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Valuing Cash Flows. Non-Contingent Payments. Non-Contingent Payouts. Given an asset with fixed payments (i.e. independent of the state of the world), the asset’s price should equal the present value of the cash flows. Treasury Notes.

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Valuing cash flows

Valuing Cash Flows

Non-Contingent Payments


Non contingent payouts
Non-Contingent Payouts

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


Treasury notes
Treasury Notes

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

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

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


Consider a 3 year Treasury note with a 6% annual coupon and a $1,000 face value.

$30

$30

$30

$30

$30

$1,030

Now

6mos

1yrs

1.5 yrs

2yrs

2.5yrs

3yrs

F(0,1)

F(1,1)

F(2,1)

F(3,1)

F(4,1)

F(5,1)

F(0,1) = 2.25%

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

F(1,1) = 2.75%

F(2,1) = 2.8%

F(3,1) = 3%

F(4,1) = 3.1%

F(5,1) = 4.1%


$30 a $1,000 face value.

$30

$30

$30

$30

$1,030

Now

6mos

1yrs

1.5 yrs

2yrs

2.5yrs

3yrs

2.25%

2.75%

2.8%

3%

3.1%

4.1%

Given an expected path for (annualized) forward rates, we can calculate the present value of future payments.

+ …

$30

$30

$30

P =

+

+

(1.01125)

(1.01125)(1.01375)

(1.01125)(1.01375)(1.014)

+ …

$1,030

= $1,084.90

+

(1.01125)………….(1.0205)


Forward rate pricing
Forward Rate Pricing a $1,000 face value.

Cash Flow at time t

Current Asset Price

Interest rate between periods t-1 and t


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

$30

$30

$30

$30

$30

$1,030

Now

6mos

1yrs

1.5 yrs

2yrs

2.5yrs

3yrs


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

$30

$30

$30

$30

$30

$1,030

Now

6mos

1yrs

1.5 yrs

2yrs

2.5yrs

3yrs

$30

$30

$30

$30

$30

$1,030

P

=

+

+

+

+

+

2

3

4

5

6

(1.0125)

(1.0125)

(1.0135)

(1.0135)

(1.015)

(1.015)

S(1)

S(2)

S(3)

2

2

2

P

=

$1,084.90


Spot rate pricing
Spot Rate Pricing curve

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.

$30

$30

$30

$30

$30

$1,030

Now

6mos

1yrs

1.5 yrs

2yrs

2.5yrs

3yrs

$30

$30

$30

$30

$30

$1,030

P

+

+

+

+

+

=

2

3

4

5

6

(1+i)

(1+i)

(1+i)

(1+i)

(1+i)

(1+i)

(1+i) = 1.015 (1.5%)

P

=

$1,084.90

Given the current ,market price of $1,084.90, this Treasury Note has an annualized Yield to Maturity of 3%


Yield to maturity
Yield to Maturity (constant) interest rate.

Cash flow at time t

Yield to Maturity

Current Market Price


Consider $1,000, 2 year STRIP selling for $942

.5

$1,000

$1,000

1.03 (3%)

$942

=

(1+Y)

=

=

$942

2

(1+Y)

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

The one year interest rate is currently 5% and is expected to stay constant. Further, there is no liquidity premium

Yield

5%

Term

$50

$50

$50

$50

$50

P

+

+

+

+

=

= $1,000

2

3

4

5

(1.05)

(1.05)

(1.05)

(1.05)

(1.05)

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

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

Yield

6%

5%

Term

$50

$50

$50

$50

$50

P

+

+

+

+

=

= $958

2

3

4

5

(1.06)

(1.06)

(1.06)

(1.06)

(1.06)

This bond sells at a discount and YTM > Coupon Rate


Price (paid annually) and a face value of $1,000

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

$1,000

$42

$958

Yield

5%

6%


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

Now, suppose that the current 1 year rate falls to 4% and is expected to remain there

Yield

5%

4%

Term

$50

$50

$50

$50

$50

P

+

+

+

+

=

= $1045

2

3

4

5

(1.04)

(1.04)

(1.04)

(1.04)

(1.04)

