Valuing Cash FlowsValuing Cash Flows

Non-Contingent PaymentsNon-Contingent Payments

Non-Contingent PayoutsNon-Contingent Payouts

Given an asset withGiven an asset with fixed payments (i.e. payments (i.e. independent of the state of the world), the independent of the state of the world), the asset’s price should equal the present asset’s price should equal the present value of the cash flows. value of the cash flows.

Treasury NotesTreasury Notes

US Treasuries notes have maturities US Treasuries notes have maturities between 2 and ten years. between 2 and ten years.

Treasury notes make biannual interest Treasury notes make biannual interest payments and then a repayment of the payments and then a repayment of the face value upon maturityface value upon maturity

US Treasury notes can be purchased in US Treasury notes can be purchased in increments of $1,000 of face value. increments of $1,000 of face value.

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 statistical model that generates the following set of (annualized) forward rates

F(0,1) F(1,1) F(2,1) F(3,1) F(5,1)F(4,1)

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 future payments.

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)+ …

Forward Rate PricingForward Rate Pricing

N

tt

i i

t

F

CFP

11 1

01

Current Asset Price Cash Flow at time t

Interest rate between periods t-1 and t

Now 6mos 1yrs 2yrs1.5 yrs 2.5yrs 3yrs

$30 $30 $30 $30 $30 $1,030

Alternatively, we can use current spot rates from the yield curve

2.5 2.7 33.5 4

0

1

2

3

4

1 yr 2 yr 3yr 4yr 5yr

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)

2S(2)

2S(3)

2

The yield curve produces the same bond price…..why?

Spot Rate PricingSpot Rate Pricing

N

tt

t

tS

CFP

10 )(1

Current Asset Price Cash flow at period t

Current spot rate for a maturity of t periods

Alternatively, given the current price, what is the 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 an annualized Yield to Maturity of 3%

Yield to MaturityYield to Maturity

N

tt

t

Y

CFP

10

1

Current Market Price

Yield to Maturity

Cash flow at time t

Yield to maturity measures the total performance Yield to maturity measures the total performance of a bond from purchase to expiration.of a bond from purchase to expiration.

Consider $1,000, 2 year STRIP selling 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

Consider a 5 year Treasury Note with a 5% annual coupon rate (paid annually) and a face value of $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 is currently 5% and is expected to stay constant. Further, there is no liquidity premium

Term

Yield

5%

This bond sells for Par Value and YTM = Coupon Rate

Consider a 5 year Treasury Note with a 5% annual coupon rate (paid annually) and a face value of $1,000

$50 $50$50$50$50= ++++

(1.06) (1.06) (1.06) (1.06) (1.06)2 3 4 5P = $958

Now, suppose that the current 1 year rate rises to 6% and is expected to remain there

Term

Yield

5%

6%

This bond sells at a discount and YTM > Coupon Rate

Price

Yield

$958

5% 6%

$1,000$42

A 1% rise in yield is associated with a $42 (4.2%) drop in price

Consider a 5 year Treasury Note with a 5% annual coupon rate (paid annually) and a face value of $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 the current 1 year rate falls to 4% and is expected to remain there

Term

Yield

5%

4%

This bond sells at a premium and YTM < Coupon Rate

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

Price

Yield

$958

5% 6%4%

$1,045

$1,000

$45

$42

Pricing Function

A bond’s pricing function shows all the combinations of yield/price

1) The bond pricing is non-linear

2) The pricing function is unique to a particular stream of cash flows

DurationDurationRecall that in general the price of a fixed Recall that in general the price of a fixed

income asset is given by the following income asset is given by the following formulaformula

Note that we are denoting price as a Note that we are denoting price as a function of yield: P(Y).function of yield: P(Y).

n

1i 1 P(Y) i

i

Y

CF

$50 $50$50$50$50= ++++(1.05) (1.05) (1.05) (1.05) (1.05)2 3 4 5P(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 the following:

Price

Yield5%

$1,000

Pricing Function

n

1i1

1

*

dY

dPii

Y

CFi

Suppose we take the derivative of the pricing function with respect to yield

65432 Y)(1

$1,0505

Y)(1

$504

Y)(1

$503

Y)(1

$502

Y)(1

$50-

dY

dP

For the 5 year, 5% Treasury, we have

Now, evaluate that derivative at a particular point (say, Y = 5%, P = $1,000)

329,4$

05).(1

$1,0505

05).(1

$504

05).(1

$503

05).(1

$502

05).(1

$50-

dY

dP65432

For every 100 basis point change in the interest rate, the value of this bond changes by $43.29 This is the dollar duration

DV01 is the change in a bond’s price per basis point shift in yield. This bond’s DV01 is $.43

Price

Yield

$958

5% 6%4%

$1,045

$1,000Error = - $1

Pricing Function

Error = $2

Duration predicted a $43 price change for every 1% change in yield. This is different from the actual price

