Financial risk management course

Financial Risk Management 2010-11 Topics

T1

Stock index futures
Duration, Convexity, Immunization

T2

Repo and reverse repo
Futures on T-bills
Futures on T-bonds
Delta, Gamma, Vega hedging

T3

Portfolio insurance
Implied volatility and volatility smiles

T4

Modelling stock prices using GBM
Interest rate derivatives (Bond options, Caps, Floors, Swaptions)

T5

Value at Risk

T6

Value at Risk: statistical issues

Monte Carlo Simulations
Principal Component Analysis
Other VaR measures

T7

Parametric volatility models (GARCH type models)
Non-parametric volatility models (Range and high frequency models)
Multivariate volatility models (Dynamic Conditional Correlation DCC models)

T8

Credit Risk Measures (credit metrics, KMV, Credit Risk Plus, CPV)

T9

Credit derivatives (credit options, total return swaps, credit default swaps)
Asset Backed Securitization
Collateralized Debt Obligations (CDO)

* This file provides you an indication of the range of topics that is planned to be covered in the
module. However, please note that the topic plans might be subject to change.


Topics

Financial Risk Management
Topic 1
Managing risk using Futures
Reading: CN(2001) chapter 3

Futures Contract:
Speculation, arbitrage, and hedging
Stock Index Futures Contract:
Hedging (minimum variance hedge
ratio)
Hedging market risks

Futures Contract
Agreement to buy or sell “something” in the future at
a price agreed today. (It provides Leverage.)
Speculation with Futures: Buy low, sell high
Futures (unlike Forwards) can be closed anytime by taking
an opposite position

Arbitrage with Futures: Spot and Futures are linked
by actions of arbitragers. So they move one for one.
Hedging with Futures: Example: In January, a farmer
wants to lock in the sale price of his hogs which will
be “fat and pretty” in September.
Sell live hog Futures contract in Jan with maturity in Sept

Speculation with Futures


Speculation with Futures

Speculation with Futures

Purchase at F0 = 100

Hope to sell at higher price later F1 = 110
Close-out position before delivery date.
Obtain Leverage (i.e. initial margin is ‘low’)

Profit/Loss per contract
Long future
$10

0

-$10

F1 = 90
F0 = 100

Example:
Example: Nick Leeson: Feb 1995
Long 61,000 Nikkei-225 index futures (underlying
value = $7bn).
Nikkei fell and he lost money (lots of it)
- he was supposed to be doing riskless ‘index
arbitrage’ not speculating

F1 = 110

Futures price

Short future

Speculation with Futures

Profit payoff (direction vectors)
F increase
then profit increases
Profit/Loss

F increase
then profit decrease
Profit/Loss

Arbitrage with Futures

-1

+1
Underlying,S

+1

or Futures, F

-1

Long Futures

Short Futures

or, Long Spot

or, Short Spot

Arbitrage with Futures
At expiry (T), FT = ST . Else we can make riskless
profit (Arbitrage).
Forward price approaches spot price at maturity
Forward price, F
Forward price ‘at a premium’ when : F > S (contango)

0

Stock price, St
T

At T, ST = FT

Forward price ‘at a discount’, when : F < S (backwardation)

Arbitrage with Futures
General formula for non-income paying security:
F0 = S0erT or F0 = S0(1+r)T
Futures price = spot price + cost of carry
For stock paying dividends, we reduce the ‘cost of
carry’ by amount of dividend payments (d)
F0 = S0e(r-d)T

For commodity futures, storage costs (v or V) is
negative income
F0 = S0e(r+v)T or F0 = (S0+V)erT

Arbitrage with Futures

Arbitrage with Futures

For currency futures, the ‘cost of carry’ will be
reduced by the riskless rate of the foreign currency
(rf)

Arbitrage at t
F0 = S0e(r-rf)T

For stock index futures, the cost of carry will be
reduced by the dividend yield
F0 = S0e(r-d)T

If F0 > S0erT then buy the asset and short the futures
contract
If F0 < S0erT then short the asset and buy the futures
contract

Example of ‘Cash and Carry’ arbitrage: S=£100,
r=4%p.a., F=£102 for delivery in 3 months.
0.04×0.25
= 101 £
We see Fɶ = 100 × e
Since Futures is over priced,
time = Now
•Sell Futures contract at £102

time = in 3 months

•Pay loan back (£101)

