“If Max gets to Heaven, he won’t last long.
He will be chucked out for trying to pull off a
merger between Heaven and Hell…after
having secured a controlling interest in key
subsidiary companies in both places, of
course.”
H.G. Wells
Saunders & Cornett, Financial
Institutions Management, 4th ed.
1
The Impact of Unanticipated
Changes in Interest Rates:
• On Profitability
– Net Interest Income (NII) = Interest Income minus
Interest Expense
– Interest rate risk of NII is measured by the repricing
model. (chap. 8)
• On Market Value of Equity
– Market Value of Equity = Market Value of Assets minus
Market Value of Debt
– Interest rate risk of equity MV is measured by the
duration model. (chap. 9)
Saunders & Cornett, Financial
Institutions Management, 4th ed.
2
The Repricing Model
• Rate Sensitive Assets (Liabilities) RSA/RSL: are
repriced within a period of time called a maturity
bucket.
– Repricing occurs whenever either maturity or a roll date
is reached.
– The roll date is the reset date specified in floating rate
instruments that determines the new market benchmark
rate used to set cash flows (eg., coupon payments).
• The Federal Reserve set the following 6 maturity
buckets: 1 day; 1day-3 months; 3-6 months; 6-12
months; 1-5 yrs; > 5 yrs.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
3
The Repricing Model
• Repricing Gap (GAP) = RSA – RSL
• R = interest rate shock
• NII = GAP x R for each maturity
bucket i
• Cumulative Gap (CGAP) = i GAPi
• NII = CGAP x Ri where Ri is the
average interest rate change on RSA & RSL
• Gap Ratio = CGAP/Assets
Saunders & Cornett, Financial
Institutions Management, 4th ed.
4
Example of Repricing Model
Assets
Short term (1 yr fixed rate)
consumer loans
Long term (2 yrs fixed rate)
consumer loans
3 mo. T-bills
6 mo. T-bills
3 yr. T-bonds
10 yr, fixed rate mortgages
30 yr. floating rate mortgages (9
mo. adjustment period)
TOTAL
$m
50
Liabilities & Net Worth
Equity Capital (fixed)
$m
20
25
Demand Deposits
40
30
35
70
20
40
Passbook savings
3 mo. CDs
3 mo. bankers acceptances
6 mo. commercial paper
1 yr. time deposits
2 yr. time deposits
TOTAL
30
40
20
60
20
40
270
270
Saunders & Cornett, Financial
Institutions Management, 4th ed.
5
Repricing Ex. (cont.)
•
•
•
•
•
•
•
•
•
•
1 day GAP = 0 – 0 = 0 (DD & passbook excluded)
(1day-3mo] GAP = 30 – (40+20) = -$30m
(3mo-6mo] GAP = 35 – 60 = -$25m
(6mo-12mo] GAP = (50+40) - 20 = $70m
(1yr-5yr] GAP = (25+70) – 40 = $55m
>5 yr GAP = 20 – (20+40+30) = -$70m
1 yr CGAP = 0-30-25+70 = $15m
1 yr Gap Ratio = 15/270 = 5.6%
5 yr CGAP = 0-30-25+70+55 = $70m
5 yr Gap Ratio = 70/270 = 25.9%
Saunders & Cornett, Financial
Institutions Management, 4th ed.
6
Assume an across the board 1%
increase in interest rates
•
•
•
•
•
•
•
•
1 day NII = 0(.01) = 0
(1day-3mo] NII = -$30m(.01) = -$300,000
(3mo-6mo] NII = -$25m(.01) = -$250,000
(6mo-12mo] NII = $70m(.01) = $700,000
(1yr-5yr] NII = $55m(.01) = $550,000
>5 yr GAP = -$70m(.01) = -$700,000
1 yr CNII = $15m(.01) = $150,000
5yr CNII = $70m(.01) = $700,000
Saunders & Cornett, Financial
Institutions Management, 4th ed.
7
Unequal Shifts in Interest Rates
• NII = (RSA x RRSA) – (RSL x RRSL)
• Even if GAP=0 (RSA=RSL) unequal shifts
in interest rates can cause NII.
• Must compare relative size of RSA and RSL
(GAPs) to relative size of interest rate
shocks (RRSA- RRSL = spread).
• The spread can be positive or negative =
Basis Risk.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
8
Strengths of Repricing Model
• Simplicity
• Low data input requirements
• Used by smaller banks to get an estimate of
cash flow effects of interest rate shocks.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
9
Weaknesses of the
Repricing Model
• Ignores market value effects.
