvestment rates will be less
than the yield to maturity at the time the bond is purchased. This risk is
referred to as reinvestment risk.
• The longer the maturity, the greater the reinvestment risk.
• The longer the coupon rate, the greater the reinvestment risk.
45
Total Return
• The total return is a measure of yield that incorporates an explicit
assumption about the reinvestment rate.
• Step 1: Compute the total coupon payments plus the interest on interest
based on the assumed reinvestment rate.
• Step 2: Determine the projected sale price at the end of the planned
investment horizon.
• Step 3: Sum the values computed in steps 1 and 2.
• Step 4: To obtain the semiannual total return by [total future
dollars/purchase price]^(1/h) -1
• Step 5: Double the interest rate found in step 4 and the resulting interest
rate is the total return.
46
An Example on Total Return
• Suppose that an investor has a six-year investment horizon. The investor is
considering q 13-year 9% coupon bond selling at par. The investor’s
expectations are as follows:
• 1. The first four semiannual coupon payments can be reinvested from the
time of receipt to the end of the investment horizon at a simple annual
interest rate of 8%.
• 2. The last eight semiannual coupon payments can be reinvested from the
time of receipt to the end of the investment horizon at a 10% simple annual
interest rate.
• 3. The required yield to maturity on seven-year bonds at the end of the
investment horizon will be 10.6%.
47
An Example on Total Return (Continued)
• Step 1: The coupon interest plus interest on interest for the first four
coupon payments is $191.09 (assume coupon payment $45). Reinvested at
4% until the end of the planned investment horizon will grow to $261.52.
For the last eight coupon payments, the coupon interest plus interest on
interest is $429.71. The total is $691.23.
• Step 2: the projected sale price of the bond is $922.31.
• Step 3: The total future dollars are $1,613.54.
• Step 4: [$1,613.54/$1,000]^(1/12) - =4.07%
• Step 5: 4.07% * 2 = 8.14%
48
Yield Changes
• Absolute yield change (in basis points) = | initial yield – new yield| * 100
• Percentage yield change = 100 * ln (new yield / initial yield)
49