chapter 9
DESCRIPTION
Financial institutions fina 481TRANSCRIPT
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Interest Rate Risk II
Chapter 9
Copyright © 2010 McGraw-Hill Ryerson Ltd., All Rights Reserved..
Prepared by Lois Tullo, Schulich School of Business, York University
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Chapter Outline
• This chapter presents the duration model and duration gap as measures of an FI’s interest rate risk: – BasiC arithmetic to calculate Duration– Economic meaning of Duration– Immunization using Duration– Problems in applying duration
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Duration: A Simple Introduction
• In general, the longer the term to maturity, the greater the sensitivity to interest rate changes.
• Example: Suppose a $100 loan with a 15% interest rate, with half repayment after ½ a year, and the remaining at the end of the year.Weights
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Duration: A Simple Introduction
• => Duration = 0.5349 x 0.5 + 0.4651 x 1 = 0.7326 years– This is the Weighted Average Time to Maturity of
this loan• Think of it this way: on a TVM basis, the bank’s
initial investment in the loan is recovered after 0.7326 years
• After that it earns a profit
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Computing duration
• Consider a 2-year, 8% coupon bond, with a face value of $1,000 and yield-to-maturity of 12%. Coupons are paid semi-annually.
• Therefore, each coupon payment is $40 and the per period YTM is (1/2) × 12% = 6%.
• Present value of each cash flow equals CFt ÷ (1+ 0.06)t where t is the period number.
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Example
– Bond A: P = $1000 = $1762.34/(1.12)5 – Bond B: P = $1000 = $3105.84/(1.12)10
• Now suppose the interest rate increases by 1%. – Bond A: P = $1762.34/(1.13)5 = $956.53– Bond B: P = $3105.84/(1.13)10 = $914.94
• The longer maturity bond has the greater drop in price because the payment is discounted a greater number of times.
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Duration: Definition and Features
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Duration Formula
N
tt
N
tt
N
ttt
N
ttt
PV
tPV
DFCF
tDFCFD
1
1
1
1
D = Duration in YearsCFt = Cash Flow at the end of Period tN = Bond MaturityDFt = Discount Factor = R = Annual Yield to Maturity
Notice that the weights correspond to the relative present values of the cash flows.
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Coupon Effect• Bonds with identical maturities will respond
differently to interest rate changes when the coupons differ.
• Think of coupon bonds as a bundle of “zero-coupon” bonds.
• With higher coupons, more of the bond’s value is generated by cash flows which take place sooner in time.
Þ Consequently, less sensitive to changes in R.
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Duration of a 6yr bond with an 8% Coupon
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2yr Bond, 8% coupon, 12% Yield
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2yr Bond, 6% coupon, 12% Yield
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2yr Bond, 8% coupon, 16% Yield
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1yr Bond, 8% coupon, 12% Yield
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Remarks on Preceding Slides
• In general, longer maturity bonds experience greater price changes in response to any change in the discount rate.
• The range of prices is greater when the coupon is lower. – The 6% bond will show a greater change in price in
response to a 2% change than the 8% bond. The 6% bond has greater interest rate risk.
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Balance Sheet Example
• Consider FI held bond with 8% YTM, 10% coupon, 12 years to maturity, FV=$1000
• Financed by issuing bond at 10% YTM, 10% coupon, 12 years to maturity, FV=$1000
• PV of each bond =?
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Balance Sheet Example
Market Value Balance Sheet:
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Impact of Maturity
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Impact of Maturity
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Impact of Coupon Rate
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Impact of Coupon Rate
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Extreme examples with equal maturities
• Consider two ten-year maturity instruments:– A ten-year zero coupon bond.– A two-cash flow “bond” that pays $999.99 almost
immediately and one penny, ten years hence.
• Small changes in yield will have a large effect on the value of the zero but essentially no impact on the hypothetical bond.
• Most bonds are between these extremes– The higher the coupon rate, the more similar the bond is
to our hypothetical bond with higher value of cash flows arriving sooner.
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Duration of Zero-coupon Bond
• For a zero coupon bond, duration equals maturity since 100% of its present value is generated by the payment of the face value, at maturity.
• For all other bonds: duration < maturity
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Duration of a Consol Bond
• Consol bonds are perpetual (never mature)– Not available in Canada
• Maturity of a consol: M = .• Duration of a consol: D = 1 + 1/R• If R = 5% => D = 1 + 1/0.05 = 21 years– Based on TVM, investors would recover their
investment in 21 years. Subsequent cash flows are pure profits.
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Duration Gap
• Suppose the bond in the previous example is the only loan asset (L) of an FI, funded by a 2-year certificate of deposit (D).
• Maturity gap: ML - MD = 2 -2 = 0
• Duration Gap: DL - DD = 1.883 - 2.0 = -0.117– Deposit has greater interest rate sensitivity than
the loan, so DGAP is negative. – FI exposed to rising interest rates.
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Features of Duration
• Duration and maturity:– D increases with M, but at a decreasing rate.
• Duration and yield-to-maturity:– D decreases as yield increases.
• Duration and coupon interest:– D decreases as coupon increases.
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Features of Duration• Duration and maturity:– D increases with M, but at a decreasing rate.
