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Transcript of Copyright © 2000 by Harcourt, Inc. All rights reserved. 16-1 Chapter 16 Interest Rate Risk...
Copyright © 2000 by Harcourt, Inc. All rights reserved.
16-1
Chapter 16Interest Rate Risk Measurements and
Immunization Using Duration
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16-2
Interest Rate Risk Defined
The potential variation in returns due to UNEXPECTED changes in interest rates.
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16-3
Fundamental Principles of Financial Asset Pricing
A single investor is unable to influence the price of a financial asset.
The supply and demand for a financial asset also influences price.
All else equal, the price of a riskier asset will be lower than that of a less risky one because financial market participants are risk averse.• Risk aversion causes investors to demand higher
expected rates of return for riskier investments.
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16-4
Financial Asset Pricing
n
t
t
y
CP
10 1
where:
y= discount rate that makes the sum of the present value of assets’ cash flows (Ct) equal to its price (P0)
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16-5
All else equal, prices and yields change simultaneously in the opposite direction.
• If y rises, the price of the security falls.
• If y falls, the price of the security rises.
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16-6
Calculating Ex Ante Effective Annual Yields (y) for Bonds
The ex ante effective annual yield (y) that an investor expects to receive over the life of the bond is the discount rate that makes the present value of the bond’s future cash flows equal to its current price.
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16-7
)Value(PVIFMaturity
)IFAPayment(PVCoupon
,y
,y
0n
nP
where:
P0 = price (market value) of the bond
PVIFA= present value of the annuity factor
[1-(1/(1+ y)n)/ y)]
PVIF= present value factor [1/(1+ y)n]
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16-8
In order to earn the ex ante effective annual yield, an investor would have to:
• hold the bond until maturity; and
• invest each coupon payment at the y rate over the life of the bond.
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16-9
Calculate y for a bond with 15 years to maturity, a coupon rate of 7.5% and a price of 116% of $1,000 par
value.
P0 = 1.160 × $1,000 = $1,160
Coupon Payment = .075 × $1,000 = $75
$1,160 = $75(PVIFA y, 15) + $1,000(PVIF y,15)
y = 5.87%
An investor will only receive an annual effective yield of 5.87% if the bond is held to maturity and coupons can be reinvested at a 5.87% rate each year.
)Value(PVIFMaturity )IFAPayment(PVCoupon ,y,y 0 nnP
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16-10
Calculating Ex Post (Actual) Effective Annual Yield
1P
FV EAY
1
0
n
where:
FV = future value of the cash flows received at the end of the life of the bond
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16-11
ValueMaturity )IFAPayment(FVCoupon FV ,y n
where:
FVIFA= future value of the annuity factor
[ ((1+ y)n - 1)/ y)]
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16-12
What is the EAY for a bond with 15 years to maturity, a coupon rate of 7.5% , bought at a price of $1,160 and for which coupons where reinvested at a 5.87% rate?
$75(FVIFA 5.87%,15) + $1000
$75(23.047) + $1000 = $2,728.50
ValueMaturity )IFAPayment(FVCoupon FV ,y n
1 EAY1
0 n
PFV
($2,728.50/$1,160)(1/15) - 1
(2.35185)0.0067 - 1 = 0.0587 or 5.87%
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16-13
If all coupon payments are reinvested at the y rate and the bond is held until maturity, the EAY will be equal to the ex ante effective annual rate, y
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16-14
The Two Sides of Interest Rate Risk
Reinvestment Rate Risk
• The risk of interest rates falling and having to reinvest coupon payment at a lower rate than y, resulting in a lower ex-post yield.
Price or Market Value Risk
• The risk of rates rising and the market price of the bond falling if the bond must be sold prior to maturity, also resulting in a lower ex-post yield.
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16-15
Example of Reinvestment Rate RiskUse the previous example to calculate EAY, assuming rates have dropped to 5% over the
life of the bond after the bond was purchased.
FV of cash flows
$75(FVIFA 5%,15) + $1000
$75(21.579) + $1000 = $2,618.39
EAY = ($2,618.39/$1,160)(1/15) - 1
= (2.2573)0.0067 - 1 = 0.0558 or 5.58%
Because of reinvestment rate risk, the investor has a lower ex post yield than the expected y of 5.87%.
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16-16
Example of Market Price RiskUsing the previous example, calculate the EAY for
the bond if rates after 5 years rise to 6.5% and remain at that level for the remainder of the life of
the bond. Assume that the investor will sell the bond after 5 years.
