Tunis Business School - Fall 2024

Interest Rate Risk Management

Eymen Errais, PhD, FRM

Definition

Interest rate risk is the risk of earnings or capital losses arising from
movement of interest rates.
 Impact of adverse interest rate change

• Increase borrowing costs for borrowers

• Reduce returns for investors
• Reduce profitability of financial services providers such as banks
• Reduce the net present value of organisations due to the effect of
changes in the discount rate on the value of financial instruments, hedges
and the return on projects.
 Types of Interest Rate Risk

• Repricing Risk: is generated by the differences between maturing or

repricing assets and liabilities

• Yield Curve Risk: is generated by the differences in rate changes

between two items with different maturities or repricing intervals.

• Basis Risk: it is generated by the variation in the spread of the indices

upon which certain financial securities are based.

• Option Risk: is generated by embedded options triggered by a change

in interest rate ( callable bonds, puttable bonds, residential mortgages.)

Interest Rate Risk

 Basic Concepts

● Bond Price: Clean vs Dirty

● Yield to Maturity.

● Effect of interest rate change on fixed income securities.

● Interest rate term structure ( Yield Curve ).

 Bond Price : Clean (Flat) VS Dirty (Invoice)

▪ Clean (flat) bond prices assume that the next coupon payment is in precisely
one payment period, either a year for an annual payment bond or 6 months for a
semiannual payment bond.

▪ Dirty (invoice) bond prices are the clean bond prices with the accrued interest
on top. It’s the actual price that a buyer pays for a bond if he buys the bond in
between coupon payments.
 Yield to maturity

▪ The yield to maturity is the single interest rate that equates the present value of a
security’s cash flows.
▪ The yield to maturity is often interpreted as an estimate of the average rate of return to an
investor who purchases a bond and holds it until maturity with the assumption that the
coupons will be reinvested at the yield.
▪ The Yield to maturity is affected by the same factors that affect the bond price. In fact, the
Bond price and its yield to maturity are effectively two sides of the same coin.
 Factors Affecting Yield

● Time to maturity: the longer the time to maturity, the higher the yield

● Coupons : the bigger the coupons, the higher the yield

● Inflation: the higher the inflation, the higher the yield

● Bond features: if the bond has unfavorable features for the investor,
the yield will be higher (example: callable bond )

● Credit Risk: the higher the credit risk, the higher the yield
 Effect of interest rate change on fixed income securities.

▪ A fundamental principle of bond investing is that market interest rates and bond prices
generally move in opposite directions. When market interest rates rise, prices of fixed-rate
bonds fall and yield to maturity falls.
Constructing yield curves
 Term Structure

● Zero-coupon yield curves depict the interest rates of similar quality

bonds at different maturities.

● Term Structure estimation is based on an assumed functional

relationship between yields on the one hand and maturities on the
other.
 Shapes of Term Structure

● Upward sloping: long term yields are higher than short term yields.

● Downward sloping: short term yields are higher than long term yields.
signifies that the economy is in, or about to enter, a recessive period

● Flat: very little variation between short and long term yields. Signals
that the market is unsure about the future direction of the economy.
 Exact method: Bootstrapping

▪ Bootstrapping is a method for constructing a (zero-coupon) fixed

income yield curve from the prices of a set of coupon-bearing
products
▪ The general methodology is as follows:

1. Define the set of yielding products - these will generally be coupon-bearing

bonds

2. Derive discount factors for the corresponding terms - these are the internal
rates of return of the bonds

3. Bootstrap' the zero-coupon curve, successively calibrating this curve such

that it returns the prices of the inputs.
Bootstrapping - Linear interpolation

R(t ) =
(Tb − t )R(Ta ) + (t − Ta )R(Tb )
(Tb − Ta )

Example: Ta = 1; Tb = 2; t = 1.25; R(Ta) = 3% et R(Tb) = 4%

R(t ) =
(2 − 1.25)3% + (1.25 − 1)4% = 3.25%
(2 − 1)
 Bootstrapping: Limitations

● You need to have market data at each maturity date

● The bootstrapping method does not perform optimization, it computes

zero-coupon yields that exactly fit the bond prices.

● It will result in over-fitting since bond prices often contain idiosyncratic

errors due to lack of liquidity, bid-ask spreads, special tax effects.

● The term structure will not be necessarily smooth.