• Overaggregation within maturity buckets
• Runoffs – even fixed rate instruments pay off
principal and interest cash flows which must be
reinvested at market rates. Must estimate cash
flows received or paid out during the maturity
bucket period. But assumes that runoffs are
independent of the level of interest rates. Not true
for mortgage prepayments.
• Ignores cash flows from off-balance sheet items.
Usually are marked to market.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
10
Measuring the Impact of Unanticipated
Interest Rate Shocks on Market Values
• E = A - L
• What determines price sensitivity to changes in
interest rates?
• The longer the time to maturity, the greater the
price impact of any given interest rate shock.
• This can be viewed in the positively sloped yield
curve. See Appendix 8A.
• But, yield curves must be drawn using pure
discount yields.
• The correct statement is: The longer the
DURATION, the greater the price impact of any
given interest rate shock.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
11
What is Duration?
• Duration is the weighted-average time to maturity on
an investment.
• Duration is the investment’s interest elasticity measures the change in price for any given change in
interest rates.
• Duration (D) equals time to maturity (M) for pure
discount instruments only.
• Duration of Floating Rate Instrument = time to first
roll date.
• For all other instruments, D < M
• Duration decreases as:
– Coupon payments increase
– Time to maturity decreases
– Yields increase.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
12
The Spreadsheet Method of
Calculating Duration
Ex. 1:5 yr. 10% p.a. coupon par value
s
Cs
1
2
3
4
5
y
100
100
100
100
1100
0.1
0.1
0.1
0.1
0.1
Price=
PV(Cs)
90.90909
82.64463
75.13148
68.30135
683.0135
1000
Duration
Saunders & Cornett, Financial
Institutions Management, 4th ed.
tPV(Cs)
90.90909
165.2893
225.3944
273.2054
3415.067
4169.865
4.169865
13
Ex. 2: Interest Rates Decrease to 9% p.a.
s
Cs
1
2
3
4
5
y
100
100
100
100
1100
0.09
0.09
0.09
0.09
0.09
Price=
PV(Cs)
91.74312
84.168
77.21835
70.84252
714.9245
1038.897
Duration
Saunders & Cornett, Financial
Institutions Management, 4th ed.
tPV(Cs)
91.74312
168.336
231.655
283.3701
3574.623
4349.727
4.186872
14
Ex. 3: Interest Rates Increase to 11% p.a.
s
Cs
1
2
3
4
5
y
100
100
100
100
1100
0.11
0.11
0.11
0.11
0.11
Price=
PV(Cs)
90.09009
81.16224
73.11914
65.8731
652.7965
963.041
Duration
Saunders & Cornett, Financial
Institutions Management, 4th ed.
tPV(Cs)
90.09009
162.3245
219.3574
263.4924
3263.982
3999.247
4.152727
15
The Duration Model
• Modified Duration = MD = D/(1+R)
• Price sensitivity (interest elasticity):
P  -D(P)R/(1+R)
• Consider a 1% increase in interest rates:
• Ex. 1: P  -(4.17)(1000)(.01)/1.10 = -$37.91
- New Price = 1000 - 37.91 = $962.09 Exact $963.04
• Ex. 2: P  -(4.19)(1038.897)(.01)/1.09 = -$39.94
– New Price = 1038.897 – 39.94 = $998.96 Exact $1000
• Ex. 3: P  -(4.15)(963.041)(.01)/1.11 = -$36.01
– New Price = 963.041 – 36.01 = $927.03 Exact $927.90
Saunders & Cornett, Financial
Institutions Management, 4th ed.
16
The Duration Model:
Using Duration to Measure the FI’s
Interest Rate Risk Exposure
•
•
•
•
•
•
•
E = A - L
A = -(DAA)RA/(1+RA)
L = -(DLL)RL/(1+RL)
Assume that RA/(1+RA) = RL/(1+RL)
E/A  -DG(R)/(1+R) where
DG = DA – (L/A)DL
DA = i=A wiDi DL = j=L wjDj
Saunders & Cornett, Financial
Institutions Management, 4th ed.
17
Consider a 2% increase in all
interest rates (ie, R/(1+R) = .02)
•
•
•
•
•
•
•
FI with DG = +5 yrs.
FI with DG = +2 yrs.
