Zvi Wiener ContTimeFin - 6 slide 1 Financial Engineering Interest Rates and Fixed Income Securities Zvi Wiener [email protected] tel: 02-588-3049
Jan 19, 2016
Zvi Wiener ContTimeFin - 6 slide 1
Financial Engineering
Interest Rates and Fixed Income Securities
tel: 02-588-3049
Zvi Wiener ContTimeFin - 6 slide 2
Bonds
A bond is a contract, paid up-front that yields
a known amount at a known date (maturity).
The bond may pay a dividend (coupon) at
fixed times during the life.
Additional options: callable, puttable,
indexed, prepayment options, etc.
Credit risk, recovery ratio, rating.
Zvi Wiener ContTimeFin - 6 slide 3
Term Structure of IR
time to maturity
r
short term IR
long term IR
spot rate
Zvi Wiener ContTimeFin - 6 slide 4
Known IR
V - value of a contract.
r(t) - short term interest rate.
If there is no risk and no coupons then
dV = rVdt
V(t) = V(T)e-rt
if there is a continuous dividend stream
dV+cVdt = rVdt
Zvi Wiener ContTimeFin - 6 slide 5
Known IR
If r is not constant, but not risky r(t)
dV = r(t)Vdt
If there is a continuous dividend stream
dV+c(t)Vdt = r(t)Vdt
T
t
dr
etV )(
)(
Zvi Wiener ContTimeFin - 6 slide 6
Known IR
Assume that there are zero coupon bonds for
all possible ttm (time to maturity).
Denote the price of these bonds by V(t,T).
T
t
dr
eTtV )(
),(
Zvi Wiener ContTimeFin - 6 slide 7
Known IR
T
t
dr
eTtV )(
),(
T
t
drTtV )(),(log
T
V
TtVTr
),(
1)(
Zvi Wiener ContTimeFin - 6 slide 8
Yield
tT
TtVTtY
),(log),(
1),( )( tTYeTtV
Zvi Wiener ContTimeFin - 6 slide 9
Typical yield curves
time to maturity
yield increasing
decreasing
humped
Zvi Wiener ContTimeFin - 6 slide 10
Typical yield curves
increasing - the most typical.
decreasing - short rates are high but expected to fall.
humped - short rates are expected to fall soon.
Zvi Wiener ContTimeFin - 6 slide 11
Term Structure Explanations
Expectation hypothesis states F0=E(PT)
this hypothesis is be true if all market participants were risk neutral.
Zvi Wiener ContTimeFin - 6 slide 12
Term Structure Explanations
Normal Backwardation (Keynes), commodities are used by hedgers to reduce risk. In order to induce speculators to take the opposite positions, the producers must offer a higher return. Thus speculators enter the long side and have the expected profit of
E(PT) – F0 > 0
Zvi Wiener ContTimeFin - 6 slide 13
Term Structure Explanations
Contango is similar to the normal backwardation, but the natural hedgers are the purchasers of a commodity, rather than suppliers. Since speculators must be paid for taking risk, the opposite relation holds:
E(PT) – F0 < 0
Zvi Wiener ContTimeFin - 6 slide 14
8% Coupon BondYield to Maturity T=1 yr. T=10 yr. T=20 yr.
8% 1,000.00 1,000.00 1,000.00
9% 990.64 934.96 907.99
Price Change 0.94% 6.50% 9.20%
Yield to Maturity T=1 yr. T=10 yr. T=20 yr.8% 924.56 456.39 208.29
9% 915.73 414.64 171.93
Price Change 0.96% 9.15% 17.46%
Zero Coupon Bond
Zvi Wiener ContTimeFin - 6 slide 15
DurationF. Macaulay (1938)
Better measurement than time to maturity.
Weighted average of all coupons with the corresponding time to payment.
Bond Price = Sum[ CFt/(1+y)t ]
suggested weight of each coupon:
wt = CFt/(1+y)t /Bond Price
What is the sum of all wt?
Zvi Wiener ContTimeFin - 6 slide 16
Macaulay Duration
A weighted sum of times to maturities of each coupon.
What is the duration of a zero coupon bond?