Dollar Duration

P

1*

dY

dP Duration Modified

Dollar duration depends on the face value of the bond (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 bonds price due to a 1% change in yield

For the 5 year, 5% Treasury, we have

3.4000,1$

329,4$1*

dY

dP MD

P

Every 100 basis point shift in yield alters this bond’s price by 4.3%

Macaulay's DurationMacaulay's Duration

P

Y )1(*

dY

dp Duration sMacaulay'

Macaulay’ duration measures the percentage change in a bond’s 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(*

dY

dP D Mac

P

Y

For bonds with one payment, Macaulay duration is equal to the term

Example: 5 year STRIP

5Y)(1

$100P(Y)

6Y)(1

(5)($100)dP

dY

)1(

5

)1(100$

Y)(1(5)($100)

1dP

5

6

YY

PdY

51dP

P

Y

dY

Dollar Duration

Modified Duration

Macaulay Duration

Think of a coupon bond as a portfolio of STRIPS. Each payment has a Macaulay duration equal to its date. The bond’s 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 5P(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

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

Duration measures Duration measures interest rate risk (the (the risk involved with a parallel shift in the yield risk involved with a parallel shift in the yield curve) This almost never happens.curve) This almost never happens.

Yield curve risk involves changes in an asset’s price due to a change in the shape of the yield curve

Key DurationKey Duration

In order to get a better idea of a Bond’s (or In order to get a better idea of a Bond’s (or portfolio’s) exposure to yield curve risk, a portfolio’s) exposure to yield curve risk, a key rate key rate duration is calculated. This duration is calculated. This measures the sensitivity of a bond/portfolio measures the sensitivity of a bond/portfolio to a particular spot rate along the yield to a particular spot rate along the yield curve holding all other spot rates constant.curve holding all other spot rates constant.

55

44

33

221

51 )S(1

$1,050

)S(1

$50

)S(1

50$

)S(1

50$

)S(1

$50 )S,...,P(S

433

51

)S(1

)50($3

dS

)S,...,dP(S

Returning to the 5 Year Treasury

A Key duration for the three year spot rate is the partial derivative with respect to S(3)

Evaluated at S(3) = 5%

41.123$)05.(1

)50($3

dS

)S,...,dP(S4

3

51

Key DurationsKey Durations

45.35

86.38

123.41

156.71

39.18

0

20

40

60

80

100

120

140

160

1Yr 2Yr 3Yr 4Yr 5Yr

Note that the individual key durations sum to $4329 – the bond’s overall duration

X 100

Yield Curve ShiftsYield Curve Shifts

0

1

2

3

4

5

6

7

1 yr 2yr 3yr 4yr 5yr

Old New

- 4%- 2%0%+1%

+1%

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

Price

Yield

$958

6%4%

$1,045

Suppose that we simply calculate the slope between the two points on the pricing function

Slope = $1,045 - $958

4% - 6% = $43.50

or

Slope =

$1,045 - $958

4% - 6%

$1,000*100

= 4.35

Price

Yield

$958

6%4%

$1,045

Pricing Function

Dollar Duration

Effective Duration

Effective duration measures interest rate sensitivity using the actual pricing function rather that the derivative. This is particularly important for pricing bonds with embedded options!!

Value At RiskValue At RiskSuppose you are a portfolio manager. The current value of your portfolio is a known quantity.

Tomorrow’s portfolio value us an unknown, but has a probability distribution with a known mean and variance

Profit/Loss = Tomorrow’s Portfolio Value – Today’s portfolio value

Known Distribution Known Constant

Probability DistributionsProbability Distributions

One Standard Deviation Around the mean encompasses 65% of the distribution

1 Std Dev = 65%

2 Std Dev = 95%

3 Std Dev = 99%

Interest Rate

Mean = 6% Std. Dev. = 2%

$1,000, 5 Year Treasury (6% coupon)

Remember, the 5 year Treasury has a MD 0f 4.3

Mean = $1,000 Std. Dev. = $86

Profit/Loss

Mean = $0 Std. Dev. = $86

One Standard Deviation Around the mean encompasses 65% of the 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 is only a 5% chance of losing more that $172

Interest Rate

Mean = 6% Std. Dev. = 2%

$1000, 30 Year Treasury (6% coupon)

A 30 year Treasury has a MD of 14

Mean = $1,000 Std. Dev. = $280

Profit/Loss

Mean = $0 Std. Dev. = $280

One Standard Deviation Around the mean encompasses 65% of the 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 is only a 5% chance of losing more that $560

Example: Orange CountyExample: Orange County

In December 1994, Orange County, CA In December 1994, Orange County, CA stunned the markets by declaring stunned the markets by declaring bankruptcy after suffering a $1.6B loss.bankruptcy after suffering a $1.6B loss.

The loss was a result of the investment The loss was a result of the investment activities of Bob Citron – the county activities of Bob Citron – the county Treasurer – who was entrusted with the Treasurer – who was entrusted with the management of a $7.5B portfolio management of a $7.5B portfolio

Example: Orange CountyExample: Orange County

Actually, up until 1994, Bob’s portfolio was doing very Actually, up until 1994, Bob’s portfolio was doing very well. well.

