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