•Borrow £100 for 3 months and buy stock •Deliver stock and get agreed price of £102


Hedging with Futures

Hedging with Futures

Hedging with Futures
Simple Hedging Example:
You long a stock and you fear falling prices over the
next 2 months, when you want to sell. Today (say
January), you observe S0=£100 and F0=£101 for
April delivery.
so r is 4%
Today: you sell one futures contract
In March: say prices fell to £90 (S1=£90). So
F1=S1e0.04x(1/12)=£90.3. You close out on Futures.
Profit on Futures: 101 – 90.3 = £10.7
Loss on stock value: 100 – 90 =£10

Net Position is +0.7 profit. Value of hedged portfolio
= S1+ (F0 - F1) = 90 + 10.7 = 100.7

F and S are positively correlated
To hedge, we need a negative correlation. So we
long one and short the other.
Hedge = long underlying + short Futures

Hedging with Futures
F1 value would have been different if r had changed.
This is Basis Risk (b1 = S1 – F1)
Final Value

= S1 + (F0 - F1 ) = £100.7
= (S1 - F1 ) + F0
=
b1
+ F0
where “Final basis” b1 = S1 - F1
At maturity of the futures contract the basis is zero
(since S1 = F1 ). In general, when contract is closed
out prior to maturity b1 = S1 - F1 may not be zero.
However, b1 will usually be small in relation to F0.


Stock Index Futures Contract
Stock Index Futures contract can be used to
eliminate market risk from a portfolio of stocks

Stock Index Futures Contract
Hedging with SIFs

F0 = S0 × e( r − d )T
If this equality does not hold then index arbitrage
(program trading) would generate riskless profits.

Risk free rate is usually greater than dividend yield
(r>d) so F>S

Hedging with Stock Index Futures

Hedging with Stock Index Futures

Example: A portfolio manager wishes to hedge her
portfolio of $1.4m held in diversified equity and
S&P500 index
Total value of spot position, TVS0=$1.4m
S0 = 1400 index point
Number of stocks, Ns = TVS0/S0 = $1.4m/1400
=1000 units
We want to hedge Δ(TVSt)= Ns . Δ(St)

The required number of Stock Index Futures contract
to short will be 3

Use Stock Index Futures, F0=1500 index point, z=
contract multiplier = $250
FVF0 = z F0 = $250 ( 1500 ) = $375,000

 TVS 0 
 $1, 400, 000 
NF = −
 = −
 = − 3.73
 $375, 000 
 FVF0 

In the above example, we have assumed that S and

F have correlation +1 (i.e. ∆ S = ∆ F )
In reality this is not the case and so we need
minimum variance hedge ratio


Hedging with Stock Index Futures

Hedging with Stock Index Futures

Minimum Variance Hedge Ratio

To obtain minimum, we differentiate with respect to Nf
2
(∂σ V / ∂N f = 0 ) and set to zero

∆V = change in spot market position + change in Index Futures position
+
Nf . (F1 - F0) z
=
Ns . (S1-S0)
Ns S0. ∆S /S0
TVS0 . ∆S /S0

=
=

+
+

Nf F0. (∆

∆F /F0) z
Nf . FVF0 . (∆
∆F /F0)

where, z = contract multiple for futures ($250 for S&P 500 Futures); ∆S =
S1 - S0, ∆F = F1 - F0

N f

( F V F0 ) 2 σ ∆2 F / F

N f

= −

2

2

2

2

2

σ V = (TVS 0 ) σ ∆S / S + ( N f ) ( FVF ) σ ∆F / F
0
+ 2N

f

TVS 0 FVF0 . σ ∆S / S , ∆F / F

2
∂σV / ∂N = 0
f

TVS 0
.β p
Nf = −
FVF0

(∆S

( σ ∆ S / S ,∆ F / F

2

σ ∆F / F

)

/ S ) = α 0 + β ∆S / ∆F ( ∆ F / F ) + ε

Hedging with Stock Index Futures
Application: Changing beta of your portfolio: “Market
Timing Strategy”
TVS

implies

 Value of Spot Position 

= − 
FaceValue
of
futures
at
t
=
0



⋅ F V F ⋅ σ ∆ S / S ,∆ F / F
0

where Ns = TVS0/S0 and beta is regression coefficient of the
regression

Hedging with Stock Index Futures
SUMMARY

0

 TVS0 
=−
 β ∆ S / S ,∆ F / F
 F V F0 

The variance of the hedged portfolio is

2

 TVS0 


 F V F0 

= −TVS

Nf =

0

FVF0

.( β h − β p )