Example: Orange CountyExample: Orange County

Given a steep yield curve, the portfolio was betting on interest rates Given a steep yield curve, the portfolio was betting on interest rates falling. A large share was invested in 5 year FNMA notes. falling. A large share was invested in 5 year FNMA notes.

Example: Orange CountyExample: Orange County

Ordinarily, the duration on a portfolio of 5 year notes would be around 4-5. Ordinarily, the duration on a portfolio of 5 year notes would be around 4-5. However, this portfolio was heavily leveraged ($7.5B as collateral for a $20.5B However, this portfolio was heavily leveraged ($7.5B as collateral for a $20.5B loan). This dramatically raises the VARloan). This dramatically raises the VAR

Example: Orange CountyExample: Orange County

In February 1994, the Fed began a series of six In February 1994, the Fed began a series of six consecutive interest rate increases. The beginning of consecutive interest rate increases. The beginning of the end!the end!

Risk vs. ReturnRisk vs. Return

As a portfolio manager, your job is to As a portfolio manager, your job is to maximize your maximize your risk adjusted return

Risk Adjusted Return = 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

$5 $5$5$5= ++++

(1.05) (1.05) (1.05) (1.05)2 3 4P = $100

Again, assume that the one year spot rate is currently 5% and is expected to stay constant. There is no liquidity premium, so the 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.

…

Available AssetsAvailable Assets

1 Year Treasury Bill (5% coupon)1 Year Treasury Bill (5% coupon)3 Year Treasury Note (5% coupon)3 Year Treasury Note (5% coupon)5 Year Treasury Note (5% coupon)5 Year Treasury Note (5% coupon)10 Year Treasury Note (5% coupon)10 Year Treasury Note (5% coupon)20 Year Treasury Bond (5% coupon)20 Year Treasury Bond (5% coupon)STRIPS of all MaturitiesSTRIPS of all Maturities

How could you maximize your risk adjusted return on a $100,000 Treasury portfolio?

20 Year$100,000

$5000 $5000$5000= ++++(1.05) (1.05) (1.05) (1.05)2 3 … 20P(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 Year Treasury

…

$5000/yr $105,000

$105,000

20 Year$50,000

Alternatively, you could buy a 20 Year Treasury and a 5 year STRIPS

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 Macaulay duration equal to their Term)

Portfolio Duration = $100,000

$50,0005 = 8.812.6 +

$100,000

$50,000

20 Year$50,000

Alternatively, you could buy a 20 Year Treasury and a 5 year Treasury

5 Year$50,000

5 Year

5 Year

5 Year

$2500/yr $52,500

(5 Year Treasuries have a Macaulay duration equal to 4.3)

Portfolio Duration = $100,000

$50,0004.3 = 8.512.6 +

$100,000

$50,000

$2500/yr $52,500

20 Year$50,000

Even better, you could buy a 20 Year Treasury, and a 1 Year T-Bill

$50,000

$2500/yr $52,500

(1 Year Treasuries have a Macaulay duration equal to 1)

Portfolio Duration = $100,000

$50,0001 = 6.312.6 +

$100,000

$50,000

1 Year

1 Year

1 Year …

$52,500 $52,500 $52,500

20 Year$25,000

Alternatively, you could buy a 20 Year Treasury, a 10 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

$25,00012.6 +

$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

$25,0004.3 +

$100,000

$25,000 2.7+

Obviously, with a flat yield curve, there is no advantage to buying longer term bonds. The optimal strategy is to buy 1 year T-Bills

$100,000

Portfolio Duration = 1

1 Year

1 Year

1 Year …

$105,000 $105,000 $105,000

However, the yield curve typically slopes up, which creates a risk/return tradeoff

Also, with an upward sloping yield curve, a bond’s price will change predictably 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

Pricing Date Coupon YTM Price ($)

IssueIssue 3.75%3.75% 3.75%3.75% 100.00100.00

20052005 3.753.75 3.693.69 100.96100.96

20062006 3.753.75 3.483.48 101.77101.77

20072007 3.753.75 3.283.28 102.20102.20

20082008 3.753.75 3.043.04 102.35102.35

20092009 3.753.75 2.782.78 102.11102.11

20102010 3.753.75 2.552.55 101.29101.29

20112011 3.753.75 MaturesMatures 100.00100.00

A Bond’s price will always approach its face value upon maturity, but will rise over its lifetime as the yield drops

Length of Bond

Initial Duration

Duration after 5 Years

Percentage Change

30 Year30 Year 15.515.5 14.214.2 -8%-8%

20 Year20 Year 12.612.6 10.510.5 -17%-17%

10 Year10 Year 7.87.8 4.44.4 -44%-44%

Also, the change is a bond’s duration is also a non-linear function

As a bond ages, its duration drops at an increasing rate