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Risk Management Interest Rate Risk Copyright © 1996-2006 Investment Analytics
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 1
¾
Agenda
Basic Concepts
Bond Values & Interest Rate Risk ¾
Interest Rate Risk Measurement
Duration Immunization Convexity Multi-factor Duration Models ¾
Advanced Interest Rate Risk Modeling
Index Rate Duration Interest Rate Options Deterministic & Simulation Analysis Copyright © 1996-2006 Investment Analytics Interest Rate Risk
Slide: 2
Bond Values and Interest Rates What is the relationship between a bond’s price and interest rates? ¾ How does this sensitivity depend on the maturity of the bond? ¾ Are coupon bonds more sensitive to interest rates than zero coupon bonds? ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 3
Interest Rate Risk - Example Interest rate changes cause bond prices to fluctuate: ¾ Example: 8% coupon bond ¾
• If rates are at 8%, it will sell at par • If rates rise to 9% , price must fall below par – no-one will want to hold the bond at par value, so price will fall – must have expected capital gain to compensate for coupon below market rate
• If rates fall to 7%, price will rise above par – everyone will bid for bond paying above market rate – forces price up & builds in expected capital loss to offset coupon above current market rate Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 4
The Price-Yield Relationship Price Price Sensitivity Slope = ∆P / ∆y
∆P
∆y Yield Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 5
Worked Exercise: Bond Values & Interest Rates Start Bond Tutor ¾ Subject: Bond Values & Interest Rates ¾ Follow worked exercise ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 6
Factors Affecting Interest Rate Sensitivity ¾
Term
Long term bonds are more sensitive than short term bonds ¾
Coupon
Low (Zero) coupon bonds are more sensitive than high coupon bonds ¾
Yield
bonds at lower yields are more sensitive than at higher yields Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 7
Duration ¾
The further away cash flows are, the more their PV is affected by interest rates:
PV = C/(1 + r)t ¾
Duration measures weighted average maturity of cash flows:
D = Σt x Wt • Wt = CFt / (1 + y)t PV • y is yield to maturity ¾
Higher duration means greater risk
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 8
Duration & Risk Impact of changes in YTM: • ∆P = -[D / (1 + y)] x P x ∆y • D / (1 + y) is known as modified duration D* • D* = [∆P / P] x (1 / ∆y) • Percentage price change [∆P / P] = D* x ∆y ¾ Limitations: ¾
Small changes in y Parallel changes in yield curve Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 9
Example of Duration Calculation: Interest Rate = 10% Cash Discount Time Flow Factor
PV of Cash Flow
PV Weight
1 2 3 4 5
90.91 82.64 75.13 68.30 62.09
0.2398 0.2180 0.1982 0.1802 0.1638
0.2398 0.4360 0.5946 0.7207 0.8190
379.07
1.0000
2.8101
100 100 100 100 100
0.9091 0.8264 0.7513 0.6830 0.6209
TOTAL ¾ ¾
PV Weight x Time
Duration = 2.81 Years Modified Duration = 2.81 / 1.1 = 2.55 years
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 10
Duration & Price-Yield Relationship Price Slope = ∆P / ∆y ∼ D
P*
y* Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Yield Slide: 11
Two Ways to Think About Duration ¾
Weighted Average Time to Maturity
Weight the time of each cashflow by proportion of total NPV it represents ¾
As the sensitivity of a security’s PV to change in interest rates
Sensitivity = δP/δy = -Σt [CFt / (1 + y)t] x 1/P
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 12
Immunization ¾
If
duration of assets = duration of liabilities value of assets = value of liabilities Portfolio is “immunized” ¾ Portfolio value will be unchanged ¾
for small, parallel changes in yield
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 13
Worked Exercise on Duration Start Bond Tutor ¾ Subject: Duration ¾ Follow worked exercise ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 14
Trading Case B04 Flat yield curve 25% ¾ Can move to: 5% to 45% ¾ You have a liability/asset which you cannot trade ¾ Must try and preserve value of portfolio ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 15
Analysis of Case B04 ¾
Position 1 • 3200 cash • 14 of sec worth 307 • -51 of sec worth 64
¾
What should you do
Sell 14 @ 307.2 Buy 29 @ 112.064 Why 29?: • asset value = 29 * 112 = 3250 • liability value = 51 * 64 = 3264 Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 16
Analysis of Case B04
Note: cash = 4250 after trade “instantaneous exposure” Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 17
Analysis of case B04 Exposure at end of period
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 18
Problems with Conventional Immunization Assumption
Empirical Evidence
¾
Yield curve shifts are parallel
Short rates move more than long rates
¾
Yield curve changes perfectly correlated along the curve
