Fixed Income portfolio management:- quantifying & measuring interest rate risk

Finance 30233, Fall 2010 S. Mann

Interest rate risk measures:DurationConvexityPVBP

Interest Rate Risk Management

3% 4% 5% 6% 7% 8% 9%

10%

11%

12%

123

57

1020

30

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

yield maturity

zero-coupon price: B(0,T)

Zero-coupon bond prices for various yields & maturities

U.S. T-Notes issued 8/15/97- prices & yields through 9/1/99

Issues: 6% of 8/15/00 & 6.125% of 8/15/07 (source: Dow Jones )

80

85

90

95

100

105

110

115

3.5%

4.5%

5.5%

6.5%

7.5%

8.5%

9.5%

07 Bid 00 Bid 07 yield 00 yield

prices

yields

Price versus yield: U.S. T-Note - 6.125% of 8/15/07daily observations 7/30/97 through 9/1/99 (source: Dow Jones )

95

100

105

110

115

4.5% 4.7% 4.9% 5.1% 5.3% 5.5% 5.7% 5.9% 6.1% 6.3% 6.5%

yield-to-maturity

Pri

ce

Duration

n

n

jj

jc y

Face

y

cyB

)1()1()(

1

Bond price (Bc) as a function of yield (y):

Small change in y, y, changes bond price by how much?Classical duration weights each cash flow by the time until receipt,

then divides by the bond price:

)(/))1()1(

(1

yBy

Facen

y

cjD cn

n

jj

jc

Define DM = Dc /(1+y) (annual coupon) = Dc /(1+y/2) (semi-annual coupon)

( modified duration) approximate % change in Price:

P/P = - DM x y

Modified Duration

example: DM = 4.5 y= + 30 bpexpected % price change= -4.5 (.0030) = -1.35%

linear approximation. Convexity matters.

Modified duration

Percentage change in bond price:

yDy

yD

yB

yBMc

c

c

)(1

)1()(

)(

)(

Change in bond price:

yDBB Mcc )(

Modified Duration (DM): DM = Dc/(1+y) (annual coupon)

DM = Dc/(1+y/2) (semiannual coupon)

Duration is linear approximation

yield-to-maturity 8.13%coupon 8%year (T) coupon principal PV T x PV

1 8 0 7.399 7.3985022 8 100 92.370 184.7402

99.769 192.1387

192.138799.769

1.9261.0813

= 1.926Classical Duration (Dc)=

Modified Duration (DM) = = 1.781

Duration for an annual coupon bond

yield-to-maturity (semi-annual) 6.00%coupon 8%year (T) coupon principal PV T x PV

0.5 4 0 3.883 1.94171 4 0 3.770 3.7704

1.5 4 0 3.661 5.49082 4 100 92.403 184.8053

103.7171 196.0083

196.0083103.717

1.8901.0300

= 1.890Classical Duration (Dc)=

Modified Duration (DM) = = 1.835

Duration for a semi-annual coupon bond

yield-to-maturity (semi-annual) 10.00%coupon 8%year (j) coupon principal PV j x PV

0.5 4 0 3.810 1.9047621 4 0 3.628 3.628118

1.5 4 0 3.455 5.1830262 4 100 85.561 171.1221

96.4540 181.838

181.83896.454

1.8851.050

= 1.885Classical Duration (Dc)=

Modified Duration (DM) = = 1.795

Duration for a semi-annual coupon bond

Example: portfolio value = $100,000; DM = 4.62PVBP = (4.62) x 100,000 x .0001 = $46.20

Exercise: estimate value of portfolio above if yieldcurve rises by 25 bp (in parallel shift).

Food for thought: what about non-parallel shifts?

Price Value of Basis Point (PVBP)

PVBP = DM x Value x .0001

1 2 3 4 5 6 7 8 9 10

-75.00

-25.00

25.00

75.00

125.00

175.00

yield

maturity

Actual vs duration-predicted value of $100 invested in zero-coupon purchased at 6% yield

Predicted % price change using duration: P/P = -Dm y

Duration is FIRST derivative of bond price.(slope of curve)

convexity is SECOND derivative of bond price(curvature: change in slope)

Using duration AND convexity, we can estimatebond percentage price change as:

P/P = - Dmy + (1/2) Convexity (y)2

(a 2nd order Taylor series expansion)(the convexity adjustment is always POSITIVE)(We will not hand-calculate convexity)

Convexity: adjusting for non-linearity

example: 30 year, 8% coupon bond with y-t-m of 8%.Modified duration = 11.26, Convexity = 212.4

What is predicted % price change for increase of yield to 10%?

Duration prediction:P/P = - Dmy = -11.26 x 2.0% = -22.52%

Duration & convexity prediction:P/P = - Dmy + (1/2) Convexity (y)2

= -11.26 x 2.0% + (1/2) 212.4 (.02)2 = -22.52% + 4.25% = -18.27%

Actual % price change: price at 8% yield = 100; price at 10% yield = 81.15. % change = -18.85%

Example using Convexity

Asset-Liability Interest Rate Rrisk ManagementExample: The BillyBob Bank

Simplified balance sheet risk analysis:

Amount Duration PVBPAssets $100 mm 6.0 100,000,000 x 6.0 x 0.0001 = $60,000

Liabilities 90 mm 2.0 90,000,000 x 2.0 x 0.0001 = 18,000

Equity 10 mm ??? PVBP(E) = PVBP(A) – PVBP(L) = 60,000 – 18,000 = $42,000

Q: What is effective duration of equity?

PVBP(E) = DE x VE x 0.0001

$42,000 = DE x ($10,000,000) x 0.0001

DE = $42,000/$1000 = 42.0

The BillyBob Bank, continued

Simplified balance sheet risk analysis:

Amount Duration PVBPAssets $100 mm 6.0 100,000,000 x 6.0 x 0.0001 = $60,000Liabilities 90 mm 2.0 90,000,000 x 2.0 x 0.0001 = 18,000Equity 10 mm 42.0 PVBP(E) = PVBP(A) – PVBP(L) = 60,000 – 18,000 = $42,000

Assume that the bank has minimum capital requirements of 8% of assets(bank equity must be at least 8% of assets)

Q: What is the largest increase in rates that the bank can survive with the current asset/liability mix?

A: Set 8% = E / A = ($10mm - $42,000 y) / (100mm – 60,000 y)and solve for y:

0.08 (100mm – 60,000 y ) = 10mm - 42,000 y $8 mm – 4800 y = 10mm - 42,000 y (42,000 – 4800) y = $2,000,000 y = $2,000,000/$37,200 = 53.76 basis points