Example: βp (=say 0.8) is your current ‘spot/cash’ portfolio of stocks

βp

But

• You are more optimistic about ‘bull market’ and desire a higher exposure of
βh (=say, 1.3)
• It’s ‘expensive’ to sell low-beta shares and purchase high-beta shares

If correlation = 1, the beta will be 1 and we just have

TVS0
Nf = −
FVF0

• Instead ‘go long’

more Nf Stock Index Futures contracts

Note: If βh= 0, then

Nf = - (TVS0 / FVF0) βp


Hedging with Stock Index Futures

Hedging with Stock Index Futures
Application: Stock Picking and hedging market risk

If you hold stock portfolio, selling futures will place a
hedge and reduce the beta of your stock portfolio.
If you want to increase your portfolio beta, go long
futures.
Example: Suppose β = 0.8 and Nf = -6 contracts would
make β = 0.
If you short 3 (-3) contracts instead, then β = 0.4
If you long 3 (+3) contracts instead, then β = 0.8+0.4
= 1.2

You hold (or purchase) 1000 undervalued shares of Sven plc

V(Sven) = $110

(e.g. Using Gordon Growth model)

P(Sven) = $100 (say)
Sven plc are underpriced by 10%.

Therefore you believe Sven will rise 10% more than the market over the next
3 months.
But you also think that the market as a whole may fall by 3%.
The beta of Sven plc (when regressed with the market return) is 2.0

Hedging with Stock Index Futures

Hedging with Stock Index Futures

Can you ‘protect’ yourself against the general fall in the market and hence any
‘knock on’ effect on Sven plc ?

Application: Future stock purchase and hedging market
risk

Yes .

You want to purchase 1000 stocks of takeover target with βp = 2, in 1
month’s time when you will have the cash.

Sell Nf index futures, using:

N

f

= −

TVS
FVF

0

.β

You fear a general rise in stock prices.
p

0

If the market falls 3% then
Sven plc will only change by about

10% - (2x3%) = +4%

But the profit from the short position in Nf index futures, will give you an
additional return of around 6%, making your total return around 10%.

Go long Stock Index Futures (SIF) contracts, so that gain on the futures will
offset the higher cost of these particular shares in 1 month’s time.

N

f

=

TVS
FVF

0

.β

p

0

SIF will protect you from market risk (ie. General rise in prices) but not from
specific risk. For example if the information that you are trying to takeover
the firm ‘leaks out’ , then price of ‘takeover target’ will move more than that
given by its ‘beta’ (i.e. the futures only hedges market risk)


Topics

Financial Risk Management
Topic 2
Managing interest rate risks
Reference: Hull(2009), Luenberger (1997), and CN(2001)

Duration, immunization, convexity
Repo (Sale and Repurchase agreement)

and Reverse Repo
Hedging using interest rate Futures
Futures on T-bills
Futures on T-bonds

Readings
Books
Hull(2009) chapters 6
CN(2001) chapters 5, 6
Luenberger (1997) chapters 3
Journal Article
Fooladi, I and Roberts, G (2000) “Risk Management with Duration Analysis”
Managerial Finance,Vol 25, no. 3

Hedging Interest rate risks: Duration


Duration

Duration (also called Macaulay Duration)

Duration measures sensitivity of price changes (volatility) with
changes in interest rates
1 Lower the coupons
for a given time to
maturity, greater
change in price to
change in interest
rates

B =

2 Greater the time to

D =

T

T
PB = ∑ C t t + ParValue
T
(1+ r )
t =1 (1+ r )

T

T
PB = ∑ C t t + ParValue
T
(1+ r )
t =1 (1+ r )

Duration of the bond is a measure that summarizes
approximate response of bond prices to change in yields.
A better approximation could be convexity of the bond .

