Risk and Asset-Liability Management

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Risk and Asset-Liability Management

• Professor Banikanta Mishra
• Xavier Institute of Management

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Risk Handling Risk Avoidance Risk
Acceptance Risk Transfer Risk Sharing Risk
Exchange Ways of Risk Management
Insurance Hedging Asset-Liability Management
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Forward Contract Buyer and seller agree to
transact (buy/sell) a specified amount of a
given commodity/asset at a pre-specified-date at
a contracted price (the forward price) Futures
Contract Forward Contract with
marking-to-market OptionOption-buyer has the
right and option-seller the obligation to
transact at a given price, on or before a given
date (up to a prefixed amount of a given
asset) Call right to buy Put right to
sell Swap Both counter-parties agree to
exchange given quantities of two given assets or
two indexed flows (e.g. for commodity for
DM fixed for floating ) Range Forward
Buyer has the right to buy at Ceiling from
Seller Seller has the right to Sell at the Floor
to Buyer (gt Call Put)
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Farmer and Bread-maker

• tT
• Distribution of ST
• 40 25
• 60 35
• Expected 31
• Farmer s worry Price may touch 25
• Bread-makers worry Price may touch 35

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• Farmers Solution
• t0
• Sell wheat forward
• Agree to sell
• wheat _at_F at tT
• FORWARD SELL SHORT
• Will strictly prefer
• F31 to selling at the
• Risky open-market price
• In fact, Farmer would
• be willing to accept a
• bit less than 31 to get ? F lt 31
• rid of selling price risk

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• Bread-makers Solution
• t0
• Buy wheat forward
• Agree to buy
• wheat _at_F at tT
• FORWARD BUY LONG
• Will strictly prefer
• F31 to buying at the
• risky open-market price
• In fact, Bread-maker
• is willing to pay a bit
• more than 31 to get ? F gt 31
• rid of buying price risk

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F versus E(ST)
If there is as much demand from bread-makers
as supply from farmers, F 31 is an ideal
price as both farmers and bread-makers strictly
prefer this to the risky open-market price So,
we would find that F E(ST) Expected Future
Spot Price at Time-T
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Contango
But, if, at t0, there is a net forward (time
T) demand for wheat (gt more willing to forward
buy than forward sell), speculators have to be
enticed to take selling position To give them an
incentive to do this, we need F gt 31 gt forward
price gt expected future spot price
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Backwardation
Similarly, if, at t0, there is a net forward
supply of wheat (gt more willing to forward sell
than forward buy), speculators have to be enticed
to take buying position To give them an
incentive to do this, we need F lt 31 gt forward
price lt expected future spot price
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Option Call

• If one has right, but not obligation, to buy _at_31,
  then at tT
• Price 25 gt Buy _at_25 in open market Gain 0
• Price 35 gt Buy _at_31 (exercise right) Gain 4
• So, option holder can only gain or not gain at
  tT,
• but cannot lose (Note Expected Gain at t1 60
  x 4 2.40)
• So, she has to pay something (the call price) for
  this
• (to the call writer, when entering into contract
  at t0)
• This price (or premium) should equal PV of 2.40.
• Note Call writer has obligation to sell _at_31 if
  holder wants to buy

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Option Put

• If one has right, but not obligation, to sell
  _at_31, then at tT
• Price 25 gt Sell _at_31 (exercise right) Gain 6
• Price 35 gt Sell _at_35 in open market Gain 0
• So, option holder can only gain or not gain at
  tT,
• but cannot lose (Note Expected Gain at t1 40
  x 6 2.40)
• So, she has to pay something (the put premium)
  for this
• (to the put writer, when entering into contract
  at t0)
• This put premium should equal PV of 2.40.
• Note Put writer has obligation to buy _at_31 if
  holder wants to sell

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Rights Obligations
Buy Sell/Write
Right to buy Obligation to Sell Right to
Sell Obligation to buy
Call Put
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Options Payoff
ST gt X ST lt X
Long Call ST - X 0
Short Call X - ST 0
Long Put 0 X - ST
Short Put 0 ST - X

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t0 t1 t2 t3 t4
t5 Firm A 80 80 80 80
80 Payments 1,000 Firm B 116
116 116 116 116 Outflows
1,600 ( US Company Loan of 1,000 _at_8.00
UK Company Loan of 1,600 _at_7.25)
116 (t1 to 5), 1,600 (t5)
Current Exch Rate 1 1.60
SWAP
Firm A
Firm B
80 (t1 to 5), 1,000 (t5)
Firm As 116 116 116 116
116 Outflows 1,600 After Swap
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Forward Price

