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Valuing Cash Flows
Non-Contingent Payments
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Non-Contingent Payouts
Given an asset with fixed payments (i.e.independent of the state of the world), theassets price should equal the presentvalue of the cash flows.
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Treasury Notes
US Treasuries notes have maturitiesbetween 2 and ten years.Treasury notes make biannual interestpayments and then a repayment of theface value upon maturityUS Treasury notes can be purchased inincrements of $1,000 of face value.
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Consider a 3 year Treasury note with a 6%annual coupon and a $1,000 face value.
Now 6mos 1yrs 2yrs1.5 yrs 2.5yrs 3yrs
$30 $30 $30 $30 $30 $1,030
F(0,1) = 2.25%
F(1,1) = 2.75%F(2,1) = 2.8%
F(3,1) = 3%
F(5,1) = 4.1%
F(4,1) = 3.1%
You have a statisticalmodel that generates thefollowing set of(annualized) forwardrates
F(0,1) F(1,1) F(2,1) F(3,1) F(5,1)F(4,1)
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Now 6mos 1yrs 2yrs1.5 yrs 2.5yrs 3yrs
$30 $30 $30 $30 $30 $1,030
2.25% 2.75% 2.8% 3% 4.1%3.1%
Given an expected path for (annualized) forward
rates, we can calculate the present value of futurepayments.
P = $30(1.01125)
+ $30(1.01125)(1.01375)
+ + $30(1.01125)(1.01375)(1.014)
= $1,084.90+ $1,030(1.01125).(1.0205)
+
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Forward Rate Pricing
N
t t
i i
t
F
CF P
1 1 1
0
1
Current Asset Price Cash Flow at time t
Interest rate betweenperiods t-1 and t
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Now 6mos 1yrs 2yrs1.5 yrs 2.5yrs 3yrs
$30 $30 $30 $30 $30 $1,030
$30 $30$30$30$30 $1,030= +++++P(1.0125) (1.0125) (1.0135) (1.0135) (1.015) (1.015)2 3 4 5 6
P = $1,084.90
S(1)2
S(2)2
S(3)2
The yield curve produces the same bondprice..why?
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Spot Rate Pricing
N
t t
t
t S
CF P
10
)(1
Current Asset Price Cash flow at period t
Current spot rate for amaturity of t periods
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Alternatively, given the current price, what isthe implied (constant) interest rate.
Now 6mos 1yrs 2yrs1.5 yrs 2.5yrs 3yrs
$30 $30 $30 $30 $30 $1,030
$30 $30$30$30$30 $1,030= +++++
(1+i) (1+i) (1+i) (1+i) (1+i) (1+i)2 3 4 5 6
P = $1,084.90
P
(1+i) = 1.015 (1.5%)
Given the current ,market price of$1,084.90, this Treasury Note has anannualized Yield to Matu ri ty of 3%
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Yield to Maturity
N
t
t
t
Y CF P
1
0
1Current Market Price
Yield to Maturity
Cash flow at time t
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Yield to maturity measures the total performanceof a bond from purchase to expiration.
Consider $1,000, 2 year STRIPselling for $942
$942 = $1,000(1+Y) 2
(1+Y) =$1,000
$942
.5
= 1.03 (3%)
For a discount (one payment) bond, the YTM is equal to the expected spot rate
For coupon bonds, YTM is cash flow specific
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Consider a 5 yearTreasury Note with a 5%annual coupon rate (paid
annually) and a face valueof $1,000
$50 $50$50$50$50= ++++
(1.05) (1.05) (1.05) (1.05) (1.05)2 3 4 5P = $1,000
The one year interest rate iscurrently 5% and is expectedto stay constant. Further,there is no liquidity premium
Term
Yield
5%
This bond sells for Par Value and YTM = Coupon Rate
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Price
Yield
$958
5% 6%
$1,000$42
A 1% rise in yield is associated with
a $42 (4.2%) drop in price
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Consider a 5 yearTreasury Note with a 5%annual coupon rate (paid
annually) and a face valueof $1,000
$50 $50$50$50$50= ++++
(1.04) (1.04) (1.04) (1.04) (1.04)2 3 4 5P = $1045
Now, suppose that thecurrent 1 year rate falls to 4%and is expected to remainthere
Term
Yield
5%
4%
This bond sells at a premium and YTM < Coupon Rate
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Price
Yield
$958
5% 6%4%
$1,045
$1,000$45
$42
A 1% drop in yield is associated
with a $45 (4.5%) rise in price
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Price
Yield
$958
5% 6%4%
$1,045
$1,000$45
$42
PricingFunction
A bonds pricing functionshows all the combinations ofyield/price
1) The bond pricing is non-linear
2) The pricing function is uniqueto a particular stream of cashflows
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Duration
Recall that in general the price of a fixedincome asset is given by the followingformula
Note that we are denoting price as afunction of yield: P(Y).
