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Academy of Professional Finance 专业金融学院
CFA Level I
FIXED INCOME:
Yield Measures, Spot Rates and Forward Rates
Lecturer: Nan Chen
Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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1. Sources of Return
Source 1
Periodic coupon interest payments;
Source 2
Recovery of principal, along with any capital gain or loss
Source 3
Reinvestment income
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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2.1 Current Yield
*Current yield=Annual Cash Coupon Payment/Bond Price
*EX1: Consider a 10-year, $1000 par value, 6% coupon,
semiannual-pay bond that is currently trading at $894.5.
Calculate the current yield.
Annual cash coupon payment = 1000 * 6%= 60
current yield = 60/894.5=6.71%
For an annual-pay bond and a semiannual-pay bond with the
same coupon rate and price, they have the same current yield.
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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2.2 Yield to Maturity (YTM)
Semiannual-pay Coupon Bond
*YTM: an annualized internal rate of return, based on a bond’s
price and its promised cash flows.
*Bond Price 
C
C
C  Par

 ...
YTM
YTM 2
YTM 2 N
1
(1 
)
(1 
)
2
2
2
C=Semiannual coupon payment;
2N=Number of semiannual periods.
*EX2: Consider a 10-year, $1000 par value, 6% coupon semiannualpay bond that is currently trading at $894.5. Calculate the YTM.
-894.5 [PV]
[FV]
[N]
30
[PMT]
[CPT] [1/Y] 得到3.76
30
30  1000
30
$894.5 

 ...
1000
YTM
YTM 2
YTM 210
(1 
)
(1 
)
1
20
2
2
2
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YTM=1/Y ×2=7.52
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Bond Equivalent Yield and Annual Pay Yield
EX3: The yield of a 3-year bond issue quoted on an annual-pay basis is
84%. The yield-to-maturity on a bond-equivalent basis is closest to:
A. 3.85%
B. 7.69%
C. 7.84%
(1+bond-equivalent yield/2)
2
=1+annual-pay yield
In this case,
(1+bond-equivalent yield/2) 2 =1+0.0784
Therefore, bond-equivalent yield=7.69%
B is the correct answer.
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2.2 Yield to Maturity (YTM)
Annual-pay Coupon Bond
EX4: Consider a 10-year, $1000 par value, 6% coupon annual-pay
bond that is currently trading at $894.5. Calculate the YTM.
-894.5 [PV]
1000
[FV]
10
[N]
60
[PMT]
[CPT] [1/Y] 得到7.54
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YTM=1/Y =7.54
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2.2 Yield to Maturity (YTM)
*Coupon Rate and YTM
Trade at par
Coupon Rate= YTM
Trade at premium
Coupon Rate> YTM
Trade at discount
Coupon Rate < YTM
*Coupon Rate, Current Yield and YTM
Bond selling at par
Coupon Rate =Current Yield= YTM
Bond selling at premium
Coupon Rate >Current Yield >YTM
Bond selling at discount
Coupon Rate <Current Yield <YTM
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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2.3 Yield to Call (YTC)
*YTC: used to calculate the yield on callable bonds that are selling
at a premium to par.
In this case, YTC < YTM ; call price < current market price
*Bond Price 
C
C
C  Call Pr ice

 ...
YTC
YTC 2
YTC 2YearstoCallDate
1
(1 
)
(1 
)
2
2
2
YTC
YTM
[FV]
Call Price
Par Value
[N]
Number of Periods to Call Date
Number of Periods to Maturity
*EX5:A semiannual-pay bond is callable in five years at $106. The bond has an 8%
upon and 15 years to maturity. If the bond is currently trading at $98 today, the yield
call is closest to:
Time to call is 5 years and semi-annual pay=> number of periods till call date N=10,
A. 8.22% Call Price = $106 => FV=106,
B. 8.49% 8% coupon and semi-annual pay=> PMT=4,
C. 9.48% PV=-98
CPT -> 1/Y=4.7386
YTC=4.7386*2=9.48
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2.3 Yield to Call (YTC)
Yield to Par Call
Yield to Par Call
YTM
[FV]
Par Value (call at par value)
Par Value
[N]
Number of Periods to Par Call Date
Number of Periods to Maturity
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Yield to Worst (YTW)
*YTW: the lowest yield of any that are possible given the call
provisions of the bond.