• Suppose Gold price now is 300 and R0T 5
• Suppose Goldsmith needs one unit of gold at tT
• What can he do?
• Borrow 300, buy gold now (at t0), and hold
• OR
• 2. Long forward _at_FT

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Cash Flows

• t0 t T
• Strategy 1
• Borrow 300 300 Repay 300 (15) 315
• Buy gold _at_300 -300 Have 1 unit of gold
• Net CF 0 Net CF -315
• Strategy 2
• Long Forward _at_FT 0 Buy 1 unit of gold
  _at_FT
• Net CF 0 Net CF -FT

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Forward Pricing Formula

• So, Law of One Price /No Arbitrage Condition gt
• FT S (1 R0T) 315
• In general, (1 R0T) (1 R)T
• where
• FT is forward price for transaction at end of T
  years,
• R0T is the interest-rate over T years (T-yearly
  rate),
• and R is the Annual Interest Rate (EAR)

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Hedger

• Long Hedge Goldsmith would long forward
• He is assured of buying gold _at_315 at tT
• Short Hedge Miner would short forward
• gt She is assured of selling gold _at_315 at tT

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Speculator

• If speculator expects spot price at T to exceed
  315,
• that is, E(ST) gt F
• he would long forward (to buy _at_F at T)
• His expected profit at T E(ST) F gt 0
• If speculator expects E(ST) lt F,
• she would short forward (to sell _at_F at T)
• gt Her expected profit at T F - E(ST) gt 0

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Arbitrageur

• Gets into action only if FT S (1 R0T) 315
• If F gt 315, then forward is overpriced
• He would sell (or short) forward
• So, to cover his position,
• he would buy asset now _at_ S 300
• And, to take care of his purchase price,
• he would borrow 300

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Arbitrageurs Cash Flows

• t0 t T
• Short Forward _at_F T Sell gold _at_ FT
• Borrow 300 300 Repay 300 (15) 315
• Buy gold _at_300 -300 Have 1 unit of gold to
  sell
• ------------------------------
  ----------------------------------
• Net CF 0 Net CF FT - 315
• ------------------------------
  ----------------------------------
• Since FT gt 315, Arbitrageur is guaranteed a ve
  CF at tT

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What if FT lt 315

• t0 t T
• Long Forward _at_F T Buy gold _at_ FT
• Short gold 300 Deliver gold to close
  position
• Lend 300 -300 Get back 300 (15) 315
• ------------------------------
  ----------------------------------
• Net CF 0 Net CF 315 - FT
• ------------------------------
  ----------------------------------
• Since FT lt 315, Arbitrageur is guaranteed a ve
  CF at tT

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Caveat

• In real world, we
• Dont borrow and lend at the same rate
• Buy and sell at the spot market at the same rate
• Buy and sell forward at the same rate
• We borrow/lend _at_ banks lending/deposit rate
• We buy/sell on spot _at_ dealers ask/bid price
• We buy/sell forward _at_ dealers ask/bid price

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No-Arbitrage F

• Then, the No-Arbitrage Forward Pricing Formula
• does not any more give
• an equation or equality
• But instead dictates
• a range
• within which F must lie

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Hedging Objective
To reduce risk, Not to increase ERR
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Hedging Basics
CF at tT State Asset 1 Asset 2 Asset 3
1 28 2 -2 2 22
-1 2
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Hedging Strategy

• What position in Asset-2 would hedge exposure?
• What position in Asset-3 would hedge exposure?