T
tt
tT
tt y
CFt
iceBondwtD
11 )1(Pr
1
Zvi Wiener ContTimeFin - 6 slide 17
Macaulay Duration(1)
Time untilpayment
(in Years)
(2)
Payment
(3)Payment
Discountedat 5%
(4)
Weight
(5)column (1)multiplied
by (4)Bond A 0.5 $40 $38.095 0.0395 0.01988% 1.0 $40 $36.281 0.0376 0.0376
1.5 $40 $34.553 0.0358 0.05372.0 $1,040 $855.611 0.8871 1.7742
Sum: $964.540 1.000 1.8853
Zvi Wiener ContTimeFin - 6 slide 18
Macaulay Duration(1)
Time untilpayment
(in Years)
(2)
Payment
(3)Payment
Discountedat 5%
(4)
Weight
(5)column (1)multiplied
by (4)Bond A 0.5 $40 $38.095 0.0395 0.01988% 1.0 $40 $36.281 0.0376 0.0376
1.5 $40 $34.553 0.0358 0.05372.0 $1,040 $855.611 0.8871 1.7742
Sum: $964.540 1.000 1.8853
Bond B 0.5-1.5 0 $0 0 0zero 2.0 $1,000 $822.70 1 2Sum $822.70 1 2
Zvi Wiener ContTimeFin - 6 slide 19
Duration
Sensitivity to IR changes:
Long term bonds are more sensitive. Lower coupon bonds are more sensitive. The sensitivity depends on levels of IR.
Zvi Wiener ContTimeFin - 6 slide 20
Duration
The bond price volatility is proportional to the bond’s duration.
Thus duration is a natural measure of interest rate risk exposure.
y
yD
P
PMC 1
)1(
Zvi Wiener ContTimeFin - 6 slide 21
Modified Duration
The percentage change in bond price is the product of modified duration and the change in the bond’s yield to maturity.
yDP
P
y
DD
*
1*
Zvi Wiener ContTimeFin - 6 slide 22
Comparison of two bonds
Coupon bond with duration 1.8853
Price (at 5% for 6m.) is $964.5405
If IR increase by 1bp
(to 5.01%), its price will fall to $964.1942, or
0.359% decline.
Zero-coupon bond with equal duration must have 1.8853 years to maturity.
At 5% semiannual its price is
($1,000/1.053.7706)=$831.9623
If IR increase to 5.01%, the price becomes:
($1,000/1.05013.7706)=$831.66
0.359% decline.
Zvi Wiener ContTimeFin - 6 slide 23
Duration
Maturity
D
0 3m 6m 1yr 3yr 5yr 10yr 30yr
15% coupon, YTM = 15%
Zero coupon bond
Zvi Wiener ContTimeFin - 6 slide 24
Example
A bond with 30-yr to maturity
Coupon 8%; paid semiannually
YTM = 9%
P0 = $897.26
D = 11.37 Yrs
if YTM = 9.1%, what will be the price?
Zvi Wiener ContTimeFin - 6 slide 25
ExampleA bond with 30-yr to maturityCoupon 8%; paid semiannuallyYTM = 9%
P0 = $897.26D = 11.37 Yrsif YTM = 9.1%, what will be the price?
P/P = - y D*
P = -(y D*)P = -$9.36
P = $897.26 - $9.36 = $887.90
Zvi Wiener ContTimeFin - 6 slide 26
What Determines Duration? Duration of a zero-coupon bond equals maturity. Holding ttm constant, duration is higher when coupons are lower.Holding other factors constant, duration is higher when ytm is lower. Duration of a perpetuity is (1+y)/y.
Zvi Wiener ContTimeFin - 6 slide 27
What Determines Duration? Holding the coupon rate constant, duration not always increases with ttm.
Zvi Wiener ContTimeFin - 6 slide 28
Duration
T
tt
t
y
CFP
1 )1(
T
tt
tMC y
CFtD
1 )1(
Zvi Wiener ContTimeFin - 6 slide 29
T
tt
t
T
tt
t
y
CFtD
y
CFP
1
1
)1(
)1(
Zvi Wiener ContTimeFin - 6 slide 30
y
PD
y
CFt
dy
dP
y
CFtD
y
CFP
T
ttt
T
tt
t
T
tt
t
1)1(
)1(
)1(
11
1
1
Zvi Wiener ContTimeFin - 6 slide 31
y
PD
y
CFt
dy
dP T
ttt
1)1(11
T
tt
t
T
tt
t
y
CFtD
y
CFP
1
1
)1(
)1(
Dyyd
PdP
)1()1(
Zvi Wiener ContTimeFin - 6 slide 32
DfactordiscountinchangePercent
pricebondinchangePercent
Dyyd
PdP
)1()1(
Duration can be regarded as the discount-rate elasticity of the bond price
Modern Approach
Zvi Wiener ContTimeFin - 6 slide 33
dyDP
dP
y
dyD
P
dP
Dyyd
PdP
*
1
)1()1(
Duration can be used to measure the price volatility of a bond:
Modern Approach
Zvi Wiener ContTimeFin - 6 slide 34
What are the natural bounds on duration?