Correlation between short and long rates much less than 1.0
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 19
Price Approximation Using Duration Price
Actual Price
Error in estimating price based on duration
P*
y1 Copyright © 1996-2006 Investment Analytics
y* Interest Rate Risk
y2
Yield Slide: 20
Convexity ¾
Duration assumes linear price-yield relationship
Duration proportional to the slope of the tangent line Accurate for small changes in yield ¾
Convexity recognizes that price-yield relationship is curvilinear
Important for large changes in yield
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 21
Convexity Formula ¾
Dollar Convexity: • δ2P / δy2 = ΣCFt x t(t+1) / (1 + y)t+2
Price change due to convexity: • ∆P = Dollar Convexity x (∆y)2 ¾
Convexity = [δ2P / δy2] x (1 / P)
Percentage price change due to convexity: • ∆P / P = 0.5 x Convexity x (∆y)2 Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 22
Convexity Adjustment: Example ¾
Straight Bond • 6% coupon, 25yr, yield 9% • Modified Duration =10.62 • Convexity = 182.92
¾
% Price Change: Yield Move +200bp -200bp
Copyright © 1996-2006 Investment Analytics
Duration
Convexity
(D* ∆y)
0.5 x C (∆y)2
-21.24% +21.24%
3.66% 3.66%
Interest Rate Risk
Total -17.58% +24.90%
Slide: 23
Summary: Interest Rates & Risk How interest rates affect bond prices ¾ Duration ¾ Immunization ¾ Convexity ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 24
A Two Factor Model of Yield Curve Changes Change in spot rate
Change Change = At x in short rate + Bt x in long rate = αt x Change in spread
+
βt x Change in long rate
Spread: (Long rate - Short rate) ¾ Two factor Model: ¾
αT : sensitivity of T-period spot rate to changes in spread βT: sensitivity of T-period spot rate to changes in long rate Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 25
Immunization with Two Factor Model ¾
Factors
Long rate Spread = long rate - short rate ¾
Durations: each asset has two durations
Long Duration: sensitivity to change in long rate Spread Duration: sensitivity to change in spread
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 26
Computing Two Factor Durations ¾
Duration formula:
DS = -ΣTi αTi[cie-RTi/PV] DL = -ΣTi βTi[cie-RTi/PV] ¾
Regression Analysis ∆RT = AT + αT∆S + βT∆L + εT
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 27
Estimated Long Rate & Spread Sensitivities (Nelson/Schaefer) Maturity (Years)
Spread Sensitivity
Long Rate Sensitivity
1.000 0.743 0.542 0.391 0.269 0.200 0.163 0.131 0.100 0.100 0.043 0.019 0.000
1.000 1.036 1.026 0.997 0.970 0.953 0.950 0.962 0.983 1.005 1.022 1.022 1.000
1 2 3 4 5 6 7 8 9 10 11 12 13 Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 28
Spread & Long Rate Sensitivities 1.200 1.000 0.800 0.600 0.400 0.200 0.000 0
2
4
6
Spread Sensitivity Copyright © 1996-2006 Investment Analytics
8
10
12
14
Long Rate Sensitivity
Interest Rate Risk
Slide: 29
Implied Spot Rates: Relative Importance of Factors % of Total Explained Variance Accounted for by Maturity 6 Months 1 year 2 years 5 years 8 years 10 years 14 years 18 years
Total Variance Explained Factor 1 Factor 2 Factor 3 99.5 79.5 17.2 3.3 99.4 89.7 10.1 0.2 98.2 93.4 2.4 4.2 98.8 98.2 1.1 0.7 98.7 95.4 4.6 0.0 98.8 92.9 6.9 0.2 98.4 86.2 11.5 2.2 93.5 80.5 14.3 5.2
Average
98.4
89.5
8.5
2.0
Source: Journal of Fixed Income, “Volatility and the Yield Curve”, Litterman, Scheinkman & Weiss Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 30
Example: Calculating Spread Duration 8% 4-year bond Spot rates 10% flat
Time 1 2 3 4 TOTAL
Cash Flow 8 8 8 108
DF 0.9091 0.8264 0.7513 0.6830
PV 7.27 6.61 6.01 73.77 93.66
Time x PV Spread x Spread Sensitivity Sensitivity 1.000 7.27 0.743 9.82 0.542 9.77 0.391 115.37 142.24
Spread Duration = 142.24 / 93.66 = 1.52 Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 31
Immunization Conditions Portfolio Weights add to One ¾ Match Spread Duration ¾
Weighted average of spread duration of assets = spread duration of liabilities ¾
Match Long Duration
Weighted average of long duration of assets = long duration of liabilities ¾
Equations
w1 + w2 + w3 = 1 w1D1S + w2D2S + w3D3S = DS w1D1L + w2D2L +Interest w3D3L = DL Copyright © 1996-2006 Investment Analytics Rate Risk
Slide: 32
When One Asset is Cash ¾
Sensitivity of cash to all interest rates is zero w1D1S + w2D2S = DS w1D1L + w2D2L = DL
¾
Cash holding is residual w3 = 1 - w1 - w2
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 33
Lab: Bond Hedging Exercise Worksheet: Bond Hedging ¾ Scenario: ¾
You have a short position in 8-year bonds Have to hedge using 3 and 15 year bonds ¾
Hedging
Create conventional duration hedge Test under 4 scenarios Create 2-factor duration hedge Repeat test & compare ¾
See Notes & Solution
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 34
Solution: Bond Hedging Exercise ¾
Hedge Structure Method Conventional Two-Factor
¾
Holdings Cash 0.00 -.0089
3yr 0.3538 0.4599
8yr -1.000 -1.000
