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Valuing Cash Flows

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Valuing Cash Flows

Non-Contingent Payments

- 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.

- 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

$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)

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?

$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

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%

Cash flow at time t

Yield to Maturity

Current Market Price

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

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

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

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%

- 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

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

$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

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’ 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

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 interest rate risk (the risk involved with a parallel shift in the yield 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

- 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

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

Evaluated at S(3) = 5%

X 100

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

+1%

0%

- 2%

- 4%

+1%

+1%

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

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

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

$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

$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

- 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

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

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

- 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

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

- 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.

- 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

$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