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BUFN 722. ch-9 Interest Rate Risk – part 2 Duration, convexity, etc. Overview. This chapter discusses a market value-based model for assessing and managing interest rate risk: Duration Computation of duration Economic interpretation Immunization using duration

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bufn 722

BUFN 722

ch-9

Interest Rate Risk – part 2

Duration, convexity, etc.

BUFN722- Financial Institutions

overview
Overview
  • This chapter discusses a market value-based model for assessing and managing interest rate risk:
    • Duration
    • Computation of duration
    • Economic interpretation
    • Immunization using duration
    • * Problems in applying duration

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price sensitivity and maturity
Price Sensitivity and Maturity
  • In general, the longer the term to maturity, the greater the sensitivity to interest rate changes.
  • Example: Suppose the zero coupon yield curve is flat at 12%. Bond A pays $1762.34 in five years. Bond B pays $3105.85 in ten years, and both are currently priced at $1000.

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example continued
Example continued...
    • 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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coupon effect
Coupon Effect
  • Bonds with identical maturities will respond differently to interest rate changes when the coupons differ. This is more readily understood by recognizing that coupon bonds consist of 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. Consequently, less sensitive to changes in R.

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price sensitivity of 6 coupon bond
Price Sensitivity of 6% Coupon Bond

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price sensitivity of 8 coupon bond
Price Sensitivity of 8% Coupon Bond

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remarks on preceding slides
Remarks on Preceding Slides
  • The 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 shows greater changes in price in response to a 2% change than the 8% bond. The first bond is has greater interest rate risk.

BUFN722- Financial Institutions

duration
Duration
  • Duration
    • Weighted average time to maturity using the relative present values of the cash flows as weights.
    • Combines the effects of differences in coupon rates and differences in maturity.
    • Based on elasticity of bond price with respect to interest rate.

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duration1
Duration
  • Duration

D = Snt=1[Ct• t/(1+r)t]/ Snt=1 [Ct/(1+r)t]

Where

D = duration

t = number of periods in the future

Ct = cash flow to be delivered in t periods

n= term-to-maturity & r = yield to maturity (per period basis).

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duration2
Duration
  • Since the price of the bond must equal the present value of all its cash flows, we can state the duration formula another way:

D = Snt=1[t  (Present Value of Ct/Price)]

  • Notice that the weights correspond to the relative present values of the cash flows.

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duration of zero coupon bond
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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computing duration
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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duration of 2 year 8 bond face value 1 000 ytm 12
Duration of 2-year, 8% bond: Face value = $1,000, YTM = 12%

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special case
Special Case
  • Maturity of a consol: M = .
  • Duration of a consol: D = 1 + 1/R

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duration gap
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 = 1.885 - 2.0 = -0.115
    • 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
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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economic interpretation
Economic Interpretation
  • Duration is a measure of interest rate sensitivity or elasticity of a liability or asset:

[dP/P]  [dR/(1+R)] = -D

Or equivalently,

dP/P = -D[dR/(1+R)] = -MD × dR

where MD is modified duration.

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economic interpretation1
Economic Interpretation
  • To estimate the change in price, we can rewrite this as:

dP = -D[dR/(1+R)]P = -(MD) × (dR) × (P)

  • Note the direct linear relationship between dP and -D.

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semi annual coupon payments
Semi-annual Coupon Payments
  • With semi-annual coupon payments:

(dP/P)/(dR/R) = -D[dR/(1+(R/2)]

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an example
An example:
  • Consider three loan plans, all of which have maturities of 2 years. The loan amount is $1,000 and the current interest rate is 3%. Loan #1, is an installment loan with two equal payments of $522.61. Loan #2 is a discount loan, which has a single payment of $1,060.90. Loan #3 is structured as a 3% annual coupon bond.

