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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

ch-9

Interest Rate Risk – part 2

Duration, convexity, etc.

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• 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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• 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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• 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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• 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 8% Coupon Bond

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• 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.

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• 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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• 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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• 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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• 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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• 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%

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• Maturity of a consol: M = .

• Duration of a consol: D = 1 + 1/R

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• 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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• 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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• 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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• 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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• With semi-annual coupon payments:

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

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• 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

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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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• The formula shows 3 effects:

• The size of the FI

• The size of the interest rate shock

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• 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

• 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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• 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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• 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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• 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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• 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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• 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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• 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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%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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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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• 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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• 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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• 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

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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• 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.

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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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Calculating Duration, R =10% 10-yr 10% annual Coupon Bond

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Calculating Duration, R = 20% 10-yr 10% annual Coupon Bond

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%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.

• 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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• Measure of the average life of a bond

• Measure of a bond’s interest rate sensitivity (elasticity)

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• 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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• 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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• 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).

• BUFN722- Financial Institutions

Exhibit 6 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 Duration versus yield (8.5% coupon)

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

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

Exhibit.12 Gains from convexity versus modified duration

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• II. Interest Rate Risk: Convexity 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

• 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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Convexity 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 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 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

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• B. Putting It All Together 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

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 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 Interest rate sensitivity

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Exhibit 11A 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 Dollar gain from convexity

Exhibit 11B DGFC for two different changes in interest rates

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Another Duration problem 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

• 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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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

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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BUFN722- Financial Institutions rates + the capital gain from a one-basis-point decrease in rates

Duration problem rates + the capital gain from a one-basis-point decrease in rates

• 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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BUFN722- Financial Institutions rates + the capital gain from a one-basis-point decrease in rates

Pertinent Websites rates + the capital gain from a one-basis-point decrease in rates

Securities Exchange Commission www.sec.gov

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