n
1i 1 P(Y) i
i
Y
CF
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$50 $50$50$50$50= ++++(1.05) (1.05) (1.05) (1.05) (1.05)2 3 4 5
P(Y=5%) = $1,000
Term
Yield
5%
This bond sells for Par Value and YTM = Coupon Rate
For the 5 year, 5% Treasury, we had thefollowing:
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Price
Yield5%
$1,000
PricingFunction
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n
1i
1
1
*
dY
dPi
i
Y
CF i
Suppose we take the derivative ofthe pricing function with respect to
yield
65432 Y)(1
$1,0505Y)(1
$504Y)(1
$503Y)(1
$502Y)(1
$50- dYdP
For the 5 year, 5% Treasury, we have
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Now, evaluate that derivative at a particularpoint (say, Y = 5%, P = $1,000)
329,4$
05).(1$1,0505
05).(1$504
05).(1$503
05).(1$502
05).(1$50-
dYdP
65432
For every 100 basis point change in theinterest rate, the value of this bond changesby $43.29 This is the dollar duration
DV01 is the change in a bonds price perbasis point shift in yield. This bondsDV01 is $.43
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Price
Yield
$958
5% 6%4%
$1,045
$1,000Error = - $1
PricingFunction
Error = $2
Duration predicted a$43 price change forevery 1% change inyield. This is differentfrom the actual price
DollarDuration
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P
1
*dY
dP
DurationModified
Dollar duration depends on the face value of thebond (a $1000 bond has a DD of $43 while a$10,000 bond has a DD of $430) modified duration
represents the percentage change in a bon dspr ice du e to a 1% ch ange in yie ld
For the 5 year, 5% Treasury, we have
3.4000,1$329,4$1*
dYdP MD
P
Every 100 basis point shift in yield alters this bonds price by4.3%
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Macaulay's Duration
P Y )1(*
dYdp DurationsMacaulay'
Macaulay duration measures the percentage change in abonds price for every 1% change in (1+Y)
(1.05)(1.01) = 1.0605
For the 5 year, 5% Treasury, we have
55.4000,1$
)05.1(329,4$)1(*
dYdP
DMac P
Y
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Think of a coupon bond as a portfolio of STRIPS. Eachpayment has a Macaulay duration equal to its date. Thebonds Macaulay duration is a weighted average of the
individual durations
Back to the 5 year Treasury
$50 $50$50$50$50= ++++(1.05) (1.05) (1.05) (1.05) (1.05)2 3 4 5
P(Y=5%) = $1,000
$47.62 $822.70$41.14$43.19$45.35
$47.62$1,000
$45.35$1,000
$43.19$1,000
$41.14$1,000
$822.70$1,000
+ + + +1 2 3 4 5
Macaulay Duration = 4.55
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Macaulay Duration = 4.55
Modified Duration =Macaulay Duration
(1+Y)
Modified Duration =4.551.05
= 4.3
Dollar Duration = Modified Duration (Price)
Dollar Duration = 4.3($1,000) = $4,300
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Duration measures interest rate risk (therisk involved with a parallel shift in the yield
curve) This almost never happens.
