DURATION LECTURE (INTEREST RATE RISK)

Interest Rate Risk: The risk that investors, fund managers, companies, banks, borrowers, lenders, etc. are exposed to.  Risk of an adverse interest rate movement.

There are two types of Interest Rate Risk::

1. CAPITAL OR PRICE RISK.  The risk that interest rate movements will adversely affect your position, or portfolio.

Example: You are holding long term (20-30 year) bonds.  If interest rates rise, the value of your portfolio will fall.  Inverse relationship between Interest Rates and Bond Prices.  Capital risk is usually associated with LONG TERM investments in Fixed Income securities, and the risk is usually that interest rates will RISE (Bond Prices will fall).  Worried? Int rates rise, bond prices fall, exposing investor to a Capital Loss (capital risk).

General Rule: The longer the term to maturity, the greater the Capital/Price risk.

Example: One year zero coupon bond (A) vs. 20 year zero coupon bond (B), Assume interest rates are intiially 10% and then go up to 20% or down to 5%.  Solve for bond prices, and compare percentage price changes of  Bonds A and B.

Int = 10%         PA = \$909               PB = \$148

Assume interest rates rise from 10% to 20%:

Int = 20%         PA = \$833               PB = \$26

% CHG            Bond  A:  -8.4%       Bond B:  -82%

Assume interest rates fall from 10% to 5%:

Int = 5%               Pa =   ?               Pb =     ?

% CHG             Bond A:+               Bond B:+

The Price of the 20 year bond is more sensitive to int rate changes, more price (capital) risk than one year bond.  Think of formula for zero coupon bond:

PRICE (PV) =    \$1000 (FV)
(1 + i )t

even a small change in i gets multiplied to the 20th power for the 20 year bond vs to the first power for the one year bond.

2. INCOME OR REINVESTMENT RATE RISK.   The risk that investment income, interest payments, dividends, maturing fixed-income securities, loan payments, early loan payoffs, etc. will have to be REINVESTED at lower interest rates.  Risk is usually associated with short term investments, or high income investments, and the risk is usually that interest rates will FALL, income has to reinvested at lower rates than desired or expected.

Example: 2 year time horizon: A) Invest for 2 years, lock in at @6%, versus b) Invest for one year at 5%, take proceeds and then reinvest at X% for the second year.  The risk is that the one year interest rates during YR 2 will fall.

Example:  Interest rates were high in the early 1980s, so you sell \$10,000 of your stocks to purchase a 14% coupon rate, 30 year bond in 1981 to lock into the high yield of 14%.  Yields are 14%, coupon rates are 14%, so the purchase price is: ____________  You are expecting a 14% yield on your investment, but that assumes that you can reinvest your coupon paymentsof __________ @ ___________ for thirty years.   You were hurt by income risk.  WHY?___________________

DURATION:

Formula for a more precise measure of interest rate sensitivity (price risk) than maturity, developed by academic finance and economics researchers in the 1950s and 1960s, particularly Fred Macaulay.  Macaulay research on duration started when he observed that investors preferred high-coupon bonds to otherwise identical bonds with low coupons.

Example:  Bond A: 10 year, 14% coupon bond, YTM = 8%
Bond B: 10 year, 4% coupon bond, YTM = 8%

Ceteris paribus, investor prefer Bond A over Bond B.  Does that suggest that investors are more concerned about price risk or income risk?  (They prefer high coupon payments to low coupon payments). ___________________________

Maturity is the same for both bonds (10 years), but the preference for Bond A over Bond B suggests that maturity is not the only or best measure of Interest Rate Risk.

Duration Example:

CF1             CF2             CF3               Maturity              Duration
Investment A:

Investment B:

Duration: Unit of Measurement: YEARS.  Weighted average approach to maturity (also measured in years).

Duration (D) is the weighted average number of years until CFs are received.

Formula:  Duration (D)        =    Σ         CFt        *   t (year that CF is received)
(1 + i)t
__________________________________________

PRICE of Bond (or PV of all CFs)

Steps: 1) Calculate the Bond Price or the PV of CFs for the denominator

2) Calculate the PV of each CF separately and multiply times the YR (t) in which it occurs (1 for YR1, 2 for YR2, etc)

3) Sum PV of all CFs multiplied by the YR.