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duration as index of interest rate risk
Duration as Index of Interest Rate Risk

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immunizing the balance sheet of an fi
Immunizing theBalance 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
    • DE = -[DA - DLk]A(DR/(1+R))

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duration and immunizing
Duration and Immunizing
  • The formula shows 3 effects:
    • Leverage adjusted D-Gap
    • The size of the FI
    • The size of the interest rate shock

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an example1
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.

  • Methods of immunizing balance sheet.
    • Adjust DA , DL or k.

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immunization and regulatory concerns
Immunization and Regulatory Concerns
  • Regulators set target ratios for a bank’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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limitations of duration
*Limitations of 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.
  • Large interest rate change effects not accurately captured.
    • Convexity
  • More complex if nonparallel shift in yield curve.

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convexity
*Convexity
  • The duration measure is a linear approximation of a non-linear function. 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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convexity1
*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
*Modified duration
  • 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
*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
*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
*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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duration gap analysis
Duration Gap Analysis

%ΔP - DUR x Δi/(1+i)

i 5%, from 10% to 15% 

ΔAsset Value = %ΔP x Assets

= -2.7 x .05/(1+.10) x $100m

= -$12.3m

ΔLiability Value = %ΔP x Liabilities

= -1.03 x .05/(1+.10) x $95m

= -$4.5m

ΔNW = -$12.3m - (-$4.5m) = -$7.8m

DURgap = DURa - [L/A x DURl]

= 2.7 - [(95/100) x 1.03]

= 1.72

%ΔNW = - DURgap x Δi/(1+i)

= - 1.72 x .05/(1+.10)

= -.078 = -7.8%

ΔNW = -.078 x $100m

= -$7.8m

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example of finance company
Example of Finance Company

Friendly Finance Company

Assets Liabilities

----------------------------------------------------------------------------------------------

Cash and Deposits $ 3 m | Commercial Paper $ 40 m

|

Securities | Bank Loans

less than 1 year $ 5 m | less than 1 year $ 3 m

1 to 2 year $ 1 m | 1 to 2 year $ 2 m

greater than 2 year $ 1 m | greater than 2 year $ 5 m

|

Consumer Loans | Long-Term Bonds

less than 1 year $ 50 m | and other long-term

1 to 2 year $ 20 m | debt $ 40 m

greater than 2 year $ 15 m |

| Capital $ 10 m

Physical capital $ 5 m |

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gap and duration analysis
Gap and Duration Analysis
  • If i 5%
  • Gap Analysis
  • GAP = RSA - RSL = $55 m - $43 m = $12 million
  • ΔIncome = GAP x Δi = $12 m x 5% = $0.6 million
  • Duration Gap Analysis
  • DURgap = DURa - [L/A x DURl]
  • = 1.16 - [90/100 x 2.77] = -1.33 years
  • %ΔNW = - DURgap X Δi /(1+i)
  • = -(-1.33) x .05/(1+.10)
  • = .061 = 6.1%

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managing interest rate risk
Managing Interest-Rate Risk
  • Problems with GAP Analysis
    • 1. Assumes slope of yield curve unchanged and flat
    • 2. Manager estimates % of fixed rate assets and liabilities that are rate sensitive

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managing interest rate risk1
Managing Interest-Rate Risk
  • Strategies for Managing Interest-Rate Risk
    • In example above, shorten duration of bank assets or lengthen duration of bank liabilities
    • To completely immunize net worth from interest-rate risk, set DURgap = 0

1. Reduce DURa = 0.98 DURgap = 0.98 - [(95/100) x 1.03] = 0

2. Raise DURl = 2.80 DURgap = 2.7 - [(95/100) x 2.80] = 0

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duration a measure of interest rate sensitivity
Duration: A Measure of Interest Rate Sensitivity

The weighted-average time to maturity on an

Investment, using the relative values of the cash flows as weights

N N

 CFt  tPVt  t

t = 1(1 + R)tt = 1

D = N = N

CFt PVt

t = 1 (1 + R)t t = 1

This measure is known as Macaulay’s Duration

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key facts about duration
Key facts about Duration
  • Everything else equal,
    • 1. When the maturity of a bond lengthens, the duration rises as well.
    • 2. When interest rates rise, the duration of a coupon bond falls
    • 3. The higher is the coupon rate on the bond, the shorter is the duration of the bond.
    • 4. Duration is additive: the duration of a portfolio of securities is the weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each.

