valuing cash flows

8/10/2019 Valuing Cash Flows

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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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$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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$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.


$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


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


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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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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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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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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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


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

valuing cash flows

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