Price of the bond in 5 years $75(PVIFA 6.5,10) + $1,000(PVIF 6.5,10 )
$75(7.188) +1000(0.532) = $1,071.89FV of cash flows
$75(FVIFA 5.87,5) + $1,071.89 $75(5.6225) + $1,071.89 = $1,493.58
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16-17
EAY
($1,493.58/$1,160)(1/5) - 1
(1.2876).2 - 1 =.0519 or 5.19%
Because of market price risk, the annual effective yield received ex post is 5.19% over 5 years, 68 basis points lower than the expected effective yield of 5.87%.
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16-18
Bond Theorems
Theorem I. Bond prices must move inversely to bond yields.
Theorem II. Holding the coupon rate constant, percentage changes in bond prices are greater the longer term to maturity of a bond.
Theorem III. The percentage price change for a bond will increase at a decreasing rate as N increases.
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16-19
Theorem IV. (Convexity Theorem) Starting with a given market yield y, holding other factors constant, the rise in price with a fall in y will be greater than the fall in price with the same absolute value rise in y.
Theorem V. Holding N constant and starting from the same y, the higher the coupon rate, the smaller the percentage change in price for a given change in yield.
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16-20
SummaryA Bond’s Interest Rate Risk Depends On
The current market yield (y) The coupon rate of the bond The maturity of the bond
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16-21
Duration Defined
It is the weighted average time over which the cash flows from an investment are expected.
Duration is a measure of the effective maturity.
• For most securities, the cash benefits are received before the maturity date.
Duration is also a measure of interest rate risk.
• Securities with higher durations will have a greater change in market value with a change in market rates.
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16-22
N
t
t
N
t
t
y
Cy
tC
Dur
1
1
)1(
)1(
where:
Ct = cash flow in time period t
t = the time each cash flow comes in
y = current market yield
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16-23
Find the duration for the following three bonds:Bond 1 has an 8% coupon rate, $1000 par, n=6
Bond 2 is a zero-coupon bond with a $1,000 par, n=6Bond 3 has a 6% coupon rate, $1,000 par, n=6
BOND 1: 8% COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO MATURITY
Year Cash Flow PVIF 8% PV Cash Flows t × PV Cash Flow
1 $80 .9259 $74.07 $74.072 80 .8573 68.58 137.163 80 .7938 63.50 190.504 80 .7350 58.80 235.205 80 .6806 54.45 272.256 1080 .6302 680.62 4,083.72
Sum $1000.00 4,992.90
Duration = $4,992.90/$1,000 = 4.993 years
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16-24
BOND 2: ZERO COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO MATURITY
Year Cash Flow PVIF 8% PV Cash Flows t × PV Cash Flow
1 $0 .9259 $0 $02 0 .8573 0 03 0 .7938 0 04 0 .7350 0 05 0 .6806 0 06 1000 .6302 630.20 3,781.20
Sum $630.20 $3,781.20
Duration = $3,781.20/$630.20 = 6 years
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16-25
BOND 3: 6% COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO MATURITY
Year Cash Flow PVIF 8% PV Cash Flows t × PV Cash Flow
1 $60 .9259 $55.55 $55.552 60 .8573 51.44 102.883 60 .7938 47.63 142.894 60 .7350 44.10 176.405 60 .6806 40.84 272.256 1060 .6302 668.01 4,008.07
Sum $907.57 4,689.97
Duration = $4,689.97/$907.57 = 5.17 years
Note that duration is a negative function of a bond’s coupon payment. The duration for zero coupon will always be equal to its maturity.
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16-26
Alternative Formula for Duration to Use with Long-term Securities
NyyNy
NDur ,PVIFA1)(P
PaymentCoupon
0
y
yn
1/(11PVIFA
where:
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16-27
For Bond 1, with N = 6 years, Coupon Payment = $80, y = 8%, P0 = $1,000 and
PVIFA factor for 8% and 6 years = 4.6229
Dur = 6 - [($80/($1000×.08)]×[6-(1.08)(4.6229)]
= 6 - (1) × (1.0073) = 4.993 years
The same as was calculated using the regular formula.
NyyNy
NDur ,PVIFA1)(P
PaymentCoupon
0
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16-28
Why is duration important for assessing an institution’s interest rate risk?
Because for a given change in market yields, percentage changes in asset prices are proportional to the assets’ duration.