*EX6: A 10% annual coupon bond with 3 years to maturity is
currently trading at $1,010. The bond is callable in one year at a call
price of $1,008 and in two years at a call price of $1,005. The
bond’s yield to worst most likely occurs when the bond is:
A. held until maturity in 3 years.
B. called in year 1.
C. called in year 2.
The yield to worst for a callable bond is the lowest of the YTCs for each possible call date and YTM
YTC if the bond is called in 1 year is 10.45%, because 1,005=(100+1,010)/1.1045
YTC if the bond is called in 2 years is 10.09% , because 1,005=100/1.1009+(100+1,008)/1.10092
YTM of the bond is 9.80%, because 1,005=100/1.0980+100/1.0980 2+(100+1,000)/1.09803
=> YTW=the lowest yield of the three = YTM =9.80%
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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2.4 Yield to Put (YTP)
YTP
YTM
[FV]
Put Price
Par Value
[N]
Number of Periods to Put Date
Number of Periods to Maturity
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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2.5 Yield to Refunding
Yield to Refunding: refers to a specific situation where a bond is
currently callable but not refundable.
Yield to Refunding
YTM
[FV]
Call Price
Par Value
[N]
Number of Periods till refund protection ends
Number of Periods to Maturity
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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2.6 Cash Flow Yield (CFY)
*CFY: the IRR that will make the present value of projected cash flows
equal to the market price of the security.
*Calculated for mortgage-backed securities and asset-backed securities
CF1
CF 2
CFn

 ...
SecurityMarket Pr ice 
2
1  CFY (1  CFY )
(1  CFY ) N
*Convert a Monthly CFY to a Bond Equivalent CFY:
Bond Equivalent CFY= 2 × Semiannual Yield= 2 ×[(1+monthly CFY)6-1]]
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Limitations of Traditional Yield Measures
Yield Measures
Limitations
Current Yield
Considers only coupon payment; does not consider capital
gains/losses or reinvestment income.
YTM
Assumes:
1). Reinvestment rate= YTM
Cash flows of each period are reinvested at YTM
Cash flows of each period are discounted at the same rate of YTM
Yield curve is flat
2). The bond will be held until maturity
YTC, YTP,
YTRefunding,
CFY
Similar measure to YTM;
Same issue as YTM except that the bond will be held until maturity.
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Limitations of Traditional Yield Measures
EX7: Elaine Wong has purchased an 8% coupon bond for $1,034.88
ith 3 years to maturity. At what rate must the coupon payments be
einvested to produce a 5% yield-to-maturity rate?
A. 8%
B. 6.5%
C. 5%
C is the correct answer. Yield-to-maturity measure assumes that the
oupon payments can be reinvested at YTM. In this case, it’s 5%.
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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3.1 Reinvestment Income
*Reinvestment Rate, YTM and Realized Yield:
When:
Reinvestment Rate< YTM
Reinvestment rate<Realized Yield < YTM
Reinvestment Rate> YTM
Reinvestment rate>Realized Yield > YTM
Realized yield is always between YTM and reinvestment rate.
*Required Reinvestment Income to achieve the target yield:
EX8:To provide an investor with a compound return of 6.5% on a
semiannual basis, how much reinvestment income must be generated
for a bond with 6% coupon, 10- year Treasury bond purchased at $96?
-To achieve the 6% return, the total value that must be generated 10 years (20
semiannual periods) from now=96×（1+6.5%/2）20=$182.00
-20 coupon payments of $3 each: $3 ×20=$60
-Payment of $100 of principal at maturity
The required reinvestment income to achieve the target yield= 182.00-60-100=$22
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3.2 Reinvestment Risk
*Reinvestment Risk: the possibility that cash flows prior to stated maturity
will be reinvested at a lower yield than pre-specified yield.