FI with DG = +0.5 yrs
FI with DG = 0
FI with DG = -0.5 yrs
FI with DG = - 2 yrs
FI with DG = - 5 yrs
E/A  -10%
E/A  -4%
E/A  -1%
E/A  0% Immunization
E/A  +1%
E/A  +4%
E/A  +10%
Saunders & Cornett, Financial
Institutions Management, 4th ed.
18
Convexity
•
•
•
•
Second order approximation
Measures curvature in the price/yield relationship.
More precise than duration’s linear approximation.
Duration is a pessimistic approximator
– Overstates price declines and understates price
increases.
– Convexity adjustment is always positive.
– Long term bonds have more convexity than short term
bonds. Zero coupon less convex than coupon bonds of
same duration.
P  -D(P)(R)/(1+R) + .5(P)(CX)(R)2
Saunders & Cornett, Financial
Institutions Management, 4th ed.
19
The Spreadsheet Method to
Calculate Convexity Ex. 1
s
Cs
1
2
3
4
5
y
100
100
100
100
1100
0.1
0.1
0.1
0.1
0.1
Price=
PV(Cs)
90.90909
82.64463
75.13148
68.30135
683.0135
1000
Duration
tPV(Cs)
90.90909
165.2893
225.3944
273.2054
3415.067
4169.865
4.169865
Saunders & Cornett, Financial
Institutions Management, 4th ed.
t(t+1)PV(Cs)
181.8182
495.8678
901.5778
1366.027
20490.4
23435.69
19.36834 CX
20
Ex. 2
s
Cs
1
2
3
4
5
y
100
100
100
100
1100
0.09
0.09
0.09
0.09
0.09
Price=
PV(Cs)
91.74312
84.168
77.21835
70.84252
714.9245
1038.897
Duration
tPV(Cs)
91.74312
168.336
231.655
283.3701
3574.623
4349.727
4.186872
Saunders & Cornett, Financial
Institutions Management, 4th ed.
t(t+1)PV(Cs)
183.4862
505.008
926.6202
1416.85
21447.74
24479.7
19.83265 CX
21
Ex. 3
s
Cs
1
2
3
4
5
y
100
100
100
100
1100
0.11
0.11
0.11
0.11
0.11
Price=
PV(Cs)
90.09009
81.16224
73.11914
65.8731
652.7965
963.041
Duration
tPV(Cs)
90.09009
162.3245
219.3574
263.4924
3263.982
3999.247
4.152727
Saunders & Cornett, Financial
Institutions Management, 4th ed.
t(t+1)PV(Cs)
180.1802
486.9735
877.4297
1317.462
19583.89
22445.94
18.91677 CX
22
How Do We Forecast Interest
Rate Shocks?
• Expectation Hypothesis
– Upward (downward) sloping yield curve
forecasts increasing (decreasing) interest rates.
– (1+0R2)2 = (1+0R1)(1+1R1)
– Spot rates: 0R2= 5.5% p.a. 0R1=4%
Implied forward rate: 1R1 = 7.02% p.a.
Forecasts 3.02% increase in 1 yr rates in 1 yr.
• Liquidity Premium Hypothesis
• Market Segmentation Hypothesis
Saunders & Cornett, Financial
Institutions Management, 4th ed.
23
Appendix 8A Calculating the Forward Zero
Yield Curve for Valuation
• Three steps:
– Decompose current spot yield curve on riskfree (US Treasury) coupon bearing instruments
into zero coupon spot risk-free yield curve.
– Calculate one year forward risk-free yield
curve.
– Add on fixed credit spreads for each maturity
and for each credit rating.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
24
Step 1: Calculation of the Spot Zero Coupon Riskfree Yield Curve Using a No Arbitrage Method
• Figure 6.6 shows spot yield curve for coupon
bearing US Treasury securities.
• Assuming par value coupon securities:
Six Month Zero: 100 = C+F = C+F = 100+(5.322/2)
1+0r1 1+0z1
1 + (.05322/2)
Therefore, the six month zero riskfree rate is: 0z1 = 5.322 percent per annum
One Year Zero: 100 = C + C+F = C + C+F
1+0r2 (1+0r2)2 1+0z1 (1+0z2)2
100 =
(5.511/2) + 100+(5.511/2) = (5.511/2) + 100+(5.511/2)
1+(.05511/2) (1+.05511/2)2 1+(.05322/2) (1+.055136/2)2
• Figure 6.7 shows the zero coupon spot yield curve.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
25
Figure 6.7
Figure 6.6
Yield to
Maturity p.a.
Yield to
Maturity
p.a.