Can duration be bigger than maturity?
Can duration be negative?
How to measure duration of a portfolio?
Modern Approach
Zvi Wiener ContTimeFin - 6 slide 35
Duration: Modern Approach
*1
1
Ddy
dP
P
Ddy
dP
P
y
Zvi Wiener ContTimeFin - 6 slide 36
Duration of a Portfolio
dy
dP
P
yD
definitiondy
dP
P
yD
BA
BABA
1
)(1
Zvi Wiener ContTimeFin - 6 slide 37
Duration of a Portfolio
BBAABA
B
BB
A
AA
BA
BA
BA
BA
BABA
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1
1
Zvi Wiener ContTimeFin - 6 slide 38
Simon Benninga, Financial Modelling, the MIT press, Cambridge, MA, ISBN 0-262-02437-3, $45
MIT Press tel: 800-356-0343http://mitpress.mit.edu/book-home.tcl?isbn=0262024373
see also my advanced lecture notes on duration
Convexity is a similar measurement but with second derivative.
Modern Approach to Duration
Zvi Wiener ContTimeFin - 6 slide 39
Implementation in Excel Duration Patterns Duration of a bond with uneven payments Calculating YTM for uneven periods Nonflat term structure and duration Immunization strategies Cheapest to deliver option and Duration
Financial Modellingby Simon Benninga
Zvi Wiener ContTimeFin - 6 slide 40
Passive Bond Management
Passive management takes bond prices as fairly set and seeks to control only the risk of the fixed-income portfolio.
Indexing strategy– attempts to replicate a bond index
Immunization– used to tailor the risk to specific needs (insurance companies, pension funds)
Zvi Wiener ContTimeFin - 6 slide 41
Bond-Index Funds
Similar to stock indexing.
Major indices: Lehman Brothers, Merill Lynch, Salomon Brothers.
Include: government, corporate, mortgage-backed, Yankee bonds (dollar denominated, SEC registered bonds of foreign issuers, sold in the US).
Zvi Wiener ContTimeFin - 6 slide 42
Bond-Index Funds
Properties:
many issues
not all are liquid
replacement of maturing issues
Tracking error is a good measurement of performance. According to Salomon Bros. With $100M one can track the index within 4bp. tracking error per month.
Zvi Wiener ContTimeFin - 6 slide 43
Cellular approach
ttm\Sector Treasury Agency MBS< 1yr 12.1%
1-3 yrs 5.4% 4.1% 3.2%
3-5 yrs 9.2% 6.1%
Zvi Wiener ContTimeFin - 6 slide 44
Immunization
Immunization techniques refer to strategies used by investors to shield their overall financial status from exposure to interest rate fluctuations.
Zvi Wiener ContTimeFin - 6 slide 45
Net Worth Immunization
Banks and thrifts have a natural mismatch between assets and liabilities. Liabilities are primarily short-term deposits (low duration), assets are typically loans or mortgages (higher duration).
When will banks lose money, when IR increase or decline?
Zvi Wiener ContTimeFin - 6 slide 46
Gap Management
ARM are used to reduce duration of bank portfolios.
Other derivative securities can be used.
Capital requirement on duration (exposure).
Basic idea:
to match duration of assets and liabilities.
Zvi Wiener ContTimeFin - 6 slide 47
Target Date Immunization
Important for pension funds and insurances.
Price risk and reinvestment risk.
What is the correlation between them?
Zvi Wiener ContTimeFin - 6 slide 48
Target Date Immunization
Accumulatedvalue
0 t* t
Original plan
Zvi Wiener ContTimeFin - 6 slide 49
Target Date Immunization
Accumulatedvalue
0 t* t
IR increased at t*
Zvi Wiener ContTimeFin - 6 slide 50
Target Date Immunization
Accumulatedvalue
0 t* D t
Zvi Wiener ContTimeFin - 6 slide 51
Target Date Immunization
Accumulatedvalue
0 t* D t
Continuous rebalancingcan keep the terminal value
unchanged
Zvi Wiener ContTimeFin - 6 slide 52
Good Versus Bad Immunization
value
0 8% r
Single payment obligation
$10,000
Zvi Wiener ContTimeFin - 6 slide 53
Good Versus Bad Immunization
value
0 8% r
Single payment obligation
Good immunizing strategy$10,000
Zvi Wiener ContTimeFin - 6 slide 54
Good Versus Bad Immunization
value
0 8% r
Single payment obligation
Good immunizing strategy$10,000
Zvi Wiener ContTimeFin - 6 slide 55
Good Versus Bad Immunization
value
0 8% r
Single payment obligation
Good immunizing strategy$10,000
Bad immunizing strategy
Zvi Wiener ContTimeFin - 6 slide 56
Standard Immunization
Is very useful but is based on the assumption of the flat term structure. Often a higher order immunization is used (convexity, etc.).
