Valuing Cash Flows

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# Valuing Cash Flows - PowerPoint PPT Presentation

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

Non-Contingent Payments

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
• 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)

Forward Rate Pricing

Cash Flow at time t

Current Asset Price

Interest rate between periods t-1 and t

\$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

Spot Rate Pricing

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

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%

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
• 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's Duration

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

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

Key Duration
• 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%

Key Durations

X 100

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

Yield Curve Shifts

+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

Value At Risk

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

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

Example: Orange County
• 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 County
• Actually, up until 1994, Bob’s portfolio was doing very well.
Example: Orange County
• 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 County
• 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 County
• In February 1994, the Fed began a series of six consecutive interest rate increases. The beginning of the end!
Risk vs. 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
• 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

\$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

\$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

\$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

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