This bond sells at a premium and YTM < Coupon Rate


Price (paid annually) and a face value of $1,000

A 1% drop in yield is associated with a $45 (4.5%) rise in price

$1,045

$45

$1,000

$42

$958

Yield

4%

5%

6%


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

Price

  • The bond pricing is non-linear

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

$1,045

$45

$1,000

$42

$958

Pricing Function

Yield

4%

5%

6%


Duration
Duration yield/price

  • Recall that in general the price of a fixed income asset is given by the following formula

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


For the 5 year, 5% Treasury, we had the following: yield/price

Yield

5%

Term

$50

$50

$50

$50

$50

P(Y=5%)

=

+

+

+

+

= $1,000

2

3

4

5

(1.05)

(1.05)

(1.05)

(1.05)

(1.05)

This bond sells for Par Value and YTM = Coupon Rate


Price yield/price

$1,000

Pricing Function

Yield

5%


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

For the 5 year, 5% Treasury, we have


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

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 = 5%, P = $1,000)

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

Error = $2

$1,045

$1,000

Error = - $1

$958

Pricing Function

Yield

4%

5%

6%

Dollar Duration


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

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


Macaulay s duration
Macaulay's Duration $1000 bond has a DD of $43 while a $10,000 bond has a DD of $430) modified duration represents the

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


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

Dollar Duration

Example: 5 year STRIP

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

P(Y=5%)

=

+

+

+

+

= $1,000

(1.05)

2

3

4

5

(1.05)

(1.05)

(1.05)

(1.05)

$47.62

$45.35

$43.19

$41.14

$822.70

$47.62

$45.35

$43.19

$41.14

$822.70

1

+

2

+

3

+

4

+

5

$1,000

$1,000

$1,000

$1,000

$1,000

Macaulay Duration = 4.55


Macaulay Duration = 4.55 payment has a Macaulay duration equal to its date. The bond’s Macaulay duration is a weighted average of the individual durations

Macaulay Duration

Modified Duration =

(1+Y)

4.55

Modified Duration =

=

4.3

1.05

Dollar Duration =

Modified Duration (Price)

Dollar Duration = 4.3($1,000) = $4,300


Duration measures payment has a Macaulay duration equal to its date. The bond’s Macaulay duration is a weighted average of the individual durationsinterest rate risk (the risk involved with a parallel shift in the yield curve) This almost never happens.


Yield curve payment has a Macaulay duration equal to its date. The bond’s Macaulay duration is a weighted average of the individual durations risk involves changes in an asset’s price due to a change in the shape of the yield curve


Key duration
Key Duration payment has a Macaulay duration equal to its date. The bond’s Macaulay duration is a weighted average of the individual durations

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


Returning to the 5 Year Treasury payment has a Macaulay duration equal to its date. The bond’s Macaulay duration is a weighted average of the individual durations

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

Evaluated at S(3) = 5%


Key durations
Key Durations payment has a Macaulay duration equal to its date. The bond’s Macaulay duration is a weighted average of the individual durations

X 100

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


Yield curve shifts
Yield Curve Shifts payment has a Macaulay duration equal to its date. The bond’s Macaulay duration is a weighted average of the individual durations

+1%

0%

- 2%

- 4%

+1%


+1% payment has a Macaulay duration equal to its date. The bond’s Macaulay duration is a weighted average of the individual durations

0%

- 2%

- 4%

+1%

$.4535

1

+

$.8638

1

+

$.12341

0

+

$.15671

(-2)

+

$39.81

(-4)

= $161

This yield curve shift would raise a five year Treasury price by $161


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

Price

$1,045 - $958

Slope =

= $43.50

4% - 6%

$1,045

or

$1,045 - $958

*100

$1,000

= 4.35

Slope =

$958

4% - 6%

Yield

4%

6%


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!!

Price

$1,045

Effective Duration

$958

Pricing Function

Yield

4%

6%

Dollar Duration


Value at risk
Value At Risk the actual pricing function rather that the derivative. This is particularly important for pricing bonds with embedded options!!

Suppose 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 distributions
Probability Distributions the actual pricing function rather that the derivative. This is particularly important for pricing bonds with embedded options!!