15yr 0.6462 0.5490
Hedge Performance (Profit/Loss) Scenario I II III IV
Conventional -27bp -29bp 28bp 25bp
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
2-Factor 3bp 3bp 2bp 2bp Slide: 35
Advanced Interest Risk Modeling ¾
Index rate contingent cash flows
Key Treasury Rate Duration ¾
Interest rate options
Option-adjusted duration ¾
Analytical methods
Deterministic Monte Carlo simulation
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 36
Duration Risk Measurement Recall: (dP/P) = - D* x dr ¾ Modified Duration D* = -(dP/dr) x 1/P ¾
For swaps & derivatives concept of duration is ambiguous ¾
Need to measure sensitivity to changes in:
¾
Index Rate DURINDEX = -(dP/drindex) x 1/P Discount Rate DURDISC = -(dP/drdisc) x 1/P
¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 37
Calculating Duration Perturbation Method ¾
IRD
Add small increment dr (1bp) to index rate Recompute PV ¾
DRD
Add small increment dr (1bp) to discount rate Recompute PV ¾
DURATION = [PVOrig - PVNew]/PVOrig x 1/dr
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 38
Discount Rate Can be found by assuming cash flows are non-contingent ¾ YTM of comparable fixed coupon note of same maturity ¾ Hence DURDISC = Duration of vanilla note ¾
E.g. for 3-yr note DRD = 2.8 yrs
¾
Exception: Note which has indeterminate maturity
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 39
Index rate E.g. 3-yr FRN ¾ Coupon = 3-month LIBOR, paid quarterly ¾ What is appropriate index rate? ¾
NOT 3-month LIBOR Aggregate of all floating rate components • 12 different IR’s in this example ¾
Solution: swap rate
Summarizes entire LIBOR cash flow stream Expressed as a spread over 3-year treasury rate Hence DURINDEX = -2.8 approx. Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 40
Net Duration ¾
3-year FRN, coupon 3-m LIBOR
DRD IRD NET DURATION
Copyright © 1996-2006 Investment Analytics
= 2.8 = -2.8 = 0
Interest Rate Risk
Slide: 41
Key Treasury Rate Duration (KTRD) Calculates change in price wrt change in one segment of the Treasury curve. ¾ Used when Index rate and Discount rate are not equal ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 42
Duration for Derivative Structures ¾
E.g. Capped FRN
Like capped floating leg of swap ¾
Option Adjusted Duration
OAD = DUR x P / PC x (1 - ∆) • • • •
DUR = Duration of uncapped FRN P = price of uncapped FRN PC = price of capped FRN ∆ = cap delta
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 43
Deterministic Analysis & Option Delta 1.0
Deterministic analysis overestimates delta
Delta
Deterministic analysis underestimates delta
Deterministic analysis Option analysis
0.0 K Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 44
Volatility Duration Applies to securities with embedded optionality ¾ DURVOL = - (1/P) x (dP/dσ) ¾
= - (1/P) x Vega ¾ Vega greatest for ATM options
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 45
Cap Vega Interest Rate Cap Vega 30
Vega
25 20 15 10 5 0 K
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 46
Evaluating Risk ¾
Deterministic Analysis
Assume know rates in advance Determines cash flows, yield • Duration estimated using perturbation method ¾
Simulation Analysis
Monte Carlo simulation model of interest rates Statistical analysis of: • Cash flows • Yield • Duration Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 47
Deterministic Analysis ¾
Forward Analysis
Assumes index spot rates move to forwards Problem of bias • Forward rates typically exceed future spot rates ¾
Expectation analysis
Projects ‘expected’ spot rates
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 48
Linear Smooth Expectation (LSE) Analysis ¾
Set final index spot rate
E.G. from forward rate Estimate intermediate index rates using linear interpolation ¾ Compute cash flows, yield, duration in normal way ¾ Repeat for range of final index rates ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 49
Monte-Carlo Methodology Simulate movement in index rates ¾ Calculate cash flows, PV’s, yield, duration ¾ Repeat large no of times ¾ Create histogram of yield, duration values ¾
Calculate average yield, duration
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 50
Generating Simulated Index Rates ¾
R + ∆R = R x Exp[(µ−σ2/2)∆t + σ∆z] • • • • •
¾
∆R is change in index rate µ is drift factor σ is volatility ∆Z = ε(∆t)1/2 ε is normal random variable, No(0,1)
Procedure:
Generate ε (random) Compute new index rates, cash flows, etc Estimate duration using perturbation method Repeat many times (10,000+) Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 51
Example YTM Probability Distribution 12.00%
Frequency
10.00% 8.00% 6.00% 4.00% 2.00%
More
5.75%
5.52%
5.29%
5.06%
4.83%
4.60%
4.37%
4.14%
3.91%
3.68%
3.45%
0.00%
YTM
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 52
Lab: Capped FRN ¾
Start with simple 3-year FRN, quarterly LIBOR
Confirm IRD = - DRD ¾
FRN Coupon LIBOR + 0.5%, 5.5% Cap