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Long Short
CF at tT Long Short State 1 Unit
Asset 1 2 Units Asset 2 TOTAL 1
28 -4 24 2 22 2 24
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Long Long
CF at tT Long Long State 1 Unit
Asset 1 1.5 Unit Asset 3 TOTAL 1
28 -3 25 2 22 3 25
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Date Price of Security y Price of Security x
1 Y1 X1 2 Y2 X2 ... . .
. . . . T YT XT Yt a b Xt
ut gt DYt b DXt et gt If price of x
goes up by 1, then price of y would go up by b
on the average gt If you are long (short)
ONE unit of y, then hedge by going short (long)
in b units of x
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Let b 1.6 Then, DYt 1.6
DXt et gt For each 1.0 change in the
price of x, there is a 1.6 change in the
price of y in the same direction Price of 1.6
units of If Price of 1 unit of Price of 1
unit of x changes by x changes by
y changes by (on the average) 1.6
1 1.6 1.6 1 1.6 3.2 2 3.2 8.0
5 8.0 Long (Short) Position Gets
Canceled By Short (Long) Position In
This In This
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If DYt b DXt et, then (DYt / Yt) b
(DXt / Xt) et gt DYt b (Yt / Xt) DXt
et So, b b (Yt / Xt) where Yt and Xt
represent the current levels of y and x resp. If
duration of Y is Dy and that of x is Dx, then,
for 1 change in interest-rate, DYt Dy and
DXt Dx gt DYt (Dy / Dx) DXt gt b
Dy / Dx Dy / Dx b (Dy / Dx) x (Yt / Xt)
(Dy x Yt) / (Dx x Xt)
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t 1 Sep t 1 Nov t 1 Dec t 1 Feb t 1
Mar Expects to Issue on 90-day CP _at_ Pcp
Sell short 90-day TB Futures To sell _at_
96.62 90-d TB _at_ f1-Dec Close
short 90-d TB Fut To buy _at_f1-Nov 90-d
TB 96.62 - f1-Nov _at_ f1-Dec Gain (loss)
in the Futures position partially/fully offsets
loss (gain) in CP issue
maturing on
maturing on
offset
cum MTM
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Possible Interest-Rates on 1 November Lower
Expected Higher
(unchanged) Sell CP _at_ 96.43 95.92 95.23
(Rate) (3.70) (4.25)
(5.00) f1-Nov 96.95
96.62 96.17 CF from -0.33 0.00
0.45 1 Futures Position 96.10 95.92
95.68 1.5 Fut -0.50 0.00
0.68 Position 95.93 95.92 95.91
0.51
-0.69
0.33
-0.45
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20 May 5 Aug 1 Sep 1 Dec 7 Feb Gets the 3.3
mill 3.3 mill News to be recd
x (1 r6m) and put in 180-day
TB Money Withdrawn
and invested in a project Go long
in To buy 90-d TB Futures TB _at_
S1-Sep _at_973,600 Close long TB Futures To
sell 90-d _at_ f5-Aug TB _at_
S1-Sep f5-Aug - 973,600 f5-Aug
- 973,600 x (1 r6m)
interest-rate r6m
offset
cum MTM
invest _at_ r6m till 7 Feb
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Possible r6m (unannualized) on 5 Aug
Lower Unchanged Higher 4.90
5.70 6.50 f5-Aug 976,400
973,600 971,000 cum MTM 2,800 0
-2,600 x 7 (contracts) 19,600 0
-18,200 Cash Flows on 7 Feb
(unannualized) Return FV of this amt 20,560
0 -19,383 Interest Recd on 3.3
mill 161,700 188,100 214,500 T O T A
L 182,260 188,100 195,117 Return (6-mth)
5.52 5.70 5.91 Stays at the
level of the forward-rate (expectation on 20 May
about r6m on 5 Aug)
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Consider a 3-month TB and a 6-month
TB D6-m 0.5 D3-m 0.25 If D r
1 DP6-m 0.50 x 1 0.50 DP3-m
0.25 x 1 0.25 If current
annual-interest-rate is 12, then P6-m
94.34 P3-m 97.09 So that b (D6-m /
D3-m) x (P6-m / P3-m ) (D6-m x P6-m ) / (D3-m
x P3-m ) (0.50 / 0.25) x (94.34 / 97.09)
(0.50 x 94.34) / (0.25 x 97.09) 1.94 So,
hedge a long (short) position in ONE 6-m TB by
a short (long) position in 1.94 3-m TBs
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Basis Risk

• t 0 t t t T
• Long Futures _at_f0
• Close Futures _at_ft
• Buy in the open _at_St
• --------------------------------------------------
  -----
• Net CF - St (ft f0) - f0 unless t T

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Basis Risk Example

• t 0 t t t T
• Long Futures _at_120
• Close Futures _at_125
• Buy in the open _at_128
• --------------------------------------------------
  ------
• Net CF - 128 (125 120) -123 -120