BUFN722- Financial Institutions

example of duration calculation
Example of Duration Calculation

Table 3

1 CFt CFt X t Percent of Initial

t CFt (1 + 4%)2t (1 + 4%)2t (1 + 4%)2t Investment Recovered

48.08

46.23

44.45

42.74

41.10

39.52

38.00

767.22

1,067.34

.5

1

1.5

2

2.5

3

3.5

4

50

50

50

50

50

50

50

1,050

0.9615

0.9246

0.8890

0.8548

0.8219

0.7903

0.7599

0.7307

24.04/1,067.34 = 0.02

46.23/1,067.34 = 0.04

66.67/1,067.34 = 0.06

85.48/1,067.34 = 0.08

102.75/1,067.34 = 0.10

118.56/1,067.34 = 0.11

133.00/1,067.34 = 0.13

3,068.88/1,067.34 = 2.88

24.04

46.23

66.67

85.48

102.75

118.56

133.00

3,068.88

3,645.61

3,645.61

1,067.34

D =

= 3.42 years

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duration and interest rate risk
Duration and Interest-Rate Risk

%ΔP - DUR x ΔR/(1+R)

Modified Duration = Duration / (1 + R)

R 10% to 11% (.10 to .11) => D = + .01:

Table 4 -10% coupon bond

%ΔP = 6.76 x .01/(1+.10)

= -.0615 = -6.15%.

Actual decline = 6.23% (need to add correction for convexity)

The duration measure is a less accurate measure of price sensitivity the larger the change in interest rates

20% required return on 10% coupon bond, DUR = 5.72 years

%ΔP = - 5.72 x .01/(1+.10)

= -.0520 = -5.20%

  • The greater is the duration of a security, the greater is the percentage change in the market value of the security for a given change in interest rates. Therefore, the greater is the duration of a security, the greater is its interest-rate risk.
features of the duration measure
Features of the Duration Measure
  • Duration and Coupon Interest
    • the higher the coupon payment, the lower its duration
  • Duration and Yield to Maturity
    • duration decreases as yield to maturity increases
  • Duration and Maturity
    • Duration increases with the maturity of a bond but at a decreasing rate

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economic meaning of duration
Economic Meaning of Duration
  • Measure of the average life of a bond
  • Measure of a bond’s interest rate sensitivity (elasticity)

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slide49
Interest Rate Risk &Value of Cash Flows
  • I. Interest Rate Risk: Duration Measures
    • A. Interest Rate Risk of Zero-Coupon Bonds
      • Value Function — the plot of a bond’s price vs. interest rate
      • Dollar duration (DD) is a measure of a bond’s absolute price sensitivity to interest rate changes:
  • Modified duration (MD) measures a bond’s relative (%) change in price due to an interest rate change.

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slide50
Changes in price or percentage can be approximated by:
      • Modified duration is the maturity of the Zero divided by one plus the spot rate.
    • B. Interest Rate Risk of Coupon Bonds and Other Fixed Cash Flows
      • Modified duration of a complex instrument is determined by summing the weighted modified durations of each of its cash flows. The weights are based upon the present value of the cash flow divided by the present value of the whole instrument.
  • .

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slide51
Two bonds of the same modified duration, but different cash flows, may exhibit different relative price changes for large interest rate changes. This is due to differences in the curvature of their price / yield relation, or convexity.
    • Relationships between bond value and duration are shown in Exhibits 6, 7, 8 on the next 2 slides
  • C. Callable Bonds and Other Interest-Sensitive Cash Flow Streams

Securities that have cash flows which are sensitive (in amount or timing) to interest rate levels should use effective duration (ED) and effective dollar duration (EDD).