Calculate IRD, DRD, Net Duration Use simulation analysis to estimate yield, duration Use LSE analysis to compute yield, duration Compare LSE & simulation analysis Compare OAD with deterministic & simulation analysis Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 53
Solution: Capped FRN Duration Estimates 2.50
2.00
1.50 Duration
LSE DUR SIM DUR OAD
1.00
0.50
0.00 3.5%
4.5%
5.5%
6.5%
7.5%
8.5%
9.5%
10.5%
11.5%
12.5%
LIBOR at Maturity
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 54
Step-up Recovery Floaters (SURFs) ¾
Objective Provide higher floating yield than CMT or LIBOR FRNs
¾
Structure Above-market floor, some upside participation Example: 5-year note • Coupon = 0.5*(10-year CMT) + 1.5% • Floor 4.5%
¾
Equivalent Position Short T-Bonds Long ITM Bond Call Options
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 55
SURF vs CMT FRN CMT FRN vs. SURF 8
Yield (%)
7 6 5 CMT FRN
4
SURF
3 3
5
7
9
11
10Yr CMT at Maturity (%)
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 56
SURF Risk Factors ¾
Net Duration A lower rates, behaves like a fixed income security • Due to coupon floor • Hence higher duration a low rates
At higher rates, behaves more like an FRN • Hence lower duration at high rates ¾
Volatility Duration Long a floor option, positive Vega Hence negative Vol. Duration Value of floor (and note) increases with volatility
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 57
SURF - Net Duration Net Duration of SURF 4.5
Dura tion
4.0 3.5 3.0 2.5 2.0 3
4
5
Copyright © 1996-2006 Investment Analytics
6 7 8 10Yr CMT at Maturity Interest Rate Risk
9
10
11
Slide: 58
SURF - Volatility Duration Volatility Duration
Volatility Duration (bps/vol)
0 3
5
7
9
11
-2 -4 -6 -8 -10 10Yr CMT at Maturity (%)
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 59
Range Floaters / LIBOR Enhanced Accrual Notes (LEANs) ¾
Typical Structure 4 Year FRN Coupon LIBOR + 50bp • Only paid if LIBOR in range
Year 1-2 range 5% - 6% Year 3-4 range 6% - 7% • Ranges increase due to upward sloping forward curve ¾
Investor has written series of binary calls and puts • Compensated by higher spread • Taking advantage of high implieds • Betting that volatility will be lower than anticipated
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 60
LEANs - Risk Factors ¾
Net Duration Close to zero within range Changes dramatically outside range • Negative below range – note value rises with rates
• Positive above range (>> maturity) – -note value falls as rates rise
¾
Volatility Duration Positive in range • Note loses value if volatility increases
Negative outside range • Note gains in value if volatility rises
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 61
Multi-Index Notes Coupon based on sum or difference between multiple indices ¾ Most common structures: ¾
CMT-LIBOR Differential Notes Prime-LIBOR Differential Notes
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 62
Example: CMT-LIBOR Diff. Note ¾
¾
¾
Note features: Issuer: US Agency Maturity: 3 years Annual Coupon: (10-year CMT - 12m LIBOR) +2.00% Discount Rate Duration DR is to-maturity Treasury rate Hence DRD = 2.8 years approx. Index Component 10-year CMT LIBOR
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 63
CMT-LIBOR Diff. Note Overview ¾ ¾
Investor Outlook Achieve higher coupon than either CMT or LIBOR Risk Yield curve flattening will rapidly erode the note’s yield advantage
¾
Equivalent Position: Long CMT FRN Long Eurodollar Futures
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 64
CMT-LIBOR Diff. Note ¾
10-Year CMT: CouponPV =
¾ ¾
T1,10 (1 + r1 )
1
+
T2,10 (1 + r2 )
2
+
T3,10 (1 + r3 )
3
1bp change in T10 produces approx. 1bp change in 10-year forward rate T1,10 Hence value of note will increase by PV01 in each year
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 65
CMT-LIBOR Diff. Note: Key Treasury Rate Durations Key Rate T10 T11 T12
Copyright © 1996-2006 Investment Analytics
PV01 -1/(1+T1)1 -1/(1+T2)2 -1/(1+T3)3
Interest Rate Risk
Duration -0.95 -0.91 -0.86
Slide: 66
CMT-LIBOR Diff. Note: LIBOR Component Equivalent to 3-year swap ¾ Corresponds to to-maturity Treasury rate ¾ Hence duration is equiv. to fixed coupon 3year note ¾ KTRD for LIBor component is 2.8 years ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 67
CMT-LIBOR Diff Note: KTRD’s Component Index
KTR
Index rate 10-yr CMT T10 T11 T12 12-m LIBOR Discounting Rate
T3
To-maturity T3 Treasury
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
KTRD -0.95 -0.91 -0.86 2.8 2.8
Slide: 68
CMT-LIBOR Diff Note: KTRD Spectrum Key Rate Duration (years)
CMT-LIBOR Diff Note - Key Rate Duration Spectrum LIBOR
6
Discounting
5
CMT
4 3 2 1 0 -1
3
4
5
6
7
8
9
10
11
12
-2 Maturity
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 69
Summary: Risk Management ¾
Risk Measurement
Duration Concepts Index Rate Duration Key Treasury Rate Duration ¾
Risk Analysis
Deterministic Simulation
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 70