CYCRF
7.6006%
ZYCRF
6.47%
6.25%
6.09%
6.2755%
5.98%
6.1079%
5.511%
5.9353%
CYCRF
5.322%
0.0647%
6.25%
5.5136%
6 1 1.5 2 2.5 3
Mos. Yr. Yr. Yr. Yr. Yr.
5.98%
6.09%
Maturity
5.511%
5.322%
Figure 6.8
6 Mos
Yield to
Maturity p.a.
1 Yr 1.5 Yrs 2 Yrs 2.5 Yrs 3 Yrs
Maturity
FYCR 1 Year Forw ard
FYCRF 1 Year Forw ard
14.8551%
14.3551%
ZYCRF
7.4475%
7.1264%
7.6006%
6.9475%
7.2813%
6.6264%
6.7813%
5.9353%
6.1079%
6.2755%
5.5136%
5.322%
6 Mos
1 Yr 1.5 Yrs 2 Yrs 2.5 Yrs 3 Yrs
Saunders & Cornett, Financial
Institutions Management, 4th ed.
Maturity
26
Step 2: Calculating the Forward Yields
• Use the expectations hypothesis to calculate
6 month maturity forward yields:
(1 + 0z2)2 = (1 + 0z1)(1 + 1z1)
(1+(.055136/2)2 = (1+.05322/2)(1+1z1)
Therefore, the rate for six months forward delivery of 6-month maturity US Treasury
securities is expected to be: 1z1 = 5.7054 percent p.a.
(1 + 0z3)3 = (1 + 0z2)2(1 + 2z1)
(1+(.059961/2)3 = (1+.055136/2)2(1+2z1)
Therefore, the rate for one year forward delivery of 6-month maturity US Treasury
securities is expected to be: 2z1 = 6.9645 percent p.a.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
27
Use the 6 month maturity forward yields to calculate
the 1 year forward risk-free yield curve
Figure 6.8
(1 + 2z2)2 = (1 + 2z1)(1 + 3z1)
Therefore, the rate for 1 year maturity US Treasury securities to be delivered in 1 year is:
2z2 = 6.703 percent p.a.
(1 + 2z3)3 = (1 + 2z1)(1 + 3z1)(1 + 4z1)
Therefore, the rate for 18-month maturity US Treasury securities to be delivered in 1 year
is: 2z3 = 6.7148 percent p.a.
(1 + 2z4)4 = (1 + 2z1)(1 + 3z1)(1 + 4z1)(1 + 5z1)
Therefore, the rate for 2 year maturity US Treasury securities to be delivered in 1 year is:
2z4 = 6.7135 percent p.a.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
28
Step 3: Add on Credit Spreads to Obtain the
Risky 1 Year Forward Zero Yield Curve
• Add on credit spreads (eg., from Bridge Information
Systems) to obtain FYCR in Figure 6.8.
Table 6.8 - Credit Spreads For Aaa Bonds
Maturity (in years, compounded annually)
2
3
5
10
15
20
Credit Spread, si
0.007071
0.008660
0.011180
0.015811
0.019365
0.022361
Saunders & Cornett, Financial
Institutions Management, 4th ed.
29
Calculating Duration if the
Yield Curve is not Flat
Ex. 1 with upward sloping yield curve
s
Cs
1
2
3
4
5
y
100
100
100
100
1100
0.1
0.102
0.107
0.115
0.12
Price=
PV(Cs)
90.90909
82.34492
73.71522
64.69944
624.1695
935.8382
Duration
tPV(Cs)
90.90909
164.6898
221.1456
258.7978
3120.848
3856.39
4.120787
Saunders & Cornett, Financial
Institutions Management, 4th ed.
t(t+1)PV(Cs)
181.8182
494.0695
884.5826
1293.989
18725.09
21579.55
19.05707 CX
30
The Barbell Strategy
• Convexity of Zero Coupon Securities: CX =
T(T+1)/(1+R)2
• Strategy 1: Invest in 15 yr zero coupon with 8% pa
yield. D=15, CX = 15(16)/1.082=206
• Strategy 2: Invest 50% in overnite FF D=0, CX =0
and 50% in 30 yr zero coupon with 8% yield
D=30, CX = 30(31)/1.082 = 797 Portfolio CX =
.5(0) + .5(797) = 398.5 > 206 Invest in Strategy 2.
But the cost of Strategy 2>Stategy 1 if CX priced.
Saunders & Cornett, Financial
Institutions Management, 4th ed.
31