)1(%%
0
00 y
yDurPPP
where:%P0 = percentage change in price%y = percentage change in market yield y = the original market rate prior to the change
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16-29
What is the percentage change in price for Bond 1 with a duration of 4.993, a P0 = $1,000, and y
= 8% if rates increase to 8.5%?
)1(%%
0
00 y
yDurPPP
- 4.99[0.50%/(1.08)] = - 2.31%
Multiplying (0.0231 × $1,000), the bond’s price would fall by approximately $23.10.
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16-30
Duration assumes a linear relationship between price changes and changes in market yields.
• Bond price changes follow a convex pattern with changes in market yields.
Duration will overestimate the percent fall in price with a rise in rates.
Duration wil1 underestimate the percent rise in price with a fall in rates.
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16-31
Estimating Interest Rate Elasticity
y 1yDur - E
Interest rate elasticity is the percentage price change expected for a 1% change in market yields.
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16-32
Calculate the interest rate elasticity for Bonds 1, 2, and 3.
y 1yDur - E
Bond 1: E = -4.993(0.08/1.08) = -0.3699
Bond 2: E = -6.000(0.08/1.08) = -0.4444
Bond 3: E = -5.170(0.08/1.08) = -0.3830
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16-33
For Bond 1, for every 1% change in the bond’s yield of 8% , the bondholder could have expected a price change of 0.3699%.
Bonds 2 and 3 have higher elasticity than Bond 1, which implies greater interest rate risk.
In this case, elasticities describe the same risk ranking as their relative duration, because each bond has the same market yield of 8%.
If market yields are different, calculating elasticities provides a better comparison of different securities’ relative interest rate risk.
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16-34
Modified Duration
y 1Dur (MD)Duration Modified
Multiplying MD by any y gives the approximate change in value of that fixed-rate asset or liability with a change in y.
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16-35
Limitations of Duration-Based Measures of Interest Rate Risk
Duration assumes that there will be a parallel shift in the yield curve so that for a given level of default risk, yields across the entire term structure change equally.
Duration assumes a flat yield curve because the same y discount rate is used for early and late cash flows.
As market rate (y ) changes, so will a security’s duration.
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16-36
Portfolio Immunization
It is a duration based strategy that makes a portfolio immune to interest rate risk over a given holding period.
Used to achieve a realized annual rate of return at the end of the holding period that is no less than the expected annual yield at the beginning of the period.
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16-37
Suppose an insurance company issues a $1 million guaranteed investment contract that guarantees an annual effective yield of 8%. The company is considering two alternatives:
1. Bond 1: six-year bonds with an 8% coupon rate at $1,000 par value which sell for $1,000 per bond, with y = 8%.
2. Bond 2: five-year bonds with an 8% coupon rate and $1,000 par value, which sell for $1,000 per bond, with y = 8%.
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16-38
Duration of Bond 1 = 4.993 (from earlier calculations)
Duration of Bond 2
NyyNy
NDur ,PVIFA1)(P
PaymentCoupon
0
5 - [(80/80)] - [5 - (1.08)(3.9927)]
5 - 0.68788 = 4.312
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16-39
ILLUSTRATION OF DURATION AND PORTFOLIO IMMUNIZATION.
STRATEGY 1: DURATION MATCH; BUY BOND 1 WITH APPROXIMATELY A
5-YEAR DUATION Coupon payments reinvested at y and bond sold at end of 5 years.Scenario (1) Market interest rates stay at y = 8% for 5 years
Ex post effective annual yield for Bond 1:FV of cash flows per bond at the end of year 5
80(FVIFA 5,8%) + (1080/1.08) = $1,469.33Ex post effective annual rate is (1469.33/1000)0.20
- 1 = .08 or 8%Scenario (2) Market interest rates stay at y = 9% for 5 years
Ex post effective annual yield for Bond 1:FV of cash flows per bond at the end of year 5
80(FVIFA 5,9%) + (1080/1.09) = $1,469.61Ex post effective annual rate is (1469.61/1000)0.20
- 1 = .08 or 8%Scenario (3) Market interest rates stay at y = 7% for 5 years
Ex post effective annual yield for Bond 1:FV of cash flows per bond at the end of year 5
80(FVIFA 5,7%) + (1080/1.07) = $1,469.41Ex post effective annual rate is (1469.41/1000)0.20
- 1 = .08 or 8%
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16-40
STRATEGY 2: MATURITY MATCH; BUY BOND 2 WITH A MATURITY OF 5 YEARS
Coupon payments reinvested at y and bond sold at end of 5 years.Scenario (1) Market interest rates stay at y = 8% for 5 years
Ex post effective annual yield for Bond 2:FV of cash flows per bond at the end of year 5
80(FVIFA 5,8%) + 1000 = $1,469.33Ex post effective annual rate is (1469.33/1000)0.20
- 1 = .08 or 8%Scenario (2) Market interest rates stay at y = 9% for 5 years
Ex post effective annual yield for Bond 2:FV of cash flows per bond at the end of year 5
80(FVIFA 5,9%) + 1000 = $1,478.78Ex post effective annual rate is (1478.78/1000)0.20
- 1 = .0814 or 8.14%Scenario (3) Market interest rates stay at y = 7% for 5 years
Ex post effective annual yield for Bond 2:FV of cash flows per bond at the end of year 5
80(FVIFA 5,7%) + 1000 = $1,460.06Ex post effective annual rate is (1460.06/1000)0.20
- 1 = .0.786 or 7.86%
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16-41
In this example, holding bonds with a duration of 5 years will lock in an annual return of 8% over the holding period.