maturity with a given
coupon, greater
change in price to
change in interest

rates

3 For a given percentage change in yield, the actual price increase is

n

∑

c i e − y ti

i =1
n

∑

i =1

weight

t i ⋅ c i e − y ti
B

=

n

∑

i =1

 c e − y ti 
ti  i

 B


Duration is weighted average of the times when payments
are made. The weight is equal to proportion of bond’s total
present value received in cash flow at time ti.
Duration is “how long” bondholder has to wait for cash flows

greater than a price decrease

Macaulay Duration
For a small change in yields ∆ y / d y
∆B =

dB
∆y
dy

Evaluating d B :
n


d y ∆ B =  − ∑ t i c i e − y ti  ∆ y
 i =1

= −B ⋅ D ⋅∆y
∆B

= −D ⋅∆y
B
D measures sensitivity of percentage change in bond
prices to (small) changes in yields
Note negative relationship between Price (B)
and yields (Y)

Modified Duration and Dollar Duration
For Macaulay Duration, y is expressed in continuous
compounding.
When we have discrete compounding, we have Modified
Duration (with these small modifications)
If y is expressed as compounding m times a year, we divide D
D
by (1+y/m) ∆B = − B ⋅
⋅ ∆y

(1 + y / m)

∆B = − B ⋅ D* ⋅ ∆y
Dollar Duration, D$ = B.D
That is, D$ = Bond Price x Duration (Macaulay or Modified)

∆B = − D$ ⋅ ∆y

So D$ is like Options Delta

D$ = −

∆B

∆y


Duration

Duration -example

Example: Consider a trader who has $1 million in
bond with modified duration of 5. This means for
every 1 bp (i.e. 0.01%) change in yield, the value of
the bond portfolio will change by $500.
∆B = − ( $1, 000, 000 × 5 ) ⋅ 0.01% = −$500

A zero coupon bond with maturity of n years has a
Duration = n
A coupon-bearing bond with maturity of n years will
have Duration < n
Duration of a bond portfolio is weighted average of
the durations of individual bonds
D

p o r tfo lio

=

∑ (B

i

/ B )⋅ D i

A

B

Year
0.5
1.0
1.5
2.0
2.5
3.0
Sum

C

D

E

Present value Weight =
Payment Discount
A× E
=B× C
D/Price
factor 8%
3.5
0.962
3.365
0.035

0.017
3.5
0.925
3.236
0.033
0.033
3.5
0.889
3.111
0.032
0.048
3.5
0.855
2.992
0.031
0.061
3.5
0.822
2.877
0.030
0.074
103.5
0.79
81.798
0.840
2.520
Price = 97.379
Duration = 2.753

Here, yield to maturity = 0.08, m = 2, y = 0.04, n = 6, Face value = 100.

i

Qualitative properties of duration
Duration of bonds with 5% yield as a function of
maturity and coupon rate.
Coupon rate

Years to
maturity
1
2
5
10
25
50
100
Infinity

Example: Consider a 7% bond with 3 years to maturity. Assume that the bond
is selling at 8% yield.

1%

2%

5%

10%

0.997
1.984
4.875
9.416
20.164
26.666
22.572
20.500

0.995
1.969
4.763
8.950
17.715
22.284
21.200
20.500

0.988
1.928
4.485
7.989
14.536
18.765
20.363
20.500

0.977
1.868
4.156

7.107
12.754
17.384
20.067
20.500

Properties of duration
1. Duration of a coupon paying bond is always less
than its maturity. Duration decreases with the increase
of coupon rate. Duration equals bond maturity for noncoupon paying bond.
2. As the time to maturity increases to infinity, the
duration do not increase to infinity but tend to a finite
limit independent of the coupon rate.
1 + mλ

where λ is the yield to maturity
λ
per annum, and m is the number of coupon

Actually, D →

payments per year.


Properties of Duration
3. Durations are not quite sensitive to increase in
coupon rate (for bonds with fixed yield). They don’t
vary huge amount since yield is held constant and
it cancels out the influence of coupons.
4. When the coupon rate is lower than the yield, the

duration first increases with maturity to some
maximum value then decreases to the asymptotic
limit value.
5. Very long durations can be achieved by bonds with
very long maturities and very low coupons.

