04/08/2023 1

BEC 3322 Financial

EconomicsSession 6

04/08/2023 2

Lecture Outline

• Interest Rate and Bond• Measuring Interest-Rate Risk• Duration and Gap Analysis – Bank– Nonbanking Financial Institution

04/08/2023 3

Interest Rate and Bond

• We saw that when interest rates change, a bond with a longer term to maturity has a larger change in its price and hence more interest rate risk than a bond with a shorter term to maturity (bond prices and interest rates are negatively related).

04/08/2023 4

Bond Price at Different interest Rate

• Time to 4% 6% 8% 10% 12%• Maturity• 1 year 1038 1029 1000 981

963• 10 years 1327 1148 1000 875

770 20 years 1547 1231 1000 828 699

• 30 years 1695 1277 1000 810 676

• 8% Coupon rate

04/08/2023 5

Cont.

• If you buy the bond at par with an 8% coupon rate, and market interest rate subsequently rise, then you suffer a loss. You tied up your money earning at 8% when alternative investments offer higher returns.

• This is reflected in a capital loss on the bond – (decrease in the market price of bond). This is why, T-Bills are considered as safest.

04/08/2023 6

Rate of Capital Gain

• Example 1• Calculate the rate of capital gain or

loss on a ten-year zero-coupon bond for which the interest rate has increased from 10% to 20%. The bond has a face value of $1,000.

• Prices and returns for long-term bonds are more volatile than those for shorter term bonds.

04/08/2023 7

Cont.

• G = (Pt+1 – Pt) / Pt

• Pt+1 = 1000/ (1+0.2)9 = 193.81

• Pt = 1000 / (1+0.1)10 = 385.54

• G = (193.81- 385.54) / 385.54 = 49.7 %

04/08/2023 8

Cont.

• Changes in interest rates make investments in long-term bonds quite risky

• The riskiness of an asset’s return that results from interest-rate changes has been given a special name, interest-rate risk

• Dealing with interest-rate risk is a major concern of managers of financial institutions and investors,

04/08/2023 9

Cont.

• There is no interest-rate risk for any bond whose time to maturity matches the holding period.

• The change in interest rates can then have no effect on the price at the end of the holding period for these bonds, and the return will therefore be equal to the yield to maturity known at the time the bond is purchased.

04/08/2023 10

Cont.• Bond is a good investment when the holding period and the

maturity of the bond are identical. Bonds whose term to• maturity is longer than the holding period are subject to

interest-rate risk.

• Changes in interest rates lead to capital gains and losses that produce substantial differences between the return and the yield to maturity known at the time the bond is purchased.

• Interest-rate risk is especially important for long-term bonds, where the capital gains and losses can be substantial. This is why long-term bonds are not considered to be safe assets with a sure return over short holding periods.

04/08/2023 11

Measuring Interest-RateRisk

• To calculate the duration or effective maturity on any debt security, Frederick M, a researcher at the National Bureau of Economic Research, invented the concept of duration.

• A zero-coupon bond makes no cash payments before the bond matures, it makes sense to define its effective maturity as equal to its actual term to maturity

04/08/2023 12

Duration of an Individual Security

• Duration is a weighted average of the maturities of the cash payments (see Table 1).

• Thus we have reached the flowing conclusions: All other factors being equal, the longer the term to maturity of a bond, the longer its duration.

• Example 2• Calculate the duration for an 11-year 10% coupon bond when

the interest rate is again 10%

• All else being equal, when interest rates rise, the duration of a coupon bond falls (see Table 3).

04/08/2023 13

Cont.

• Example 3• Consider a ten-year 20% coupon bond when the interest

rate is 10%.

• All else being equal, the higher the coupon rate on the bond, the shorter the bond’s duration.

• Example 4 • A manager of a financial institution is holding 25% of a

portfolio in a bond with a five year duration and 75% in a bond with a ten-year duration. What is the duration of the portfolio?

04/08/2023 14

Duration of a Portfolio

• The duration of a portfolio of securities is the weighted average of the durations of the individual securities, with the weights reflecting the proportion of the portfolio invested in each.

.

04/08/2023 15

Interest Rate Risk-Examples

• Example 5• A pension fund manager is holding a ten-year 10% coupon

bond in the fund’s portfolio and the interest rate is currently 10%. What loss would the fund be exposed to if the interest rate rises to 11% tomorrow?

• Example 6

• Now the pension manager has the option to hold a ten-year coupon bond with a coupon rate of 20% instead of 10%. As mentioned earlier, the duration for this 20% coupon bond is 5.98 years when the interest rate is 10%. Find the approximate change in the bond price when the interest rate increases from 10% to 11%.

04/08/2023 16

Cont.