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Yield curve risk involves changes in an assets pricedue to a change in the shape of the yield curve
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Key Duration
In order to get a better idea of a Bonds (orportfolios) exposure to yield curve risk, akey rate duration is calculated. Thismeasures the sensitivity of a bond/portfolioto a particular spot rate along the yieldcurve holding all other spot rates constant.
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Key Durations
45.35
86.38
123.41
156.71
39.18
0
20
40
6080
100
120
140
160
1Yr 2Yr 3Yr 4Yr 5Yr
Note that the individual key durations sum to $4329 the bonds overall duration
X 100
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0
1
2
3
4
5
6
7
1 yr 2yr 3yr 4yr 5yr
- 4%- 2%0%+1%
+1%
+ + + +1 1 0 (-2) (-4)$.4535 $.8638 $.12341 $.15671 $39.81
This yield curve shift would raise a five year Treasury price by$161
= $161
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Price
Yield
$958
6%4%
$1,045
Suppose that we simply calculatethe slope between the two pointson the pricing function
Slope = $1,045 - $958
4% - 6%= $43.50
or
Slope =
$1,045 - $958
4% - 6%
$1,000*100
= 4.35
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Price
Yield
$958
6%4%
$1,045
PricingFunction
Dollar
Duration
EffectiveDuration
Effective duration measures interest ratesensitivity using the actual pricing functionrather that the derivative. This is particularlyimportant for pricing bonds with embeddedoptions!!
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Value At RiskSuppose you are a portfolio manager.The current value of your portfolio is aknown quantity.
Tomorrows portfolio value us an
unknown, but has a probabilitydistribution with a known mean andvariance
Profit/Loss = Tomorrows Portfolio Value Todays portfolio value
Known Distribution Known Constant
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Probability Distributions
One Standard Deviation Around
the mean encompasses 65% ofthe distribution
1 Std Dev = 65%
2 Std Dev = 95%
3 Std Dev = 99%
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Interest RateMean = 6%
Std. Dev. = 2%
$1,000, 5 Year Treasury (6% coupon)
Remember, the5 year Treasury
has a MD 0f 4.3
Mean = $1,000Std. Dev. = $86
Profit/LossMean = $0Std. Dev. = $86
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One Standard Deviation Aroundthe mean encompasses 65% ofthe distribution
1 Std Dev = 65%
2 Std Dev = 95%
3 Std Dev = 99%
The VAR(65) for a $1,000, 5 Year Treasury(assuming the distribution of interest rates) would
be $86. The VAR(95) would be $172
In other words, there isonly a 5% chance oflosing more that $172
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Interest RateMean = 6%
Std. Dev. = 2%
$1000, 30 Year Treasury (6% coupon)
A 30 yearTreasury has a
MD of 14
Mean = $1,000Std. Dev. = $280
Profit/LossMean = $0Std. Dev. = $280
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One Standard Deviation Aroundthe mean encompasses 65% ofthe distribution
The VAR(65) for a $1,000, 30 Year Treasury(assuming the distribution of interest rates) would
be $280. The VAR(95) would be $560
In other words, there isonly a 5% chance oflosing more that $560
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Example: Orange County
In December 1994, Orange County, CAstunned the markets by declaringbankruptcy after suffering a $1.6B loss.The loss was a result of the investmentactivities of Bob Citron the countyTreasurer who was entrusted with themanagement of a $7.5B portfolio
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Example: Orange County
Given a steep yield curve, the portfolio was betting oninterest rates falling. A large share was invested in 5
year FNMA notes.
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Example: Orange County
Ordinarily, the duration on a portfolio of 5 year noteswould be around 4-5. However, this portfolio washeavily leveraged ($7.5B as collateral for a $20.5B loan).This dramatically raises the VAR
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Example: Orange County
In February 1994, the Fed began a series of sixconsecutive interest rate increases. The beginning ofthe end!
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Risk vs. Return
As a portfolio manager, your job is tomaximize your risk adjusted return
Risk AdjustedReturn = Nominal Return Risk Penalty
You can accomplish this by 1 of two methods:
1) Maximize the nominal return for a given level of risk
2) Minimize Risk for a given nominal return
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$5 $5$5$5= ++++(1.05) (1.05) (1.05) (1.05)2 3 4
P = $100
Again, assume that the one year spot rate is currently 5% andis expected to stay constant. There is no liquidity premium, sothe yield curve is flat.