Immunization (or Duration matching)
This is widely implemented by Fixed Income Practitioners.
time 0

time 1

time 2

time 3

0
pay $
pay $
pay $
You want to safeguard against interest rate increases.
A few ideas:
1. Buy zero coupon bond with maturities matching timing of
cash flows (*Not available) [Rolling hedge has reinv. risk]
2. Keep portfolio of assets and sell parts of it when cash is
needed & reinvest in more assets when surplus (* difficult as
Δ value of in portfolio and Δ value of obligations will not
identical)
3. Immunization - matching duration and present
values

of portfolio and obligations (*YES)

Changing Portfolio Duration
Changing Duration of your portfolio:
If prices are rising (yields are falling), a bond
trader might want to switch from shorter
duration bonds to longer duration bonds as
longer duration bonds have larger price
changes.
Alternatively, you can leverage shorter
maturities. Effective portfolio duration =
ordinary duration x leverage ratio.

Immunization
Matching present values (PV) of portfolio and obligations
This means that you will meet your obligations with the cash
from the portfolio.
If yields don’t change, then you are fine.
If yields change, then the portfolio value and PV will both change
by varied amounts. So we match also Duration (interest rate risk)

PV1 + PV2 = PVobligation

Matching duration
Here both portfolio and obligations have the same sensitivity to
interest rate changes.
If yields increase then PV of portfolio will decrease (so will the PV
of the obligation streams)
If yields decrease then PV of portfolio will increase (so will the PV
of the obligation streams)

D1 PV1 + D 2 PV2 = Dobligation PVobligation


Immunization

Immunization

Example

Suppose only the following bonds are available for its choice.

Suppose Company A has an obligation to
pay $1 million in 10 years. How to invest
in bonds now so as to meet the future
obligation?
• An obvious solution is the purchase of a
simple zero-coupon bond with maturity 10
years.
* This example is from Leunberger (1998) page 64-65. The numbers
are rounded up by the author so replication would give different
numbers.

Bond 1
Bond 2
Bond 3

coupon rate
6%
11%

9%

maturity
30 yr
10 yr
20 yr

yield
9%
9%
9%

duration
11.44
6.54
9.61

•

Present value of obligation at 9% yield is $414,642.86.

•

Since Bonds 2 and 3 have durations shorter than 10 years, it is not
possible to attain a portfolio with duration 10 years using these
two bonds.

Suppose we use Bond 1 and Bond 2 of amounts V1 & V2,
V1 + V2 = PV
P1V1 + D2V2 = 10 × PV

giving V1 = $292,788.64,

Immunization

price
69.04
113.01
100.00

V2 = $121,854.78.

Immunization

Yield
9.0
Bond 1
Price
Shares
Value

8.0

10.0

69.04
77.38
62.14
4241
4241
4241

292798.64 328168.58 263535.74

Bond 2
Price
113.01
120.39
106.23
Shares
1078
1078
1078
Value
121824.78 129780.42 114515.94
Obligation
value
414642.86 456386.95 376889.48
Surplus
-19.44
1562.05 1162.20

Observation: At different yields (8% and 10%), the value of the
portfolio almost agrees with that of the obligation.

Difficulties with immunization procedure
1. It is necessary to rebalance or re-immunize the
portfolio from time to time since the duration depends
on yield.
2. The immunization method assumes that all yields
are equal (not quite realistic to have bonds with
different maturities to have the same yield).

3. When the prevailing interest rate changes, it is
unlikely that the yields on all bonds change by the
same amount.


Duration for term structure
We want to measure sensitivity to parallel shifts in the spot
rate curve
For continuous compounding, duration is called FisherFisher-Weil
duration.
duration
If x0, x1,…, xn is cash flow sequence and spot curve is st where
t = t0,…,tn then present value of cash flow is
PV =

n

∑x
i=0

The Fisher-Weil duration is
1
D FW =
PV

ti

n

∑t

i=0

i

⋅e

− sti ⋅ti

⋅ x ti ⋅ e

− sti ⋅ti

Duration for term structure

(

Consider parallel shift in term structure: sti changes to sti + ∆y
Then PV becomes
P ( ∆y ) =

∑x
i=0

ti

⋅e

(

)

− sti + ∆ y ⋅ti

Taking differential w.r.t ∆y in the point ∆y=0 we get
n
dP ( ∆ y )
− s ⋅t
| ∆ y = 0 = − ∑ t i x t i ⋅ e ti i
d ∆y
i=0
So we find relative price sensitivity is given by DFW
1
dP (0)
⋅
= − D FW
P (0) d ∆ y