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 1
¾
Agenda
Basic Concepts
Bond Values & Interest Rate Risk ¾
Interest Rate Risk Measurement
Duration Immunization Convexity Multi-factor Duration Models ¾
Advanced Interest Rate Risk Modeling
Index Rate Duration Interest Rate Options Deterministic & Simulation Analysis Copyright © 1996-2006 Investment Analytics Interest Rate Risk
Slide: 2
Bond Values and Interest Rates What is the relationship between a bond’s price and interest rates? ¾ How does this sensitivity depend on the maturity of the bond? ¾ Are coupon bonds more sensitive to interest rates than zero coupon bonds? ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 3
Interest Rate Risk - Example Interest rate changes cause bond prices to fluctuate: ¾ Example: 8% coupon bond ¾
• If rates are at 8%, it will sell at par • If rates rise to 9% , price must fall below par – no-one will want to hold the bond at par value, so price will fall – must have expected capital gain to compensate for coupon below market rate
• If rates fall to 7%, price will rise above par – everyone will bid for bond paying above market rate – forces price up & builds in expected capital loss to offset coupon above current market rate Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 4
The Price-Yield Relationship Price Price Sensitivity Slope = ∆P / ∆y
∆P
∆y Yield Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 5
Worked Exercise: Bond Values & Interest Rates Start Bond Tutor ¾ Subject: Bond Values & Interest Rates ¾ Follow worked exercise ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 6
Factors Affecting Interest Rate Sensitivity ¾
Term
Long term bonds are more sensitive than short term bonds ¾
Coupon
Low (Zero) coupon bonds are more sensitive than high coupon bonds ¾
Yield
bonds at lower yields are more sensitive than at higher yields Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 7
Duration ¾
The further away cash flows are, the more their PV is affected by interest rates:
PV = C/(1 + r)t ¾
Duration measures weighted average maturity of cash flows:
D = Σt x Wt • Wt = CFt / (1 + y)t PV • y is yield to maturity ¾
Higher duration means greater risk
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 8
Duration & Risk Impact of changes in YTM: • ∆P = -[D / (1 + y)] x P x ∆y • D / (1 + y) is known as modified duration D* • D* = [∆P / P] x (1 / ∆y) • Percentage price change [∆P / P] = D* x ∆y ¾ Limitations: ¾
Small changes in y Parallel changes in yield curve Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 9
Example of Duration Calculation: Interest Rate = 10% Cash Discount Time Flow Factor
PV of Cash Flow
PV Weight
1 2 3 4 5
90.91 82.64 75.13 68.30 62.09
0.2398 0.2180 0.1982 0.1802 0.1638
0.2398 0.4360 0.5946 0.7207 0.8190
379.07
1.0000
2.8101
100 100 100 100 100
0.9091 0.8264 0.7513 0.6830 0.6209
TOTAL ¾ ¾
PV Weight x Time
Duration = 2.81 Years Modified Duration = 2.81 / 1.1 = 2.55 years
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 10
Duration & Price-Yield Relationship Price Slope = ∆P / ∆y ∼ D
P*
y* Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Yield Slide: 11
Two Ways to Think About Duration ¾
Weighted Average Time to Maturity
Weight the time of each cashflow by proportion of total NPV it represents ¾
As the sensitivity of a security’s PV to change in interest rates
Sensitivity = δP/δy = -Σt [CFt / (1 + y)t] x 1/P
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 12
Immunization ¾
If
duration of assets = duration of liabilities value of assets = value of liabilities Portfolio is “immunized” ¾ Portfolio value will be unchanged ¾
for small, parallel changes in yield
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 13
Worked Exercise on Duration Start Bond Tutor ¾ Subject: Duration ¾ Follow worked exercise ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 14
Trading Case B04 Flat yield curve 25% ¾ Can move to: 5% to 45% ¾ You have a liability/asset which you cannot trade ¾ Must try and preserve value of portfolio ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 15
Analysis of Case B04 ¾
Position 1 • 3200 cash • 14 of sec worth 307 • -51 of sec worth 64
¾
What should you do
Sell 14 @ 307.2 Buy 29 @ 112.064 Why 29?: • asset value = 29 * 112 = 3250 • liability value = 51 * 64 = 3264 Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 16
Analysis of Case B04
Note: cash = 4250 after trade “instantaneous exposure” Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 17
Analysis of case B04 Exposure at end of period
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 18
Problems with Conventional Immunization Assumption
Empirical Evidence
¾
Yield curve shifts are parallel
Short rates move more than long rates
¾
Yield curve changes perfectly correlated along the curve
Correlation between short and long rates much less than 1.0
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Interest Rate Risk
Slide: 19
