Valuing Cash Flows

1
Valuing Cash Flows

• Non-Contingent Payments

2
Non-Contingent Payouts

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

3
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.

4
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
2yrs
1.5 yrs
2.5yrs
3yrs
F(0,1)
F(1,1)
F(2,1)
F(3,1)
F(5,1)
F(4,1)
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
5
30
30
30
30
30
1,030
Now
6mos
1yrs
2yrs
1.5 yrs
2.5yrs
3yrs
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.

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)
6
Forward Rate Pricing
Cash Flow at time t
Current Asset Price
Interest rate between periods t-1 and t
7
Alternatively, we can use current spot rates
from the yield curve
30
30
30
30
30
1,030
Now
6mos
1yrs
2yrs
1.5 yrs
2.5yrs
3yrs
8
The yield curve produces the same bond
price..why?
30
30
30
30
30
1,030
Now
6mos
1yrs
2yrs
1.5 yrs
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
9
Spot Rate Pricing
Current Asset Price
Cash flow at period t
Current spot rate for a maturity of t periods
10
Alternatively, given the current price, what is
the implied (constant) interest rate.
30
30
30
30
30
1,030
Now
6mos
1yrs
2yrs
1.5 yrs
2.5yrs
3yrs
30
30
30
30
30
1,030





P

2
3
4
5
6
(1i)
(1i)
(1i)
(1i)
(1i)
(1i)
(1i) 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
11
Yield to Maturity
Cash flow at time t
Yield to Maturity
Current Market Price
12

• 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

(1Y)


942
2
(1Y)
For a discount (one payment) bond, the YTM is
equal to the expected spot rate
For coupon bonds, YTM is cash flow specific
13
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
14
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 gt Coupon
Rate
15
Price
A 1 rise in yield is associated with a 42
(4.2) drop in price
1,000
42
958
Yield
5
6
16
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 lt Coupon Rate
17
Price
A 1 drop in yield is associated with a 45
(4.5) rise in price
1,045
45
1,000
42
958
Yield
5
6
4
18
A bonds pricing function shows all the
combinations of yield/price
Price

1. The bond pricing is non-linear
2. The pricing function is unique to a particular
   stream of cash flows

1,045
45
1,000
42
958
Pricing Function
Yield
5
6
4
19
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).

20
For the 5 year, 5 Treasury, we had the
following
Yield
5
Term
50
50
50
50
50





P(Y5)
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
21
Price
1,000
Pricing Function
Yield
5
22
Suppose we take the derivative of the pricing
function with respect to yield
For the 5 year, 5 Treasury, we have
23
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 bonds price per basis
point shift in yield. This bonds DV01 is .43
24
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
5
6
4
Dollar Duration
25
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
bonds price by 4.3
26
Macaulay's Duration
Macaulay duration measures the percentage change
in a bonds price for every 1 change in (1Y)
(1.05)(1.01) 1.0605
For the 5 year, 5 Treasury, we have
27
For bonds with one payment, Macaulay duration is
equal to the term
Dollar Duration
Example 5 year STRIP
Modified Duration
Macaulay Duration
28
Think of a coupon bond as a portfolio of STRIPS.
Each payment has a Macaulay duration equal to its
date. The bonds Macaulay duration is a weighted
average of the individual durations
Back to the 5 year Treasury
50
50
50
50
50





P(Y5)
1,000
(1.05)
2
3
4
5
(1.05)
(1.05)
(1.05)
(1.05)
47.62
822.70
41.14
43.19
45.35
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
29
Macaulay Duration 4.55
Macaulay Duration
Modified Duration
(1Y)
4.55
Modified Duration

4.3
1.05
Dollar Duration
Modified Duration (Price)
Dollar Duration 4.3(1,000) 4,300
30
Duration measures interest rate risk (the risk
involved with a parallel shift in the yield
curve) This almost never happens.
31
Yield curve risk involves changes in an assets
price due to a change in the shape of the yield
curve
32
Key Duration

• In order to get a better idea of a Bonds (or
  portfolios) 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.

33
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
34
Key Durations
X 100
Note that the individual key durations sum to
4329 the bonds overall duration
35
Yield Curve Shifts
0
1
- 2
- 4
1
36
0
1
- 2
- 4
1




1
1
0
(-2)
(-4)
.4535
.8638
.12341
.15671
39.81
161
This yield curve shift would raise a five year
Treasury price by 161
37
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
6
4
38
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
6
4
Dollar Duration
39
Value At Risk
Suppose you are a portfolio manager. The current
value of your portfolio is a known quantity.
Tomorrows portfolio value us an unknown, but has
a probability distribution with a known mean and
variance
Profit/Loss Tomorrows Portfolio Value
Todays portfolio value
Known Distribution
Known Constant
40
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
41
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
42
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
43
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
44
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
45
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

46
Example Orange County

• Actually, up until 1994, Bobs portfolio was
  doing very well.

47
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.

48
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

49
Example Orange County

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

50
Risk vs. Return

• 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
51
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.
52
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?
53
Suppose you buy a 20 Year Treasury
5000/yr
105,000
20 Year
100,000
5000
5000
5000
105,000






P(Y5)
(1.05)
2
3
20
(1.05)
(1.05)
(1.05)
4,762
39,573
4,319
4,535
4,762
4,535
4,319
82,270





1
2
3
20
100,000
100,000
100,000
100,000
Macaulay Duration 12.6
54
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
5 8.8
12.6
100,000
100,000
55
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
5 Year
5 Year
5 Year
2500/yr
50,000
(5 Year Treasuries have a Macaulay duration equal
to 4.3)
50,000
50,000
Portfolio Duration
4.3 8.5
12.6
100,000
100,000
56
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
1 6.3
12.6
100,000
100,000
57
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
58
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
59
Also, with an upward sloping yield curve, a
bonds price will change predictably over its
lifetime
60
A Bonds price will always approach its face
value upon maturity, but will rise over its
lifetime as the yield drops
Pricing Date Coupon YTM Price ()
Issue 3.75 3.75 100.00
2005 3.75 3.69 100.96
2006 3.75 3.48 101.77
2007 3.75 3.28 102.20
2008 3.75 3.04 102.35
2009 3.75 2.78 102.11
2010 3.75 2.55 101.29
2011 3.75 Matures 100.00
61
Also, the change is a bonds duration is also a
non-linear function
Length of Bond Initial Duration Duration after 5 Years Percentage Change
30 Year 15.5 14.2 -8
20 Year 12.6 10.5 -17
10 Year 7.8 4.4 -44
As a bond ages, its duration drops at an
increasing rate