FIS Ch 3: Basics of Interest Rate Management

Question 1

Define Duration

Answer 1

The duration of a security represents the sensitivity of its price, P, to a small parallel
shift in the level of interest rates, dr :
Duration DP
1
P
dP
dr

Question 2

Using duration to approximate change in portfolio value from changes in the interest
rate

Answer 2

Given a duration DP of a security with price P, the change in portfolio value due to a
small change in rates is approximated by:
dP DP P dr

This approximation is only valid when dr is small!

Question 3

Duration of a Zero-Coupon Bond

Answer 3

The price of a zero coupon bond is:
Pzpr , t, Tq er pTtq
Thus, the duration of the zero-coupon bond is:
Dz,T
1
Pzpr , t, Tq

dPzpr , t, Tq
dr


1
Pzpr , t, Tq
rpT tq Pzpr , t, Tqs
T t X

Question 4

Duration of a Portfolio of Bonds

Answer 4

The duration of a portfolio of n securities is given by:
DW
¸n
i1
wiDi

wi = fraction of portfolio invested in security i

Di = duration of security i

Question 5

Duration of a Coupon Bond

Answer 5

The value of the coupon bond is:
Pcp0, Tnq
n¸1
i1
c
2 Pzp0, Ti q 􀀀

1 􀀀
c
2

Pzp0, Tnq
By interpreting the coupon bond as a weighted average of zero-coupon bonds and
noting that the duration of a zero-coupon maturing at Ti is simply Ti , we obtain the
duration of the coupon bond as:
Dc
¸n
i1
wiTi

The weights the coupon bond has on each zero-coupon bond is:
wi
c{2 Pzp0, Ti q
Pcp0, Tnq
for i 1, .., n 1
wn
p1 􀀀 c{2q Pzp0, Tnq
Pcp0, Tnq

Question 6

Warnings for Duration Interpretation

Answer 6

Duration is often also defined as the average time of payments

However, this interpretation is only correct for fixed rate bonds

For bonds that don’t have fixed payments, such as floating-rate bonds and inverse
floaters, this interpretation is incorrect

Question 7

Duration of Floating Rate Bond

Answer 7

The duration of the floating rate bond, denoted by DFR, is:
DFR
1
PFRpt, Tq
dPFR
dr

1
PFRpt, Tq

dZpt, Ti􀀀1q
dr

100

1 􀀀
r2pTi q
2


1
PFRpt, Tq
rpTi􀀀1 tqs Zpt, Ti􀀀1q 100

1 􀀀
r2pTi q
2

Ti􀀀1 t X

The duration of a floating rate bond is simply equal to the time left to the next
coupon date, Ti􀀀1 t

Question 8

Properties of Duration

Answer 8

1. The higher the coupon rate c, the lower the duration
   Ÿ The higher the coupon rate, the larger are cashflows in the near future compared to
   the long-term future
   Ÿ Cashflows that arrive sooner are less sensitive to changes in interest rates
2. The higher the interest rate, the lower the duration
   Ÿ Higher interest rates implies that short-term cash flows have a relatively higher
   weight in the value of the bond, and thus a lower sensitivity to changes in interest
   rates

Question 9

Define Dollar Duration

Answer 9

The dollar duration, denoted by D$, of a security P is defined by:
D$
P
dP
dr
The relation between duration and dollar duration is:
D$
P P DP

Question 10

Define VaR

Answer 10

Let be a percentile (i.e. 5%) and T be a time horizon (i.e. 1 year)

The p100 q%T Value-at-Risk of a portfolio P is the maximum loss the portfolio
can suffer over the horizon T with a p100 q% probability

If we let LT be a random variable denoting the loss of a portfolio over the horizon
T, then VaR is the number such that:
PpLT ¡ VaRq %

Question 11

List two methods for calculating VaR

Answer 11

1. Historical distribution approach
2. Normal distribution approach

Question 12

Historical distribution approach for calculating VaR

Answer 12

This method generates a historical distribution of changes in the portfolio value
dP based on historical data of changes in the level of interest rates, dr

From the histogram of portfolio profits/losses, we can compute the VaR by ranking
the portfolio P&L realizations from worst to best, and then pick the % worst case

Question 13

Normal distribution approach for calculating VaR

Answer 13

Assume dr is distributed as Np, q

Under the assumption that dr is small, we can approximate the change in the
bond portfolio value as: dP DP P dr

Thus, dP also has a normal distribution with mean and standard deviation given
by:
P DP P
P DP P

We can then calculate the p1 q% VaR of portfolio loss by calculating the
corresponding z-score of the standard normal distribution for

For example, the 95% VaR of portfolio loss is given by:
95% VaR pP 1.645 Pq

Question 14

Describe Some Potential Issues with Using VaR

Answer 14

VaR is a statistical measure of risk, so it depends on the distributional
assumptions and the sample data used

The duration approximation for dP, given by dP DP P dr , is only
appropriate for small parallel changes in the level of interest rates
Ÿ However, VaR focuses on large changes in interest rates

VaR measures the maximum loss with p1 q% probability (i.e. 95% probability),
but does not quantify the tails of the loss distribution

The VaR formula based on the normal distribution approach includes the
expected change in the portfolio value, given by P DP P Erdr s
Ÿ However, the computation of Erdr s is imprecise and has large standard errors
Ÿ This can lead to large errors in the VaR computation
Ÿ This, it is often more accurate to consider only the unexpected VaR, which focuses
on the 95% loss relative to the expected P&L, P
Ÿ This is equivalent to setting P to 0 in the VaR calculation

Question 15

Describe Expected Shortfall

Answer 15

Expected shortfall ErLT |LT ¡ VaRs

The expected shortfall measures what the expected loss is, conditional on the
loss exceeding the VaR loss threshold

It is better than VaR at capturing the tail of the loss distribution

Question 16

Describe two common methods of interest rate risk management

Answer 16

1. Cash Flow Matching: The bank attempts to replicate its liability cashflows exactly
2. Immunization: The bank purchases a portfolio of securities with the same
   present value and duration as the liability
   Ÿ This method is preferred over cash flow matching because the bank can choose
   bonds that have favorable properties in terms of liquidity and transaction costs

Question 17

Asset Liability Management (ALM)

Answer 17

The aim of asset-liability management is to minimize the impact of a change in
interest rates on the equity of the firm/bank
The value of equity is total assets (A) minus total liability (Lq:
E A L
It then follows that the duration of the equity is given by:
DE
A
A L DA
L
A L DL

To remove sensitivity from interest rates, we need: D$
A ADA LDL D$
L

One can eliminate a duration mismatch by using derivative securities, such as
swaps