Price Approximation Using Duration Price
Actual Price
Error in estimating price based on duration
P*
y1 Copyright © 1996-2006 Investment Analytics
y* Interest Rate Risk
y2
Yield Slide: 20
Convexity ¾
Duration assumes linear price-yield relationship
Duration proportional to the slope of the tangent line Accurate for small changes in yield ¾
Convexity recognizes that price-yield relationship is curvilinear
Important for large changes in yield
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Interest Rate Risk
Slide: 21
Convexity Formula ¾
Dollar Convexity: • δ2P / δy2 = ΣCFt x t(t+1) / (1 + y)t+2
Price change due to convexity: • ∆P = Dollar Convexity x (∆y)2 ¾
Convexity = [δ2P / δy2] x (1 / P)
Percentage price change due to convexity: • ∆P / P = 0.5 x Convexity x (∆y)2 Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 22
Convexity Adjustment: Example ¾
Straight Bond • 6% coupon, 25yr, yield 9% • Modified Duration =10.62 • Convexity = 182.92
¾
% Price Change: Yield Move +200bp -200bp
Copyright © 1996-2006 Investment Analytics
Duration
Convexity
(D* ∆y)
0.5 x C (∆y)2
-21.24% +21.24%
3.66% 3.66%
Interest Rate Risk
Total -17.58% +24.90%
Slide: 23
Summary: Interest Rates & Risk How interest rates affect bond prices ¾ Duration ¾ Immunization ¾ Convexity ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 24
A Two Factor Model of Yield Curve Changes Change in spot rate
Change Change = At x in short rate + Bt x in long rate = αt x Change in spread
+
βt x Change in long rate
Spread: (Long rate - Short rate) ¾ Two factor Model: ¾
αT : sensitivity of T-period spot rate to changes in spread βT: sensitivity of T-period spot rate to changes in long rate Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 25
Immunization with Two Factor Model ¾
Factors
Long rate Spread = long rate - short rate ¾
Durations: each asset has two durations
Long Duration: sensitivity to change in long rate Spread Duration: sensitivity to change in spread
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 26
Computing Two Factor Durations ¾
Duration formula:
DS = -ΣTi αTi[cie-RTi/PV] DL = -ΣTi βTi[cie-RTi/PV] ¾
Regression Analysis ∆RT = AT + αT∆S + βT∆L + εT
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 27
Estimated Long Rate & Spread Sensitivities (Nelson/Schaefer) Maturity (Years)
Spread Sensitivity
Long Rate Sensitivity
1.000 0.743 0.542 0.391 0.269 0.200 0.163 0.131 0.100 0.100 0.043 0.019 0.000
1.000 1.036 1.026 0.997 0.970 0.953 0.950 0.962 0.983 1.005 1.022 1.022 1.000
1 2 3 4 5 6 7 8 9 10 11 12 13 Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 28
Spread & Long Rate Sensitivities 1.200 1.000 0.800 0.600 0.400 0.200 0.000 0
2
4
6
Spread Sensitivity Copyright © 1996-2006 Investment Analytics
8
10
12
14
Long Rate Sensitivity
Interest Rate Risk
Slide: 29
Implied Spot Rates: Relative Importance of Factors % of Total Explained Variance Accounted for by Maturity 6 Months 1 year 2 years 5 years 8 years 10 years 14 years 18 years
Total Variance Explained Factor 1 Factor 2 Factor 3 99.5 79.5 17.2 3.3 99.4 89.7 10.1 0.2 98.2 93.4 2.4 4.2 98.8 98.2 1.1 0.7 98.7 95.4 4.6 0.0 98.8 92.9 6.9 0.2 98.4 86.2 11.5 2.2 93.5 80.5 14.3 5.2
Average
98.4
89.5
8.5
2.0
Source: Journal of Fixed Income, “Volatility and the Yield Curve”, Litterman, Scheinkman & Weiss Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 30
Example: Calculating Spread Duration 8% 4-year bond Spot rates 10% flat
Time 1 2 3 4 TOTAL
Cash Flow 8 8 8 108
DF 0.9091 0.8264 0.7513 0.6830
PV 7.27 6.61 6.01 73.77 93.66
Time x PV Spread x Spread Sensitivity Sensitivity 1.000 7.27 0.743 9.82 0.542 9.77 0.391 115.37 142.24
Spread Duration = 142.24 / 93.66 = 1.52 Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 31
Immunization Conditions Portfolio Weights add to One ¾ Match Spread Duration ¾
Weighted average of spread duration of assets = spread duration of liabilities ¾
Match Long Duration
Weighted average of long duration of assets = long duration of liabilities ¾
Equations
w1 + w2 + w3 = 1 w1D1S + w2D2S + w3D3S = DS w1D1L + w2D2L +Interest w3D3L = DL Copyright © 1996-2006 Investment Analytics Rate Risk
Slide: 32
When One Asset is Cash ¾
Sensitivity of cash to all interest rates is zero w1D1S + w2D2S = DS w1D1L + w2D2L = DL
¾
Cash holding is residual w3 = 1 - w1 - w2
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 33
Lab: Bond Hedging Exercise Worksheet: Bond Hedging ¾ Scenario: ¾
You have a short position in 8-year bonds Have to hedge using 3 and 15 year bonds ¾
Hedging
Create conventional duration hedge Test under 4 scenarios Create 2-factor duration hedge Repeat test & compare ¾
See Notes & Solution
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 34
Solution: Bond Hedging Exercise ¾
Hedge Structure Method Conventional Two-Factor
¾
Holdings Cash 0.00 -.0089
3yr 0.3538 0.4599
8yr -1.000 -1.000
15yr 0.6462 0.5490
Hedge Performance (Profit/Loss) Scenario I II III IV
Conventional -27bp -29bp 28bp 25bp