*Factors that contribute to a higher reinvestment risk:
Characteristic
Reinvestment Risk
Higher coupon rate
Higher
Longer maturities
Higher
Has a call feature
Higher
An amortizing security
Higher
Contains a prepayment option
Higher
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3.2 Reinvestment Risk
EX9: Consider the three bonds in the following table. Which of the
three bonds is most likely to have the greatest reinvestment risk?
Bond
YTM
Time to Maturity
Current Price
A
8%
15
$980
B
8%
15
$1,000
C
8%
15
$1,098
A. Bond A
B. Bond B
C. Bond C
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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4. Theoretical Spot Rate Curve
*Spot Rate: the yield of a default-free zero-coupon bond.
*Treasury securities are considered default-free.
=> YTM of treasury securities of different maturities are spot rates.
*The US government issues zero-coupon bonds at maturities under 1 year.
=> We need to calculate the spot rates for periods longer than 1 year.
=> Spot rates calculated by ourselves are theoretical spot rates.
*Bootstrapping: we will solve for the spot rates by knowing the
prices of the coupon bonds.
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Bootstrapping
Annual-pay Bond
Prices and Yield of Two Annual-pay Bonds
Maturity
Coupon
Yield
Price
1 year
3%
3%
$1,000
2 years
4%
4%
$1,000
=>2-year spot rate = 4.019%
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Bootstrapping
Semiannual-Pay Bond
Prices and Yields of Two Semiannual-pay Bonds
Maturity
Coupon
Yield
Price
6 months
5%
5%
$100
1 year
6%
6%
$100
=>1-year Bond Equivalent spot rate S1.0= 2×0.030076=6.0152%
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4. Theoretical Spot Rate Curve
EX10: Tina Mo, a fixed income analyst, is asked to value a single,
default-free cash flow of $60,000. She is given the information in
the following table:
Period
Years
Annual Par Yield to Maturity BEY
Theoretical Spot Rate BEY
6‐month Forward Rates BEY
1
0.5
2.00%
2.00%
2.00%
2
1.0
2.40 %
2.40%
2.71%
3
1.5
2.70%
2.71%
3.12%
4
2.0
3.20%
3.23%
4.55%
The value of this single cash flow at the end of Period 4 is closest to:
A. $56,427
B. $56,309
C. $56,276
The theoretical spot rate for Treasury securities represent the appropriate set of interest
rates that should be used to value single, default-free cash flows.
Therefore, the current value of the$60,000 at the end of Period 4 should be discounted
at the Period 4 spot rate= $60,000/(1+0.0323/2)4=$56,276
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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5.1 Nominal Spread
*Nominal Spread
= YTM of an issue - YTM of a treasury security of similar maturity
*Nominal Spread ignores the shape of the spot yield curve.
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5.2 Zero-volatility Spread
Maturity
Coupon
Nominal Spread
The equal amount added to the Treasury YTM that produces a bond
value equal to the market price of the bond
Z-Spread
The equal amount added to each rate on the Treasury spot yield curve
that makes the PV of a risky bond’s cash flows equal to its market price
EX11:Consider a 3-year, 9% annual coupon corporate bond trading at
$89.464. 1-, 2-, 3- year spot rates on Treasuries are 4%, 8.167%, and
12.377%. Compute the zero-volatility spread of the corporate bond.
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5.2 Zero-volatility Spread
EX12:The zero-volatility spread is a measure of the spread off:
A. one point on the Treasury yield curve.
B. all points on the Treasury yield curve.
C. all points on the Treasury spot curve.
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Nominal Spread an Z-Spread
Three factors that influence the difference between nominal spread and Z-spread:
* The steeper the benchmark spot rate curve, the greater the difference between
the two spread measures.