Convexity
Duration applies to only small changes in y
Two bonds with same duration can have different
change in value of their portfolio (for large changes
in yields)

n

Convexity
Convexity for a bond is
n

1 d 2B

C =
=
B dy 2

∑

i =1

t i2 ⋅ c i e − y t i
B

=

 c e − y ti 
t i2  i

 B


n

∑

i =1

Convexity is the weighted average of the ‘times squared’
when payments are made.
From Taylor series expansion

First order approximation cannot capture this, so we

take second order approximation (convexity)

1 d 2B
dB
∆ y +
dy
2 dy 2
1
= − D ⋅ ∆ y +
C ⋅
2

∆ B =

(∆

y

)

∆ B
B

(∆

y

)

2

2

So Dollar convexity is like Gamma measure in
options.

)


Short term risk management using Repo
Repo is where a security is sold with agreement to buy it back at

REPO and REVERSE REPO

a later date (at the price agreed now)
Difference in prices is the interest earned (called repo rate)
rate
It is form of collateralized short term borrowing (mostly overnight)
Example: a trader buys a bond and repo it overnight. The
money from repo is used to pay for the bond. The cost of this
deal is repo rate but trader may earn increase in bond prices
and any coupon payments on the bond.

There is credit risk of the borrower. Lender may ask for
margin costs (called haircut) to provide default protection.
Example: A 1% haircut would mean only 99% of the value of
collateral is lend in cash. Additional ‘margin calls’ are made if
market value of collateral falls below some level.

Short term risk management using Repo

Hedge funds usually speculate on bond price differentials
using REPO and REVERSE REPO
Example: Assume two bonds A and B with different prices (say price(A)similar characteristics. Hedge Fund (HF) would like to buy A and sell B
simultaneously.This can be financed with repo as follows:
(Long position) Buy Bond A and repo it. The cash obtained is used to pay for
the bond. At repo termination date, sell the bond and with the cash buy
bond back (simultaneously). HF would benefit from the price increase in
bond and low repo rate
(short position) Enter into reverse repo by borrowing the Bond B (as
collateral for money lend) and simultaneously sell Bond B in the market. At
repo termination date, buy bond back and get your loan back (+ repo
rate). HF would benefit from the high repo rate and a decrease in price of
the bond.

Interest Rate Futures
(Futures on T-Bills)


Interest Rate Futures
In this section we will look at how Futures contract written on a
Treasury Bill (T-Bill) help in hedging interest rate risks
Review - What is T-Bill?
T-Bills are issued by government, and quoted at a discount
Prices are quoted using a discount rate (interest earned as % of
face value)
Example: 90-day T-Bill is quoted at 0.08.
0.08 This means annualized
return is 8% of FV. So we can work out the price, as we know FV.


 d   90 
P = F V 1 − 


 100   360 


Day Counts convention (in US)
Actual/Actual
2. 30/360
3. Actual/360
1.

(for treasury bonds)
(for corporate and municipal bonds)
(for other instruments such as LIBOR)

Hedge decisions
When do we use these futures contract to hedge?
Examples:
1) You hold 3m T-Bills to sell in 1-month’s time ~ fear price fall
~ sell/short T-Bill futures
2) You will receive $10m in 3m time and wish to place it on a Eurodollar bank
deposit for 90 days ~ fear a fall in interest rates
~ go long a Eurodollar futures contract
3) Have to issue $100m of 180-day Commercial Paper in 3 months time (I.e.
borrow money) ~ fear a rise in interest rates
~ sell/short a T-bill futures contract as there is no commercial bill futures
contract (cross hedge)

Interest Rate Futures
So what is a 3-month T-Bill Futures contract?
At expiry, (T), which may be in say 2 months time
the (long) futures delivers a T-Bill which matures at
T+90 days, with face value M=$100.
As we shall see, this allows you to ‘lock in’ at t=0, the forward
rate, f12