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Interest Rate Risk
2-Factor 3bp 3bp 2bp 2bp Slide: 35
Advanced Interest Risk Modeling ¾
Index rate contingent cash flows
Key Treasury Rate Duration ¾
Interest rate options
Option-adjusted duration ¾
Analytical methods
Deterministic Monte Carlo simulation
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 36
Duration Risk Measurement Recall: (dP/P) = - D* x dr ¾ Modified Duration D* = -(dP/dr) x 1/P ¾
For swaps & derivatives concept of duration is ambiguous ¾
Need to measure sensitivity to changes in:
¾
Index Rate DURINDEX = -(dP/drindex) x 1/P Discount Rate DURDISC = -(dP/drdisc) x 1/P
¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 37
Calculating Duration Perturbation Method ¾
IRD
Add small increment dr (1bp) to index rate Recompute PV ¾
DRD
Add small increment dr (1bp) to discount rate Recompute PV ¾
DURATION = [PVOrig - PVNew]/PVOrig x 1/dr
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 38
Discount Rate Can be found by assuming cash flows are non-contingent ¾ YTM of comparable fixed coupon note of same maturity ¾ Hence DURDISC = Duration of vanilla note ¾
E.g. for 3-yr note DRD = 2.8 yrs
¾
Exception: Note which has indeterminate maturity
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 39
Index rate E.g. 3-yr FRN ¾ Coupon = 3-month LIBOR, paid quarterly ¾ What is appropriate index rate? ¾
NOT 3-month LIBOR Aggregate of all floating rate components • 12 different IR’s in this example ¾
Solution: swap rate
Summarizes entire LIBOR cash flow stream Expressed as a spread over 3-year treasury rate Hence DURINDEX = -2.8 approx. Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 40
Net Duration ¾
3-year FRN, coupon 3-m LIBOR
DRD IRD NET DURATION
Copyright © 1996-2006 Investment Analytics
= 2.8 = -2.8 = 0
Interest Rate Risk
Slide: 41
Key Treasury Rate Duration (KTRD) Calculates change in price wrt change in one segment of the Treasury curve. ¾ Used when Index rate and Discount rate are not equal ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 42
Duration for Derivative Structures ¾
E.g. Capped FRN
Like capped floating leg of swap ¾
Option Adjusted Duration
OAD = DUR x P / PC x (1 - ∆) • • • •
DUR = Duration of uncapped FRN P = price of uncapped FRN PC = price of capped FRN ∆ = cap delta
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 43
Deterministic Analysis & Option Delta 1.0
Deterministic analysis overestimates delta
Delta
Deterministic analysis underestimates delta
Deterministic analysis Option analysis
0.0 K Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 44
Volatility Duration Applies to securities with embedded optionality ¾ DURVOL = - (1/P) x (dP/dσ) ¾
= - (1/P) x Vega ¾ Vega greatest for ATM options
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 45
Cap Vega Interest Rate Cap Vega 30
Vega
25 20 15 10 5 0 K
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 46
Evaluating Risk ¾
Deterministic Analysis
Assume know rates in advance Determines cash flows, yield • Duration estimated using perturbation method ¾
Simulation Analysis
Monte Carlo simulation model of interest rates Statistical analysis of: • Cash flows • Yield • Duration Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 47
Deterministic Analysis ¾
Forward Analysis
Assumes index spot rates move to forwards Problem of bias • Forward rates typically exceed future spot rates ¾
Expectation analysis
Projects ‘expected’ spot rates
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 48
Linear Smooth Expectation (LSE) Analysis ¾
Set final index spot rate
E.G. from forward rate Estimate intermediate index rates using linear interpolation ¾ Compute cash flows, yield, duration in normal way ¾ Repeat for range of final index rates ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 49
Monte-Carlo Methodology Simulate movement in index rates ¾ Calculate cash flows, PV’s, yield, duration ¾ Repeat large no of times ¾ Create histogram of yield, duration values ¾
Calculate average yield, duration
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 50
Generating Simulated Index Rates ¾
R + ∆R = R x Exp[(µ−σ2/2)∆t + σ∆z] • • • • •
¾
∆R is change in index rate µ is drift factor σ is volatility ∆Z = ε(∆t)1/2 ε is normal random variable, No(0,1)
Procedure:
Generate ε (random) Compute new index rates, cash flows, etc Estimate duration using perturbation method Repeat many times (10,000+) Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 51
Example YTM Probability Distribution 12.00%
Frequency
10.00% 8.00% 6.00% 4.00% 2.00%
More
5.75%
5.52%
5.29%
5.06%
4.83%
4.60%
4.37%
4.14%
3.91%
3.68%
3.45%
0.00%
YTM
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 52
Lab: Capped FRN ¾
Start with simple 3-year FRN, quarterly LIBOR
Confirm IRD = - DRD ¾
FRN Coupon LIBOR + 0.5%, 5.5% Cap