1 Std Dev = 65%

2 Std Dev = 95%

3 Std Dev = 99%

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


Remember, the 5 year Treasury has a MD 0f 4.3 the actual pricing function rather that the derivative. This is particularly important for pricing bonds with embedded options!!

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

Interest Rate

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

Mean = 6% Std. Dev. = 2%

Profit/Loss

Mean = $0 Std. Dev. = $86


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

1 Std Dev = 65%

2 Std Dev = 95%

3 Std Dev = 99%

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


A 30 year Treasury has a MD of 14 distribution of interest rates) would be $86. The VAR(95) would be $172

$1000, 30 Year Treasury (6% coupon)

Interest Rate

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

Mean = 6% Std. Dev. = 2%

Profit/Loss

Mean = $0 Std. Dev. = $280


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

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


Example orange county
Example: Orange County distribution of interest rates) would be $280. The VAR(95) would be $560

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

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


Example orange county1
Example: Orange County distribution of interest rates) would be $280. The VAR(95) would be $560

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


Example orange county2
Example: Orange County distribution of interest rates) would be $280. The VAR(95) would be $560

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


Example orange county3
Example: Orange County distribution of interest rates) would be $280. The VAR(95) would be $560

  • 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 loan). This dramatically raises the VAR


Example orange county4
Example: Orange County distribution of interest rates) would be $280. The VAR(95) would be $560

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


Risk vs return
Risk vs. Return distribution of interest rates) would be $280. The VAR(95) would be $560

  • As a portfolio manager, your job is to 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


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.

Yield

5%

Term

$5

$5

$5

$5

P

+

+

+

+

=

= $100

2

3

4

(1.05)

(1.05)

(1.05)

(1.05)

All 5% coupon bonds sell for Par Value and YTM = Coupon Rate = Spot Rate = 5%. Further, bond prices are constant throughout their lifetime.


Available assets
Available Assets and is expected to stay constant. There is no liquidity premium, so the yield curve is flat.

  • 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 $100,000 Treasury portfolio?


Suppose you buy a 20 Year Treasury and is expected to stay constant. There is no liquidity premium, so the yield curve is flat.

$5000/yr

$105,000

20 Year

$100,000

$5000

$5000

$5000

$105,000

P(Y=5%)

=

+

+

+

+

(1.05)

2

3

20

(1.05)

(1.05)

(1.05)

$4,762

$4,535

$4,319

$39,573

$4,762

$4,535

$4,319

$82,270

1

+

2

+

3

+

+

20

$100,000

$100,000

$100,000

$100,000

Macaulay Duration = 12.6


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

$2500/yr

$52,500

20 Year

$50,000

5 Year

5 Year

5 Year

5 Year

$50,000

$63,814

$63,814

$63,814

$63,814

(Remember, STRIPS have a Macaulay duration equal to their Term)

$50,000

$50,000

Portfolio Duration =

12.6 +

5 = 8.8

$100,000

$100,000


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

$2500/yr

$52,500

20 Year

$50,000

$52,500

5 Year

$2500/yr

5 Year

5 Year

5 Year

$50,000

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

$50,000

$50,000

Portfolio Duration =

12.6 +

4.3 = 8.5

$100,000

$100,000


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

$2500/yr

$52,500

20 Year

$50,000

1 Year

1 Year

1 Year

$50,000

$52,500

$52,500

$52,500

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

$50,000

$50,000

Portfolio Duration =

12.6 +

1 = 6.3

$100,000

$100,000


Alternatively, you could buy a 20 Year Treasury, a 10 Year Treasury, 5 year Treasury, and a 3 Year Treasury

$1250/yr

20 Year

$25,000

D = 12.6

$1250/yr

10 Year

$25,000

D = 7.7

$1250/yr

5 Year

$25,000

D = 4.3

$1250/yr

3 Year

$25,000

D = 2.7

$25,000

$25,000

$25,000

$25,000

7.7

+

2.7

12.6 +

4.3 +

$100,000

$100,000

$100,000

$100,000

Portfolio Duration = 6.08


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

1 Year

1 Year

1 Year

$100,000

$105,000

$105,000

$105,000

Portfolio Duration = 1

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


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


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


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