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Economic Meaning of Duration
• Duration is a measure of interest rate sensitivity or elasticity of a liability or asset:
Or equivalently,ΔP/P = -D[ΔR/(1+R)] = -MD × ΔRwhere MD is modified duration:
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Economic Meaning of Duration
• To estimate the change in price, we can rewrite this as:ΔP = -D[ΔR/(1+R)]P = -(MD) × (ΔR) × (P)
• Note the direct linear relationship between ΔP and -D.
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Semi-annual Coupon Payments
• With semi-annual coupon payments:
(ΔP/P)/(ΔR/R) = -D[ΔR/(1+(R/2)]
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Immunization example:• In 2010 an insurer guarantees a lump sum
payment in 2015 of $1,469 equal to an annually compounded rate of 8% over 5 years. Consider 2 immunization strategies:
• Discounted Bond –P = 680.58 = 1000 / (1.08) 5
– Buy 1.469 of these bonds• Six Year Coupon Bond (Table 9.2)– P5 = 1080 / 1.08 = $1000– Duration = 4.993 years (same as liability)
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Immunizing the Balance Sheet of an FI
• Duration Gap: – From the balance sheet, E=A-L. Therefore,
DE=DA-DL. In the same manner used to determine the change in bond prices, we can find the change in value of equity using duration.
– DE = [-DAA + DLL] DR/(1+R) or
– Or: DE = -[DA - DLk]A(DR/(1+R))
where k = L/A
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Duration and Immunizing
• The formula shows 3 effects:– Leverage adjusted D-Gap (in years)• Measure of exposure to interest rate risk
– The size of the FI• Measured by Assets
– The size of the interest rate shock
– E = [Leverage adjusted Duration Gap] x Assets x Interest Rate Shock
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An example:• Suppose DA = 5 years, DL = 3 years and rates
are expected to rise from 10% to 11%. (Rates change by 1%).
• Also, A = 100, L = 90 and E = 10. • Find change in E.• = -[DE DA - DLk]A[DR/(1+R)]
= -[5 - 3(90/100)]100[.01/1.1] = - $2.09 million• Methods of immunizing balance sheet.– Adjust DA , DL or k.
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An example:• If rates rise by 1%:– ∆A/A = -5 x (0.01)/1.1 = -0.04545– A = 100 + (-0.04545 x 100) = 95.45– ∆L/L = -3 x (0.01)/1.1 = -0.02727– L = 90 + (-0.02727 x 90) = 87.54– E = A – L = 7.91
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Immunization and Regulatory Concerns
• Regulators set target ratios for an FI’s capital (net worth): – Capital (Net worth) ratio = E/A
• If target is to set (E/A) = 0:– DA = DL
• But, to set E = 0:– DA = kDL
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Immunization and Regulatory Concerns
• If target is to set (E/A) = 0:– DA = DL
– E = -(5-(0.9) x 5) x $100m x (0.01/1.1) = -$0.45m
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Immunization and Regulatory Concerns
• Set E = 0:– DA = kDL
• Ex: Set DL=5.55– E = -(5-(0.9) x 5.55) x $100m x (0.01/1.1) = 0
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Difficulties in Applying Duration
– Immunizing the entire balance sheet need not be costly. Duration can be employed in combination with hedge positions to immunize.
– Immunization is a dynamic process since duration depends on instantaneous R.
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Difficulties in Applying Duration
– Large interest rate change effects not accurately captured.• Convexity
– More complex if nonparallel shift in yield curve.
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Convexity
• The duration measure is a linear approximation of a non-linear function. Þ If there are large changes in R, the approximation
is much less accurate. • All fixed-income securities are convex.• Convexity is desirable, but greater convexity
causes larger errors in the duration-based estimate of price changes.
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Convexity
• Recall that duration involves only the first derivative of the price function. – We can improve on the estimate using a Taylor
expansion. • In practice, the expansion rarely goes beyond
second order (using the second derivative).
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Modified duration & Convexity
– DP/P = -D[DR/(1+R)] + (1/2) CX (DR)2 or DP/P = -MD DR + (1/2) CX (DR)2
– Where MD implies modified duration and CX is a measure of the curvature effect.
CX = Scaling factor × [capital loss from 1bp rise in yield + capital gain from 1bp fall in yield]
– Commonly used scaling factor is 108.
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Calculation of CX
• Example: convexity of 8% coupon, 8% yield, six-year maturity Eurobond priced at $1,000.
CX = 108[DP-/P + DP+/P] = 108[(999.53785-1,000)/1,000 +
(1,000.46243-1,000)/1,000)]= 28.
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Duration Measure: Other Issues
• Default risk.• Floating-rate loans and bonds.• Duration of demand deposits and passbook
savings.• Mortgage-backed securities and mortgages– Duration relationship affected by call or
prepayment provisions.
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Contingent Claims
• Interest rate changes also affect value of off-balance sheet claims.– Duration gap hedging strategy must include the
effects on off-balance sheet items such as futures, options, swaps, caps, and other contingent claims.
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Chapter Summary
• This chapter presented the duration model and duration gap as measures of an FI’s interest rate risk: – Basis arithmetic to calculate Duration– Economic meaning of Duration– Immunization using Duration– Problems in applying duration
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Pertinent Websites
Bank for International Settlements www.bis.org Securities Exchange Commission www.sec.govThe Wall Street Journal www.wsj.com