 Parametric models

● Function-based models: Specification of a single-piece function that

is defined over the entire maturity domain.

● Model parameters are determined through the minimisation of the

squared deviations of theoretical prices from observed prices.
Spline-based models: Smoothing spline

● Fit the yield curve by relying on a piecewise polynomial: the spline

function. This method is inspired from Principal Component Analysis

R(t ) = at 3 + bt 2 + ct + d

 R(t1 ) = at13 + bt12 + ct1 + d


 R(t ) = at 2 + bt 2 + ct 2 + d
3 2


 R (t ) = at 3
3
+ bt 3 + ct3 + d
2


 R (t ) = at 4
3
+ bt 4 + ct 4 + d
2
 Parametric models: Nelson Siegel Model

● Fitting for a point in time t a discount function to bond price data by

assuming explicitly the following function:
1 − exp (−  ) 1 − exp (−  ) 
R( ) =  0 + 1   +   − exp (−  )
 
2
   
s −t
=

▪ R: Zero coupon rate
▪  0 Long term interest rate
▪  1 Slope factor
▪  2 Convexity factor
▪  Scaling factor
 Parametric models: Svensson Model
● Improved fit and curves flexibility and extra precision . The Nelson and
Siegel model is extended by adding two more parameters to the function
1 − exp (−  ) 1 − exp (−  )  1 − exp (−  ') 
R( ) =  0 + 1   +   − exp (−  ) +   − exp (−  ' )
  '
2 3

s −t
=
 ▪ R: Zero coupon rate
s −t
 '= ▪  0 Long term interest rate
'
▪  1 Slope factor
▪  2 Convexity factor
▪  1st scaling factor
▪  ' 2nd scaling factor
Managing Interest Rate Risk
 Opening question

Question: I would like to invest my money in a bond, but I am

worried that bond yields may change in the future. Should I invest
in a short or long maturity bond to reduce the effects of changes in
yield on the value of my bond?
What about coupons? Should I choose a bond with large coupons
or small coupons?

Answer: We need a way to compute the sensitivity of bond prices to

yield

Page 21
 Opening question

Page 22
 Duration

▪ Maturity time is related to the sensitivity of the bond price to its

yield.

▪ When there are coupons, maturity time does not exactly

correspond to sensitivity.

▪ We will define something called duration to try to “generalize” the

idea that for a zero coupon bond the time to maturity captures its
sensitivity to yield.

Page 23 Duration
 Macaulay Duration

Idea: Coupons mess things up... so, why don’t I try to take them into account:
C
F
C C C C 
0 t1 t2 tn
Definition: Macaulay Duration:

PV (t1 )t1 + PV (t 2 )t 2 + ... + PV (t n )t n

D=
PVtotal
PV (ti )
= wt1 t1 + wt2 t2 + ... + wtn tn where w t = i
PVtotal
PV(ti) is the present value of the cash flow at time tk.
n
PVtotal =  PV (t ) is the present value of the entire cash flow.
i =0
i

Page 24
 Macaulay Duration

Macaulay duration is a weighted average of times. It is quoted in

years.
T
D=  t w
t =1
t

wt = CF t (1 + y )
t
Price
CFt = Cash Flow for period t

In practice, when people refer to duration, they often mean

Macaulay duration.
Page 25
 Macaulay Duration and Maturity

Zero-Coupon bond:

Macaulay Duration = Maturity Date

Coupon bond:

Macaulay Duration < Maturity Date

Page 26
 Rules of Duration

▪ Rule 1 The duration of a zero-coupon bond equals its time to maturity

▪ Rule 2 Holding maturity constant, a bond’s duration is higher when the coupon rate
is lower

▪ Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with
its time to maturity

▪ Rule 4 Holding other factors constant, the duration of a coupon bond is higher when
the bond’s yield to maturity is lower

▪ Rules 5 The duration of a level perpetuity is equal to: (1+y) / y

Page 27
 Bonds Duration vs Bonds maturity

Page 28
 Macaulay Duration Formula

C/m
F
C/m C/m C/m C/m 
0 t1 t2 tn
The Macaulay duration for a bond with:
coupon rate/year = C
yield =
periods per year = m
periods remaining = n

1+ 
1+ m
+ n( mc − m

)
D= m
−
 c[(1 + m ) − 1] + 
 n

Page 29
 Modified Duration

Why not calculate the sensitivity of a bond to the yield exactly?