• Examples 5 and 6 have led the pension fund manager to an important conclusion about the relationship of duration and interest-rate risk:

• The greater the duration of a security, the greater the percentage change in the market value of the security for a given change in interest rates. Therefore, the greater the duration of a security, the greater its interest-rate risk.

04/08/2023 17

Cont.

• Duration is a useful concept because it provides a good approximation of the sensitivity of a security’s market value to a change in its interest rate

04/08/2023 18

Gap Analysis

• With the increased volatility of interest rates that occurred in the 1980s, banks and other financial institutions became more concerned about their exposure to interest rate risk, the riskiness of earnings and returns that is associated with changes in interest rates.

04/08/2023 19

Cont.

• The sensitivity of bank profits to changes in interest rates can be measured more directly using gap analysis, in which the amount of rate-sensitive liabilities is subtracted from the amount of rate-sensitive assets.

04/08/2023 20

Cont.

• Gap = the amount of rate-sensitive assets - the amount of rate-sensitive liabilities (Example 7)

• If a bank has more rate-sensitive liabilities than assets, a rise in interest rates will reduce bank profits and a decline in interest rates will raise bank profits.

04/08/2023 21

Duration Analysis for Banks

• An alternative method for measuring interest-rate risk, called duration analysis, examines the sensitivity of the market value of the bank’s total assets and liabilities to changes in interest rates.

• Duration analysis and gap analysis are thus useful tools for telling a manager of a financial institution its degree of exposure to interest-rate risk

04/08/2023 22

Cont.

• Duration analysis involves using the average (weighted) duration of a financial institution’s assets and of its liabilities to see how its net worth responds to a change in interest rates.

04/08/2023 23

Examples

• Example 8• The bank manager wants to know what

happens when interest rates rise from 10% to 11%. The total asset value is $100 million, and the total liability value is $95 million.

• The result is that the net worth of the bank would decline by $1.6 million ($2.5 million - $0.9 million = $1.6 million).

04/08/2023 24

Cont.

• Example 9• Based on the information provided in

Example 8, determine the duration gap for First National Bank and calculate the change in the market value of net worth as a percentage of assets as a result of change in interest rates.

• Example 10• If interest rates fall from 10% to 5%, what

will happen to the net worth of the bank

04/08/2023 25

Conclusion of Duration Gap Analysis

• Duration gap analysis indicate that the Bank will suffer from a rise in interest rates. In this example, we have seen that a rise in interest rates from 10% to 11% will cause the market value of net worth to fall by $1.6 million, which is one-third the initial amount of bank capital. Thus the bank manager realizes that the bank faces substantial interest-rate risk because a rise in interest rates could cause it to lose a lot of its capital. Clearly, this analysis is useful tools for understanding a manager of the financial institution to degree of exposure to interest rate risk.

04/08/2023 26

Nonbanking Financial Institution and Duration Gap Analysis

• Some financial institutions have and duration gaps that is opposite in sign to those of banks, so that when interest rates rise, net worth rise rather than fall.

04/08/2023 27

Cont.

• In the finance company, the manager finds that company’s income will rise by 120,000 when interest rates rise by 1%.

• The reason that the company has benefited from the interest-rate rise, in contrast to the Bank, whose profits suffer from the rise in interest rates. On the other hand, the Friendly Finance Company has a positive income gap because it has more rate-sensitive assets than liabilities.

04/08/2023 28

Cont.

• Even though the income and duration gap analysis indicates that the Friendly Finance Company gains from a rise in interest rates, the manager realizes that if interest rates go in the other direction, the company will suffer a fall in income and market value of net worth.

• Thus the finance company manager, like the bank manager, realizes that the institution is subject to substantial interest-rate risk.

04/08/2023 29

Conclusion

• Duration gap analysis, examines the sensitivity of the market value of the financial institution’s net worth to changes in interest rates

04/08/2023 30

Strategies for Managing Interest-Rate Risk

• Assume that you are a manager of the First National Bank. You have done the duration and income gap analysis for this bank, and you must decide which alternative strategies to pursue under the following conditions

• interest rates will fall in the future

• interest rates will rise in the future

04/08/2023 31

Question 1

• Suppose that you are the manager of a bank that has $15 million of fixed-rate assets, $30 million of rate sensitive assets, $25 million of fixed-rate liabilities, and $20 million of rate-sensitive liabilities. Conduct a gap analysis for the bank, and show what will happen to bank profits if interest rates rise by 5 percentage points. What actions could you take to reduce the bank’s interest-rate risk?

04/08/2023 32

Question 2

• Suppose that you are the manager of a bank whose $100 billion of assets have an average duration of four years and whose $90 billion of liabilities have an average duration of six years. Conduct a duration analysis for the bank, and show what will happen to the net worth of the bank if interest rates rise by 2 percentage points. What actions could you take to reduce the bank’s interest-rate risk ?

04/08/2023 33

Thank you