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exhibit 6 duration versus yield 8 5 coupon
Exhibit 6Duration versus yield (8.5% coupon)

Exhibit 7Modified duration versus coupon (8.5% yield to maturity)

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exhibit 8 modified duration versus term to maturity 8 5 to yield to maturity
Exhibit 8Modified duration versus term to maturity (8.5% to yield to maturity)

Exhibit.12 Gains from convexity versus modified duration

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slide54
II. Interest Rate Risk: Convexity
      • The relationship between price and interest rate is nonlinear. Because of this, duration measures of price sensitivity are inaccurate for large interest rate movements.
      • Gain from convexity attempts to rectify this situation.
    • A. Measuring Convexity
      • To calculate the gain from convexity we need the current price and the prices associated with 50 b.p. upward and downward movements of the interest rate.
  • Dollar Gain From Convexity (DGFC)
  • Gain From Convexity in Percentage Terms (GFC)

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convexity2
Convexity
  • Convexity is calculated with the following formula
  • N
  • å t* ( t + 1) *(CFt)
  • t=1( 1 + R)t
  • 2 ( 1 + R) 2(P)
  • This is used as a correction for the duration method to calculate price changes
  • Convexity is very important consideration in the bond market - approximate calculation:
  • P- + P+ - 2P0
  • 2 ( P0 ) (D R)2

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convexity formula
Convexity formula

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slide57
B. Putting It All Together

General Formulas Incorporating Both Duration and Convexity

  • III. Limits to Interest Rate Risk Measures
      • This Chapter’s methods are prone to error when the value curve is asymmetrical.
  • I
      • Changes in the yield curve may not involve parallel shifts.
      • The possibility of default is not addressed.
  • IV. Summary
      • The concepts of duration and convexity were developed to aid in the measurement of the sensitivity of a security’s value to changes in the interest rate.
      • While not perfect, these tools have proven to be useful and are widely employed.

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exhibit 10 interest rate sensitivity
Exhibit .10 Interest rate sensitivity

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exhibit 11a dollar gain from convexity
Exhibit 11A Dollar gain from convexity

Exhibit 11B DGFC for two different changes in interest rates

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another duration problem
Another Duration problem
  • example: $100 par, 6 % coupon, 5 years (10 semi-annual payments of 3% each six months) in an 8 % market ( 4 % semi-annual yield)
  • per. (t) CFPV(1) 1/(1.04)tPV of CF t x PV of CF
  • 1 $3.0 .9615 2.8845 2.8845
  • 2 3.0 .9246 2.7738 5.5476
  • 3 3.0 .8890 2.667 8.001
  • 4 3.0 .8548 2.5664 10.2576
  • 5 3.0 .8219 2.4657 12.3285
  • . . . . . . . . . . . .
  • 10 103.0 .6756 69.5868 695.868

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slide61

example - estimate both the duration and convexity of a 6-year bond paying 5 percent coupon annually and the annual yield to maturity is 6 % 6-year Coupon Bond Par value =$1,000 Coupon =0.05 YTM =0.06 Maturity =6

slide62
CX = the capital loss from a one-basis-point increase in rates + the capital gain from a one-basis-point decrease in rates
  • Scaling factor = 108

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duration problem
Duration problem
  • Duration of this security = 801.5775 / 91.8927 = 8.7229726 half years

= 4.36 years

  • so the five year 6% coupon security has a duration of 4.36 years in a market where expected yield to maturity = 8%
  • find duration if R = 4 %
  • is it the same as when R = 8 % ? (hint: no!)
  • also find Modified Duration = duration / (1 + R )

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pertinent websites
Pertinent Websites

Securities Exchange Commission www.sec.gov

BUFN722- Financial Institutions

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