Definition: Let P be the price of a bond. Then the

modified duration Dm of the bond is

1 dP ( )
Dm = −
P (0 ) d  = 0

Hints: Roughly, you should think Sensitivity = Derivative.

The minus sign is so it won’t be a negative quantity.
Dividing by P makes it a relative sensitivity.

Page 30
 Modified Duration

Modified duration determines the percentage change in the price of a bond given a
change in yield:

1 dP ( ) 1 P
Dm = − −
P (0 ) d  = 0
P 

If the yield increases by 1%, what percentage change will occur in a bond price with
modified duration of 5?

Answer, the price will decrease by 5%!

Page 31
 Modified Duration

How does modified duration relate changes in yield to absolute changes in prices?

1 dP ( ) 1 P
Dm = − −
P (0 ) d  = 0
P 

Rearranging gives:
P  − Dm P

If the yield increases by 1%, how much will the price of a bond with current price
$100 and modified duration of 5 change by?

P  − Dm P = −5($100)(0.01) = −5
Page 32
 Relationship between modified and Macaulay duration

D
Dm =
 
 1 + 
 m 

Where  is the yield, and m is the number of compounding

periods per year.

(i) They differ only by a constant factor

(ii) In the case of continuous compounding
Dm=D.
Page 33
 Relationship between modified and Macaulay duration

Assume: m periods per year

 is the yield

m periods
1 year

k periods
Time in years
Single cash flow:

ck dPVk −k  1
PVk = = PVk 
 d  m  1 +  
k
  
1 + 
 m   m 

Page 34
 Relationship between modified and Macaulay duration

Assume: m periods per year

 is the yield

Single cash flow:

ck dPVk −k  1
PVk = = PVk 
 d  m  1 +  
k
  
1 + 
 m   m
n

 PV
Cash flow stream:
P= k Looks like Macaulay Duration
k =1
k 
n   PVk
−1
dP  m −1 n
k 
= − =    PVk = DP = − Dm P
d       m  
k =1
1 +  1 +  k =1 1 + 
 m  m  m 
Page 35
 Example

A 10%, 30 year bond whose price is $100 has Macaulay

duration D = 9.94.

(a) What is the modified duration of this bond? A 1% increase in yield will cause approximately what percent change in
price?

Given a price of $100, we can compute the yield

as =10%.

1 1
Dm = D= (9.94) = 9.47
  1.05
1 + 
 2
Hence, a 1% increase causes a 9.47% drop in price.

Page 36
 Example

A 10%, 30 year bond whose price is $100 has Macaulay

duration D = 9.94.

(b) Estimate the change in price if the yield moves to 11%.

P  − Dm P
= −9.47(100)(0.01) = −9.47

Hence,
P  $90.53

Page 37
 Duration of a Portfolio

Given fixed income securities with prices Pi and duration Di, i=1...m.
The portfolio consisting of the aggregate of these has price P and
duration D given by

P = P1 + P2 + ... + Pm
D = w1 D1 + w2 D2 + ... + wm Dm
Pi
where
wi = , i = 1,..., m
P
Formula works for both Macaulay and all forms of Modified duration.

Note: This looks just like the Macaulay duration formula except with time replaced by
duration!

Page 38
 Duration of a Portfolio

Assume we have two cash flow streams.

n
A

0 n
D A
= 
k =0
t k PVkA
PA

n
B
0 n
D B
= 
k =0
t k PVkB
PB

Page 39
 Duration of a Portfolio

What is the duration of A+B.

n
D A+ B
= 
k =0
t k ( PVkA + PVkB )
P A +PB
n n
= 
k =0
t k PVkA
P A +PB
+
k =0
t k PVkB
P A +PB

( )+  ( )
n n
=  t k PVkA t k PVkB
A
P PB
P A
P +PB
A
P B
P +PB
A
k =0 k =0

= ( PA
P A +PB
)D A
+ ( PB
P A +PB
)D B

Page 40
 Convexity

▪The relationship between bond

prices and yields is not linear

▪Duration rule is a good

approximation for only small
changes in bond yields

1 n
 CFt 
Convexity =
P  (1 + y ) 2
 
t =1  (1 + y )
t
(t 2
+ t ) 


Page 41 Convexity & Duration

What is the real idea behind immunization: Taylor Expansions!