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Fabricators Hedge

• S0 140 f 120 Need 100,000 lbs Copper
• Suppose go long in futures _at_120. Then
• If ST 125, then futures MTM (125-120) 5
• gt Total Cost 125 5 120
• If ST 105, then futures MTM (105-120) -15
• gt Total Cost 105 (-15) 120

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Hedging Equity PF Basics
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Hedging Equity PF

• Value of Equity PF 5,000,000
• Value of SP 500 1,000
• RF 4 (1 per quarter)
• DYindex 1 (0.25 per quarter)
• F4-m 1,010 (Delivers 250 times the index)
• 1.5 gt HR 1.5 x (5,000,000 / 250,000) 30

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What Happens When ?

• At t, let St 900 and ft 902
• (Cumulative) MTM from Futures
• (1010 902) x 250 x 30 810,000
• Since Index falls by 10
• gt DM Div Change in Index 2.5 (-100)
• (DM /M) DYCGY 0.25(-10) -9.75
• PF Value falls by 15.125 756,250
• Net Gain 810,000 756,250 53,750

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To Change the PF Beta to b?

• Number of contracts required to be shorted is
• (b - b)
• x
• Value of Portfolio / Value of Futures Contract

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To Change b to 2.3

• N Short (1.5 2.3) x (5000000/250000) -16
• Or Long 16 contracts

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1 or 100 bp
Fixed 10.0 Floating LIBOR 70 bp
Fixed 11.0 Floating LIBOR 120 bp
50 bp
LIBOR 120 on Notional
Well Known
NSF
10.8 on Notional Amount
NET COST 10.8
NET COST L 40b
10 on Fixed
LIBOR 120 on Floating
CAPITAL MARKET
CAPITAL MARKET
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1 or 100 bp
Fixed 10.0 Floating LIBOR 70 bp
Fixed 11.0 Floating LIBOR 120 bp
50 bp
S W A P F I
LIBOR
LIBOR
Well Known
NSF
9.55
9.65
NET COST 10.85
NET COST L 45b
10 on Fixed
LIBOR 120 on Floating
CAPITAL MARKET
CAPITAL MARKET
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t0 t1 t2 t3 t4
t5 Firm A 80 80 80 80
80 Payments 1,000 Firm B 116
116 116 116 116 Payments
1,600 ( US Company Loan of 1,000 _at_8.00
UK Company Loan of 1,600 _at_7.25)
116 (t1 to 5), 1,600 (t5)
Current Exch Rate 1 1.60
SWAP
Firm A
Firm B
80 (t1 to 5), 1,000 (t5)
Firm As 116 116 116 116
116 Outflows 1,600 After Swap
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1 or 100 bp
Re Loan 15.0 Loan 6.0
Re Loan 16.0 Loan 5.5
-50 bp
5.3 in on Notional
Well Known
NSF
15 in Re on Re Notional
NET COST 15.2 Re
NET COST 5.3
15 on Re Loan
5.5 on Loan
CAPITAL MARKET
CAPITAL MARKET
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Current Data 1.60, r 5.5, r 6.0
LIBOR 20 bp in on Notional
SWAP BANK
FIRM A
LIBOR on Notional
LIBOR on Notional
Net Cost LIBOR in
Capital Market
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FIXED PRICE
SWAP Counter- party
Producer
SPOT PRICE
SPOT PRICE
ON THE NET PRODUCER BUYS HER RAW-MATERIAL
INPUT AT A PRE-FIXED PRICE
R/M SPOT MARKET
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FIXED PRICE
SWAP Counter- party
Producer
SPOT PRICE
SPOT PRICE
ON THE NET, PRODUCER SELLS HIS OUTPUT AT A
PRE-FIXED PRICE.
OUTPUT SPOT MARKET
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OUTPUT SPOT MARKET
INPUT SPOT MARKET
INPUT SPOT PR
OUTPUT SPOT PR
PRODUCER
PRESPECIFIED SPREAD
OUTPUT SPOT - INPUT SPOT
SWAP DEALER
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ALM for a Leasing Company

• Liabs Interest Pmt of Rs.1,790 for next 3
  years
• Principal repayment of Rs.20,337 at t3
• Assets Lease Payment Receivable of Rs.5000 for
  next five years
• Machine worth Rs.699.68 to be leased
  out at zero NPV for N (?) years
• Cash Rs.83.26