T-Bill Futures prices are quoted in terms of quoted index, Q
(unlike discount rate for underlying)
Q = $100 – futures discount rate (df)
So we can work out the price as

 d f   90 
F = F V 1 − 


 100   360 


Cross Hedge: US T-Bill Futures
Example:
Today is May. Funds of $1m will be available in August to
invest for further 6 months in bank deposit (or commercial bills)
~ spot asset is a 6-month interest rate
Fear a fall in spot interest rates before August, so today BUY Tbill futures
Assume parallel shift in the yield curve. (Hence all interest rates
move by the same amount.)
~ BUT the futures price will move less than the price of the
higher the maturity, more

commercial bill - this is duration at work!
sensitive are changes in
prices to interest rates

Use Sept ‘3m T-bill’ Futures, ‘nearby’ contract
~ underlying this futures contract is a 3-month interest rate


Cross Hedge: US T-Bill Futures
3 month
exposure period

Cross Hedge: US T-Bill Futures
Question: How many T-bill futures contract should I purchase?

Desired investment/protection
period = 6-months

We should take into account the fact that:
to hedge exposure of 3 months, we have used T-bill futures
with 4 months time-to-maturity
2. the Futures and spot prices may not move one-to-one
1.

May

Aug.

Dec.

Sept.

Feb.

Maturity of ‘Underlying’
in Futures contract

Purchase T-Bill Known $1m Maturity date of Sept.
future with Sept. cash receipts T-Bill futures contract
delivery date

We could use the minimum variance hedge ratio:

Nf =

TVS0
.β p
FVF0

However, we can link price changes to interest rate
changes using Duration based hedge ratio

Question: How many T-bill futures contract should I purchase?

Duration based hedge ratio
Using duration formulae for spot rates and futures:

∆S
= − DS ⋅ ∆ys
S

∆F
= − DF ⋅ ∆yF
F

So we can say volatility is proportional to Duration:

 ∆S 
2
2
 = DS ⋅ σ ( ∆ys )
 S 

σ2

 ∆F 
2
2
 = DF ⋅ σ ( ∆yF )
 F 

σ2

 ∆S ∆F 
Cov 
,
 = Ε ( − DS ⋅ ∆ys )( − DF ⋅ ∆yF ) 
 S F 
= DS ⋅ DF ⋅ σ ( ∆ys ∆yF )

Duration based hedge ratio
Expressing Beta in terms of Duration:

 TVS0 
Nf = 
 .β p
FVF
0 

 ∆S ∆F 
Cov 
,

 TVS0 
S F 

=

 FVF0  σ 2  ∆F 


 F 
 TVS0  Ds  σ ( ∆ys ∆yF ) 
=
 2


∆
FVF
D

y
σ
(
)
0 
F 
F



We can obtain
last term by
regressing
∆yS = α0 + βy∆yF + ε


Duration based hedge ratio
Summary:

 TVS0   Ds

βy 
Nf =
. 
 FVF0   DF 

Cross Hedge: US T-Bill Futures
Example
REVISITED

3 month
exposure period

Aug.

May

where beta is obtained from the regression of yields

∆yS = α0 + β y ∆yF + ε

Desired investment/protection
period = 6-months

Dec.

Sept.

Feb.

Maturity of ‘Underlying’
in Futures contract

Purchase T-Bill Known $1m Maturity date of Sept.
future with Sept. cash receipts T-Bill futures contract
delivery date
Question: How many T-bill futures contract should I purchase?

Cross Hedge: US T-Bill Futures
May (Today). Funds of $1m accrue in August to be invested for 6- months

in bank deposit or commercial bills( Ds = 6 )

Cross Hedge: US T-Bill Futures
Suppose now we are in August:
3 month US T-Bill Futures : Sept Maturity

Use Sept ‘3m T-bill’ Futures ‘nearby’ contract ( DF = 3)
Cross-hedge.
Here assume parallel shift in the yield curve

Spot Market(May)

CME Index

Futures Price, F

Face Value of $1m

(T-Bill yields)

Quote Qf

(per $100)

Contract, FVF

May

y0 (6m) = 11%

Qf,0 = 89.2

97.30

$973,000

August

y1(6m) = 9.6%

Qf,1 = 90.3

97.58

$975,750

Change

-1.4%

1.10 (110 ticks)

0.28

$2,750
(per contract)

Qf = 89.2 (per $100 nominal) hence:
F0
= 100 – (10.8 / 4)

= 97.30
F

FVF0 = $1m (F0/100)

= $973,000

Nf = (TVS0 / FVF0) (Ds / DF )
= ($1m / 973,000) ( 0.5 / 0.25) = 2.05 (=2)