Reasoning: With immunization, the expected return on a portfolio is protected from both reinvestment and market value risk.
• Reinvestment risk exactly offsets market value risk.
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16-42
Immunization Assumptions and Limitations
Assumes that there is only one parallel shift in the yield curve immediately after the beginning of the period.
• Market shifts may occur in the middle of the investment period and an investor may be less than perfectly immunized.
It is difficult to find an investment with a duration exceeding 10 years.
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16-43
Differences in Funding Gap and Duration
A financial institution’s funding gap serves as a measure of reinvestment risk for a given time period.
Funding gap ignores:• time value of money; and• the effects of changes in the market value of assets and
liabilities on a financial institution’s balance sheet. Duration gap provides a better overall measure of
interest rate risk including market price as well as reinvestment risk.
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16-44
Duration Gap
AssetssLiabilitieDurL -DurA DGAP
where:
DurA = duration of assets which is the weighted average of the duration of assets on the balance sheet
DurL = duration of liabilities which is the weighted average of the duration of liabilities on the balance sheet
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16-45
DGAP > 0 Increase Decrease > Decrease Fall
Decrease Increase > IncreaseRise
DGAP < 0 Increase Decrease <Decrease Rise
Decrease Increase < IncreaseFall
DGAP = 0 Increase Decrease =Decrease Zero
Decrease Decrease =Decrease Zero
Position
y
in Market Value
Assets Liabilities Equity
DURATION GAP AND CHANGES IN THE MARKET VALUE OF EQUITY
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16-46
Change in the Value of a Depository Institution’s Equity-to-Asset Ratio
)]y 1/(yDGAP[( Assets) tal(Equity/To Value
where:
y = average rate on assets
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16-47
Calculating Excelsior Bank’s DGAP
AssetsCash Dur = 0 $10,0006-month T-bills y = 6% Dur = .5 years $40,00010 year Fixed-Rate Loan Dur = ? $50,000
Total Assets $100,000
Liabilities and Equity
Transactions Deposit Dur = 0 $40,000(rate =2%)
1-year CD, rate = 7% Dur = 1 $50,000Equity $10,000
Total Liabilities and Equity $100,000
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16-48
Duration of a Loan
NyyNy
NDur ,PVIFA1)Amount(Loan
PaymentCoupon
Coupon Payment (Loan Payment)
Loan Amount/(PVIFA 10%, 10) = $50,000/6.1446 = $8,137.27 per year
Dur = 10 - [($8,137.27/($50,000×.10)] ×
[10 - (1.10)(6.1446)] = 10 - (1.6275)(3.241) =
4.73 years
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16-49
Duration of Assets
($10,000/$100,000)0 + ($40,000/$100,000)0.5 +($50,000/$100,000)4.73 = 2.565 years
Duration of Liabilities
($40,000/$90,000)0 + ($50,000/$90,000)1 = 0.555 years
2.565 yrs - ($90,000/$100,000)0.5556 yrs = 2.065 years
AssetssLiabilitieDurL -DurA DGAP
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16-50
Change in the Value of Equity from a 1% Rise in Rates
- 2.065[0.01/(1.074)] = - 0.0192 or -1.92%
)]y 1/(yDGAP[( Assets) tal(Equity/To Value
y = ($10,000/$100,000)0 + ($40,000/$100,000)0.06 + ($50,000/$100,000)0.10 = 0.074 or 7.4%