Another reason for goal oriented mutual funds
(retirement, education, housing, medical expenses).
Zvi Wiener ContTimeFin - 6 slide 57
Duration Immunization Duration protects against small IR changes. Duration assumes a parallel change in the TS. Immunization is based on nominal IR. Immunization is very conservative and is inappropriate for many portfolio managers. The passage of time changes both duration and horizon date, one need to rebalance. Duration changes if yields change. Obtaining bonds for immunization can be difficult.
Zvi Wiener ContTimeFin - 6 slide 58
Cash Flow Matching and Dedication
Is a very reasonable strategy, but not always realizable.
Uncertainty of payments.
Lack of perfect match
Saving on transaction fees.
Zvi Wiener ContTimeFin - 6 slide 59
Active Bond Management
Mainly speculative approach based on ability to predict IR or credit enhancement or market imperfections (identifying mispriced loans).
Zvi Wiener ContTimeFin - 6 slide 60
Contingent Immunization
0 5 yr t
value
$10,000
$12,000
Zvi Wiener ContTimeFin - 6 slide 61
Contingent Immunization
0 5 yr t
value
$10,000
$12,000
Stop boundary
Zvi Wiener ContTimeFin - 6 slide 62
Contingent Immunization
0 5 yr t
value
$10,000
$12,000
Stop boundary
Zvi Wiener ContTimeFin - 6 slide 63
Contingent Immunization
0 5 yr t
value
$10,000
$12,000
Stop boundary
Zvi Wiener ContTimeFin - 6 slide 64
Interest Rate Swap
One of the major fixed-income tools.
Example: 6m LIBOR versus 7% fixed.
Exchange of net cash flows.
Risk involved: IR risk, default risk (small).
Why the default risk on IR swaps is small?
Zvi Wiener ContTimeFin - 6 slide 65
Interest Rate Swap
Company A Company BSwap dealer
6.95% 7.05%
LIBOR LIBOR
No need in an actual loan.Can be used as a speculative tool or for hedging.
Zvi Wiener ContTimeFin - 6 slide 66
Interest Rate Swap
Can not be priced as an exchange of two loans (old method).
Why?
Zvi Wiener ContTimeFin - 6 slide 67
Currency Swap
A similar exchange of two loans in different currencies.
Subject to a higher default risk, because of the principal.
Is useful for international companies to hedge currency risk.
Zvi Wiener ContTimeFin - 6 slide 68
Modeling a Swap
A simple fixed versus floating swap.
Current fixed rate on a 30 years loan is 7% with semi annual payments for simplicity.
Current floating rate is 6%. Notional amount is 1,000.
How can we model our future payments?
Zvi Wiener ContTimeFin - 6 slide 69
Modeling a Swap
There are two flows of cash. At maturity they cancel each other. The fixed part has payments known in advance. The only uncertainty is with the floating part.
We need a simple model of interest rates.
Zvi Wiener ContTimeFin - 6 slide 70
Modeling a Swap
0 1 2 3 60
6%
Floating IR
Zvi Wiener ContTimeFin - 6 slide 71
Modeling a Swap
0 1 2 3 60
6%
Floating IR
1 1 0 1 0
Zvi Wiener ContTimeFin - 6 slide 72
Modeling a Swap
0 1 2 3 60
6%
Floating IR
1 1 0 1 0
Zvi Wiener ContTimeFin - 6 slide 73
Modeling a Swap
0 1 2 3 60
6%
Floating IR
1 1 0 1 0
Zvi Wiener ContTimeFin - 6 slide 74
Modeling a Swap
0 1 2 3 60
6%
Floating IR
1 1 0 1 0
Zvi Wiener ContTimeFin - 6 slide 75
Modeling a Swap
0 1 2 3 60
6%
Floating IR
1 1 0 1 0
Zvi Wiener ContTimeFin - 6 slide 76
Modeling a Swap
0 1 2 3 60
6%
Floating IR
Arithmetical BM – all jumps of the same size,direction is defined by the sequence of randomvariables that you have prepared.
1 1 0 1 0
Zvi Wiener ContTimeFin - 6 slide 77
Modeling a Swap
0 1 2 3 60
6%
Floating IR
Geometrical BM – for an up jump you multiplythe current level by a constant u > 1, for a downward jump you multiply by d < 1.