Term
Yield
5%
All 5% coupon bonds sell for Par Value and YTM = Coupon Rate =Spot Rate = 5%. Further, bond prices are constant throughout their
lifetime.
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Available Assets
1 Year Treasury Bill (5% coupon)3 Year Treasury Note (5% coupon)5 Year Treasury Note (5% coupon)10 Year Treasury Note (5% coupon)20 Year Treasury Bond (5% coupon)
STRIPS of all Maturities
How could you maximize your risk adjusted return on a $100,000Treasury portfolio?
Suppose you buy a 20 Year
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20 Year$100,000
$5000 $5000$5000= ++++(1.05) (1.05) (1.05) (1.05)2 3
20
P(Y=5%)
$4,762 $39,573$4,319$4,535
$4,762$100,000
$4,535$100,000
$4,319$100,000
$82,270$100,000
+ + + +1 2 3 20
Macaulay Duration = 12.6
Suppose you buy a 20 YearTreasury
$5000/yr $105,000
$105,000
Alternatively, you could buy a
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20 Year$50,000
Alternatively, you could buy a20 Year Treasury and a 5 yearSTRIPS
5 Year$50,000
$63,814
5 Year
5 Year
5 Year
$63,814 $63,814 $63,814
$2500/yr $52,500
(Remember, STRIPS have a Macaulayduration equal to their Term)
Portfolio Duration =$100,000$50,000 5 = 8.812.6 +
$100,000$50,000
Alternatively, you could buy a
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20 Year$50,000
Alternatively, you could buy a20 Year Treasury and a 5 yearTreasury
5 Year$50,000
5 Year
5 Year
5 Year
$2500/yr $52,500
(5 Year Treasuries have a Macaulay durationequal to 4.3)
Portfolio Duration =$100,000$50,000 4.3 = 8.512.6 +
$100,000$50,000
$2500/yr $52,500
Even better, you could buy a 20 Year
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20 Year$50,000
, y yTreasury, and a 1 Year T-Bill
$50,000
$2500/yr $52,500
(1 Year Treasuries have a Macaulay durationequal to 1)
Portfolio Duration =$100,000$50,000 1 = 6.312.6 +
$100,000$50,000
1 Year
1 Year
1 Year
$52,500 $52,500 $52,500
Alternatively, you could buy a 20 Year Treasury, a 10
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20 Year$25,000
y, y y y, Year Treasury, 5 year Treasury, and a 3 Year Treasury
10 Year
$25,000
5 Year
3 Year
$1250/yr
Portfolio Duration = 6.08
$100,000$25,000
12.6 +$100,000$25,000
$1250/yr
$1250/yr
$1250/yr$25,000
$25,000
D = 12.6
D = 7.7
D = 4.3
D = 2.7
7.7$100,000$25,000
4.3 +$100,000$25,000 2.7+
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Also with an upward sloping yield
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Also, with an upward sloping yieldcurve, a bonds price will changepredictably over its lifetime
2.552.78
3.043.28
3.483.69 3.75
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
1 Yr 2 Yr 3 Yr 4 Yr 5 Yr 6 Yr 7 Yr
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Pricing Date Coupon YTM Price ($)
Issue 3.75% 3.75% 100.00
2005 3.75 3.69 100.962006 3.75 3.48 101.772007 3.75 3.28 102.20
2008 3.75 3.04 102.352009 3.75 2.78 102.112010 3.75 2.55 101.29
2011 3.75 Matures 100.00
A Bonds price will always approach its facevalue upon maturity, but will rise over its lifetimeas the yield drops
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Length ofBond
InitialDuration
Durationafter 5 Years
PercentageChange
30 Year 15.5 14.2 -8%20 Year 12.6 10.5 -17%10 Year 7.8 4.4 -44%
Also, the change is a bondsduration is also a non-linear function
As a bond ages, its duration drops at an increasingrate