Calculate IRD, DRD, Net Duration Use simulation analysis to estimate yield, duration Use LSE analysis to compute yield, duration Compare LSE & simulation analysis Compare OAD with deterministic & simulation analysis Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 53
Solution: Capped FRN Duration Estimates 2.50
2.00
1.50 Duration
LSE DUR SIM DUR OAD
1.00
0.50
0.00 3.5%
4.5%
5.5%
6.5%
7.5%
8.5%
9.5%
10.5%
11.5%
12.5%
LIBOR at Maturity
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 54
Step-up Recovery Floaters (SURFs) ¾
Objective Provide higher floating yield than CMT or LIBOR FRNs
¾
Structure Above-market floor, some upside participation Example: 5-year note • Coupon = 0.5*(10-year CMT) + 1.5% • Floor 4.5%
¾
Equivalent Position Short T-Bonds Long ITM Bond Call Options
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 55
SURF vs CMT FRN CMT FRN vs. SURF 8
Yield (%)
7 6 5 CMT FRN
4
SURF
3 3
5
7
9
11
10Yr CMT at Maturity (%)
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 56
SURF Risk Factors ¾
Net Duration A lower rates, behaves like a fixed income security • Due to coupon floor • Hence higher duration a low rates
At higher rates, behaves more like an FRN • Hence lower duration at high rates ¾
Volatility Duration Long a floor option, positive Vega Hence negative Vol. Duration Value of floor (and note) increases with volatility
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 57
SURF - Net Duration Net Duration of SURF 4.5
Dura tion
4.0 3.5 3.0 2.5 2.0 3
4
5
Copyright © 1996-2006 Investment Analytics
6 7 8 10Yr CMT at Maturity Interest Rate Risk
9
10
11
Slide: 58
SURF - Volatility Duration Volatility Duration
Volatility Duration (bps/vol)
0 3
5
7
9
11
-2 -4 -6 -8 -10 10Yr CMT at Maturity (%)
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 59
Range Floaters / LIBOR Enhanced Accrual Notes (LEANs) ¾
Typical Structure 4 Year FRN Coupon LIBOR + 50bp • Only paid if LIBOR in range
Year 1-2 range 5% - 6% Year 3-4 range 6% - 7% • Ranges increase due to upward sloping forward curve ¾
Investor has written series of binary calls and puts • Compensated by higher spread • Taking advantage of high implieds • Betting that volatility will be lower than anticipated
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 60
LEANs - Risk Factors ¾
Net Duration Close to zero within range Changes dramatically outside range • Negative below range – note value rises with rates
• Positive above range (>> maturity) – -note value falls as rates rise
¾
Volatility Duration Positive in range • Note loses value if volatility increases
Negative outside range • Note gains in value if volatility rises
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 61
Multi-Index Notes Coupon based on sum or difference between multiple indices ¾ Most common structures: ¾
CMT-LIBOR Differential Notes Prime-LIBOR Differential Notes
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 62
Example: CMT-LIBOR Diff. Note ¾
¾
¾
Note features: Issuer: US Agency Maturity: 3 years Annual Coupon: (10-year CMT - 12m LIBOR) +2.00% Discount Rate Duration DR is to-maturity Treasury rate Hence DRD = 2.8 years approx. Index Component 10-year CMT LIBOR
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 63
CMT-LIBOR Diff. Note Overview ¾ ¾
Investor Outlook Achieve higher coupon than either CMT or LIBOR Risk Yield curve flattening will rapidly erode the note’s yield advantage
¾
Equivalent Position: Long CMT FRN Long Eurodollar Futures
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 64
CMT-LIBOR Diff. Note ¾
10-Year CMT: CouponPV =
¾ ¾
T1,10 (1 + r1 )
1
+
T2,10 (1 + r2 )
2
+
T3,10 (1 + r3 )
3
1bp change in T10 produces approx. 1bp change in 10-year forward rate T1,10 Hence value of note will increase by PV01 in each year
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 65
CMT-LIBOR Diff. Note: Key Treasury Rate Durations Key Rate T10 T11 T12
Copyright © 1996-2006 Investment Analytics
PV01 -1/(1+T1)1 -1/(1+T2)2 -1/(1+T3)3
Interest Rate Risk
Duration -0.95 -0.91 -0.86
Slide: 66
CMT-LIBOR Diff. Note: LIBOR Component Equivalent to 3-year swap ¾ Corresponds to to-maturity Treasury rate ¾ Hence duration is equiv. to fixed coupon 3year note ¾ KTRD for LIBor component is 2.8 years ¾
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 67
CMT-LIBOR Diff Note: KTRD’s Component Index
KTR
Index rate 10-yr CMT T10 T11 T12 12-m LIBOR Discounting Rate
T3
To-maturity T3 Treasury
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
KTRD -0.95 -0.91 -0.86 2.8 2.8
Slide: 68
CMT-LIBOR Diff Note: KTRD Spectrum Key Rate Duration (years)
CMT-LIBOR Diff Note - Key Rate Duration Spectrum LIBOR
6
Discounting
5
CMT
4 3 2 1 0 -1
3
4
5
6
7
8
9
10
11
12
-2 Maturity
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 69
Summary: Risk Management ¾
Risk Measurement
Duration Concepts Index Rate Duration Key Treasury Rate Duration ¾
Risk Analysis
Deterministic Simulation
Copyright © 1996-2006 Investment Analytics
Interest Rate Risk
Slide: 70
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