1 1 0 1 0
Zvi Wiener ContTimeFin - 6 slide 78
Modeling a Swap
0 1 2 3 60
6%
Floating IR
Geometrical BM – jumps have different sizesbut up*down = down*up – an important property!
1 1 0 1 0
Zvi Wiener ContTimeFin - 6 slide 79
Home Assignment
Evaluate the swap with your sequence of random or pseudo-random numbers using both approaches arithmetical and geometrical.
Up jumps are 10 bp., and 1.1
Down -10bp., ans 0.9
Your side is fixed, discount at 7% annually.
You do not have to submit, but bring it to the class, we will discuss it.
Zvi Wiener ContTimeFin - 6 slide 80
Financial Engineering
New securities created:
IO (negative duration)
PO
CMO
Swaptions
Caps and Caplets
Floors
Ratchets
Zvi Wiener ContTimeFin - 6 slide 81
Why TS is not flat?
Assume that TS is flat, but varies with time.
Then the price of a zero coupon bond maturing in time is e-r.
How one can form an arbitrage portfolio?
Requirements:
zero investment,
never losses,
sometimes gains.
Zvi Wiener ContTimeFin - 6 slide 82
Why TS is not flat?
Take 3 bonds, maturing in 1,2, and 3 years.
The current prices are:
P1 = e-r, P2 = e-2r, P3 = e-3r.
We want to form a portfolio with a one-year bonds, b two-years, c three-years.
So the first requirement is
ae-r + be-2r + ce-3r=0
Zvi Wiener ContTimeFin - 6 slide 83
Why TS is not flat?
So the second requirement is that there are no possible losses
Equate duration of long and short sides.
-ae-r - 2be-2r - 3ce-3r=0
The two equations can be solved simultaneously.
Solution is a zero-investment, zero-loss portfolio - arbitrage.
Zvi Wiener ContTimeFin - 6 slide 84
Why TS is not flat?
r
price
rnow
Zvi Wiener ContTimeFin - 6 slide 85
Why TS is not flat?
So lve[ { -e-r - 2be-2r - 3ce-3r == 0,
-e-r - 2be-2r - 3ce-3r == 0}, {b,c}]
r
price
rnow
Zvi Wiener ContTimeFin - 6 slide 86
Term Structure Models
Let V(t,T) be the price at time t of an asset
paying $1 at time T.
Obviously V(T,T) =1.
Under the equivalent martingale measure
the discounted price is a martingale, so
)(),()(),( TTTVEtTtV Qt
Zvi Wiener ContTimeFin - 6 slide 87
Term Structure Models
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)(
)(),(),(
t
TTTVETtV Q
t
T
t
Qt dssrTTVETtV )(exp),(),(
Zvi Wiener ContTimeFin - 6 slide 88
One Factor Models
Assume that the short rate is the only factor.
dZtrdttrdr ttt ),(),(
Zvi Wiener ContTimeFin - 6 slide 89
One Factor Models
),(),( 21 TtVTtV
Consider a riskless portfolio consisting of two bonds: V1, and V2 (with ttm T1 and T2).
The riskless portfolio can be formed as
How to choose and so that the portfolio is riskless?
Zvi Wiener ContTimeFin - 6 slide 90
One Factor Models
x
V
x
V
12 ,
x
V
x
V
x
P
21
This portfolio is riskless, so it earns the risk free interest.
Zvi Wiener ContTimeFin - 6 slide 91
One Factor Models
21
12 V
x
VV
x
VP
x
V
x
V
x
V
x
V
x
P
2112 0
Zvi Wiener ContTimeFin - 6 slide 92
One Factor Models
dtt
Pdx
x
Pdx
x
PdP
22
2
)(2
1
rPdtdPE
Zvi Wiener ContTimeFin - 6 slide 93
One Factor Models
dtt
V
x
Vdt
t
V
x
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dxx
V
x
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x
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dxx
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x
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2
22
212
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1
2
1
Zvi Wiener ContTimeFin - 6 slide 94
One Factor Models
dtVx
VV
x
Vrdt
t
V
x
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t
V
x
V
dtx
V
x
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Zvi Wiener ContTimeFin - 6 slide 95
One Factor Models
22
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11
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222
2
2
rVt
V
x
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x
V
rVt
V
x
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x
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Zvi Wiener ContTimeFin - 6 slide 96
One Factor Models
xV
rVt
VxV
xV
rVt
VxV
22
222
22
11
121
22
2
2
Zvi Wiener ContTimeFin - 6 slide 97
One Factor Models
qxV
rVt
VxV
1
11
21
22
2
x
VqrV
t
V
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