Durations are : Ds = 0.5, Df = 0.25
Amount to be hedged = $1m. No. of contracts held = 2

Key figure is F1 = 97.575 (rounded 97.58)
Gain on the futures position
= TVS0 (F1 - F0) NF = $1m (0.97575 – 0.973) 2 = $5,500


Cross Hedge: US T-Bill Futures
Invest this profit of $5500 for 6 months (Aug-Feb) at y1=9.6%:
= $5500 + (0.096/2) = $5764
Loss of interest in 6-month spot market (y0=11%, y1=9.6%)
= $1m x [0.11 – 0.096] x (1/2) = $7000
Net Loss on hedged position $7000 - $5764 = $1236
(so the company lost $1236 than $7000 without the hedge)

Interest Rate Futures
(Futures on T-Bonds)

Potential Problems with this hedge:

1. Margin calls may be required
2. Nearby contracts may be maturing before September. So we may have to roll
over the hedge
3. Cross hedge instrument may have different driving factors of risk

US T-Bond Futures
Contract specifications of US T-Bond Futures at CBOT:
Contract size

$100,000 nominal, notional US Treasury bond with 8% coupon

Delivery months

March, June, September, December

Quotation

Per $100 nominal

Tick size (value)

1/32 ($31.25)

Last trading day

7 working days prior to last business day in expiry month

Delivery day

Any business day in delivery month (seller’s choice)

Settlement

Any US Treasury bond maturing at least 15 years from the
contract month (or not callable for 15 years)

US T-Bond Futures
Conversion Factor (CF):
(CF): CF adjusts price of actual bond to be
delivered by assuming it has a 8% yield (matching the bond to
the notional bond specified in the futures contract)
Price = (most recent settlement price x CF) + accrued interest
Example: Possible bond for delivery is a 10% coupon (semiannual) T-bond with maturity 20 years.
The theoretical price (say, r=8%):
40

Notional is 8% coupon bond. However, Short can choose to
deliver any other bond. So Conversion Factor adjusts “delivery
price” to reflect type of bond delivered
T-bond must have at least 15 years time-to-maturity
Quote ‘98‘98-14’ means 98.(14/32)=$98.4375 per $100 nominal

5
100
+
= 119.794
i
1.0440
i =1 1.04

P=∑

Dividing by Face Value, CF = 119.794/100 = 1.19794 (per
$100 nominal) If Coupon rate > 8% then CF>1

If Coupon rate < 8% then CF<1


US T-Bond Futures

Hedging using US T-Bond Futures

Cheapest to deliver:
deliver:
In the maturity month, Short party can choose to deliver any
bond from the existing bonds with varying coupons and
maturity. So the short party delivers the cheapest one.

Hedging is the same as in the case of T-bill Futures (except
Conversion Factor).

Short receives:
(most recent settlement price x CF) + accrued interest
Cost of purchasing the bond is:
Quoted bond price + accrued interest
The cheapest to deliver bond is the one with the smallest:
Quoted bond price - (most recent settlement price x CF)

For long T-bond Futures, duration based hedge ratio is given
by:

 TVS0   Ds

Nf =
.
β
 
y  ⋅ CFCTD
FVF
D

0   F

where we have an additional term for conversion factor for
the cheapest to deliver bond.


Financial Risk Management
Topic 3a
Managing risk using Options
Readings: CN(2001) chapters 9, 13; Hull Chapter 17


Topics

Financial Engineering with Options
Black Scholes
Delta, Gamma, Vega Hedging
Portfolio Insurance

Options Contract - Review
An option (not an obligation), American and European

-

Put Premium

-


Financial Engineering with options
Synthetic call option

Put-Call Parity: P + S = C + Cash
Example: Pension Fund wants to hedge its stock holding
against falling stock prices (over the next 6 months) and
wishes to temporarily establish a “floor value” (=K) but also
wants to benefit from any stock price rises.


Financial Engineering with options
Nick Leeson’s short straddle

You are initially credited with the call and put premia C + P (at t=0) but if at expiry
there is either a large fall or a large rise in S (relative to the strike price K ) then you
will make a loss
(.eg. Leeson’s short straddle: Kobe Earthquake which led to a fall in S
(S = “Nikkei-225”) and thus large losses).