A flat spot yield curve
Z-Spread = Nominal Spread
An upward sloping spot yield curve
Z- Spread > Nominal Spread
A downward sloping spot yield curve
Z- Spread < Nominal Spread
* The earlier the bond principal is paid, the greater the difference between the
two spread measures.
An upward sloping spot yield curve
The difference between the two spread measures is greater
for an amortizing security than for a bullet security.
* The longer the maturity, the greater the difference between the two spread
measures.
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Nominal Spread an Z-Spread
EX13: The difference between Z-spread and nominal spread will most
likely be the most significant for a:
A. Treasury security with short maturity in a flat yield curve environment
B. zero coupon Treasury security.
C. mortgage-backed security in a steep upward-sloping yield curve
environment
The difference between the two spread measures grows with the maturity of
the security.
=> A is incorrect.
The difference between the two spread measures is greater for issues in
which the principal is repaid over time rather than only at maturity.
=>B is incorrect and C is the correct answer.
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5.3. Option-adjusted Spread(OAS)
*OAS is used when a bond has embedded options.
*OAS takes the option yield component out of the Z-spread measure.
OAS= Z-spread - option cost in percent
For callable bonds and MBS, option cost >0, OAS < Z-spread
For putable bonds,
option cost <0, OAS > Z-spread
*OAS is the spread for non-option characteristics like
credit risk, liquidity risk and interest rate risk.
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5.3. Option-adjusted Spread(OAS)
EX14: Which of the following statement is correct about the option
adjusted spread ( OAS ):
A. OAS is Z-Spread minus the option cost.
B. OAS is the value of the embedded option.
C. OAS is Z-spread plus the option cost.
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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6.1 Relationship between Forward Rates and Spot Rates
Measure:
Definition
Forward Rate
The borrowing/lending rate of a loan to be made at some future date.
Spot Rate
The borrowing/lending rate of a loan to be made today.
1f2:
the one-year forward rate, two years from now.
2-year bond (today): S2
1-year bond (today): S1=1f0
(1+S2)=(1+0f1)(1+1f1)
1-year bond (one year from today): 1f1
forward rate => spot rate
spot rate
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=> forward rate
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6.1 Relationship between Forward Rates and Spot Rates
EX15: Using the BEY (bond-equivalent yield) spot rates for U.S. Treasury
yields provided in the following table, the 6-month forward rate one year
from now on a bond-equivalent yield basis is closest to:
Period
Years
Spot Rate
1
0.5
1.40%
2
1.0
2.30 %
3
1.5
3.00%
4
2.0
3.50%
A. 4.41%
B. 2.20%
C. 2.30%
Answer:
xfy represents x-period forward rate y periods from now;
Z x+y represents (x+y)-period spot rate;
Z y represents y-period spot rate.
We have (1+Z x+y)x+y=(1+Zy)y (1+xfy)x
6-month forward rate one year from now in this case is 1 period forward rate 2 periods from now.
All spot rates are given on a BEY basis and must be divided by 2 in the calculation:
(1+0.03/2)3= (1+0.023/2)2 (1+1f 2)1
1f 2=0.022038
On a BEY basis, the forward rate is 0.022038*2=4.41%
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6.2 Valuing a Bond using Forward Rates
EX16: The current 1-year rate is 4%, the 1-year forward rate for
lending from time=1 to time=2 is 1f1=5%, and the 1-year forward
rate for lending from time=2 to time=3 is 1f2=6%. Value a 3- year
annual-pay bond with a 5% coupon and a par value of $1,000.
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Framework
Coupon Interest Payments
1. Sources of Return
Capital Gains or Losses
Reinvestment Income
Current Yield
Yield to Maturity
2. Traditional Yield Measures
Yield to Call
Yield to Put
Yield to Refunding
Cash Flow Yield
3. Reinvestment Income and Reinvestment Risk
4. Theoretical Spot Rate Curve
5. Nominal Spread, Z-Spread, and Option-adjusted Spread
6. Spot Rates, Forward Rates and Value of Bonds
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