0 is the current value of the yield

We can expand the price as a function of the yield around the current value.

P( ) = P(0 ) + P(0 )( − 0 ) + 1

2 P(0 )( − 0 ) 2 + ...

Price Convexity
Duration (unnormalized)
(unnormalized)

You can match as many terms as you like to make the Taylor series of your
portfolio look like the Taylor series of your cash flow stream.

Page 42
Non Parallel Shifts
Non- parallel shifts in the yield curve and approaches to account for
them

➢ One of the main assumptions we have made when deriving duration is that
the yield curve is flat and the shifts in the yield curve are parallel.

➢ However, in reality, two portfolios that have the same duration can perform
quite differently if the yield curve does not shift in the parallel manner

➢ The yield curve reshaping duration approach focuses on the sensitivity of a

portfolio to a change in the slope of the yield curve

➢ Key rate duration approach determines the sensitivity of a portfolio to a

change in ‘key’ yield rate.

Page 44
Yield curve slopes

Page 45
Parallel vs. Non-parallel shifts

Not all portfolios with the same Duration (=Investment horizon) will behave the same if the
shifts in the yield curve are not parallel
Common non-parallel shifts in the yield curve:
Upward shifts Downward shifts

Page 46
Yield curve shifts and bond portfolios

➢ Different types of yield curve shifts will have different impact on different
types of bond portfolios

▪ Bullet
▪ Barbell
▪ Ladder

➢ Even if they have the same duration, their convexities will be different so
they will behave differently

Page 47
 Types of bond portfolios: Bullet vs. Barbell Portfolio

Bullet (Focused) portfolio: Maturities/durations of bonds in the bullet portfolio are centered
around one point of the yield curve (e.g. the investment horizon):

Inv. horizon

Barbell portfolio: Portfolio is a combination of bonds concentrated at two extreme maturities:

Inv. horizon

Page 48
Flattening of the yield curve and barbell vs. bullet

➢ If the yield on a short-term bond rises more than

the yield on a bullet portfolio (medium term bond)
and yield on a long term bond rises less than a
yield on a bullet portfolio, we have flattening of the
yield curve.

➢ For this kind of shift of the yield curve, barbell

portfolio will always outperform the bullet one.

Page 49
Flattening of the yield curve and barbell vs. bullet

➢ The yield on a short-term bond in the barbell

portfolio can rise less than the yield on a bullet
portfolio and the yield on a long-term bond in
the barbell portfolio can rise more that the yield
of a bullet portfolio.
➢ In this case, the yield curve steepens, and
bullet portfolio will outperform barbell portfolio
▪ True for most realistic yield changes, but for
changes in yields of 3% or more, it can be vice
versa!
➢ It is best to perform scenario analysis to
assess which portfolio will perform better

Page 50
Implications for portfolio Immunization

➢ The losses in this case of steepening would be substantially higher for the
barbell portfolio for two reasons:
1. The lower investment rate experienced by the barbell portfolio on the
short maturities end
2. The capital loss on the barbell portfolio on the long end of maturities
would also be higher
▪ longer maturity bonds will have to be sold once the end of investment
horizon is reached - but since the longer interest rates have risen in
this example, their prices will considerably fall

Page 51
Implications for Immunization

➢ For single-period immunization, a bullet maturity structure with tight cash

flows around the liability (horizon) date generally is preferred to a ladder or
barbell portfolio because of the reduced risk exposure to the yield curve
becoming steeper or twisting

➢ To eliminate the risk of shifts in yields, the investors could purchase a zero-
coupon bond to cash-flow-match the single-period liability.

➢ In absence of zero-coupon bonds maturity-matching investment horizon, a

bullet structure is less risky, and the barbell is the most risky

Page 52
Takeaways
 Important Concepts

▪ Types of interest rate risks

▪ Bond Price
▪ YTM and the factors affecting it
▪ Term Structure & Boostrapping
▪ Macaulay Duration
▪ Modified Duration
▪ Convexity

Page 54
 Important Formulas

PV (t1 )t1 + PV (t 2 )t 2 + ... + PV (t n )t n

D=
PVtotal

1 dP ( )
Dm = −
P (0 ) d  = 0

1 n
 CFt 
Convexity =
P  (1 + y ) 2
  (1 + y) t
(t 2 + t )
t =1  

Page 55