4. Divide SUM from part 3 by the PRICE from part 1.

Example: 2 year security with \$100 CFs.

CF               \$100              \$100
|------------------|-----------------|
YR 0               1                   2

Assume interest rates are 0.  PV = ?

Duration (D)  =    \$100 (1)   +   \$100 (2)   =    1.5 years
\$200

or Duration (D)    =   \$100 (1)     +    \$100  (2)
\$200                \$200

D      =   .5 (1)   +    .5 (2)  =   1. 5 years.

Logic:  You get 50% of your payback at the end of YR1 and 50% payback at the end of YR2.  Duration is a weighted average approach, so the D = 1.5 YRS.

Assume now that i = 10%

1. Solve for the PV of the CFs    =   \$100 / 1.1   +   \$100 / (1.1)2   =

D = (\$90.91 / \$173.55) * 1    +    (\$82.64 / \$173.55) * 2     =

= .52 (1)   +   .48 (2)   =   1.47 years.

On a PV basis, you receive 52% of total CF in YR 1 and 48% of CF in YR 2, D = 1.47 years.

or D  =      (90.91) (1)    + (82.64) (2)          =   1.47 years.
\$173.55

Set up the Duration formla for a bond problem: Assume a 4 year bond, with 8% coupon payments, \$1000 face value, current YTM = 10%.   Price = ?    Coupon PMTS = ?  DURATION(D) = ?

General Duration Rules:

a) Duration (D) is ALWAYS less than maturity (D < M), except for what type of bond?
b) The HIGHER the coupon payments, the LOWER the duration.

Example:  T-Bond A, 6%,  2010
T-Bond B, 12%, 2010

Everything is the same except for the coupon rate.  Which has lower duration?

USING DURATION (D) TO MEASURE CAPITAL RISK

Knowing duration (D), we can calculate the PRICE SENSITIVITY as follows:

%P =  %PV =     -   D    *   Δ ( i )
( 1 +  i )

where D is the duration, i is the original interest rate, and  d ( i ) is the change in the interest rate.  The minus sign reflects the negative relationship between interest rates and bond prices. %P is the percentage change in the value of the security, or the %PV.

When interest rates are low, (1 + i ) will be close to one, so approximately:

%P (%PV)    =    - D   *   Δ ( i )

Example: 2 year security from above, D = 1.47 years.  i = 10%, P = \$173.55.

Assume that interest rates go up 11%, then the d ( i ) = +1%, so that:

%P = -1.47  *  +1%  =  -1.47%

New Price (PV) =  \$173.55 - 1.47%  =  \$171.00

Assume Interest rates fall to 8%:  Δ ( i ) = - 2%  (HINT: use whole numbers for percents in this forumla)

%P  =  -1.47  *  -2%   =  +2.94%

New Price (PV) = \$173.55 + 2.94% = \$178.65

USING DURATION TO ANALYZE THE S&L CRISIS

S&L Problem: Duration of Assets (DA) is much greater the Duration of Liabilities (DL),   DA > DL.

Long term assets (mortgages) and short term liabilities (deposits).

Example: Typical S&L, in PV dollars.

ASSETS     =     LIAB + EQUITY
-----------------------------------------
D=Infinity   RES      \$5m          \$96m  DEPOSITS      D = 1 year

D=8 yrs      LOANS \$95           \$4m    EQUITY

VBANK = PVA - PVL

V    =  \$100 - \$96m   =   \$4m

DA = 8 years
DL = 1 year

% PV  =  - D  *  Δ ( i )

Assume interest rates rise by 1%:

%PVA = - 8   *   +1%   =  -8%
%PVL = - 1   *   +1%    =  -1%

PVA = \$100 - 8% =  \$92m
PVL =  \$96 - 1% =  \$95.04

NEW VALUE       -\$3.04m

Value went from \$4m to -\$3.04m with a one percent increase in interest rates, i.e. over a \$7m change in value (-176% change) with just a one percent change!  Interest rates went up by about 10 percentage points in the 1970s,  so figure the impact on S&Ls:

Assets: - 80%  (100 - 80% = \$20m)
Liab:  - 10%   ( 96 - 10% = \$86.4),  New Value = \$20m - \$86.4 = -\$66.40m!!!

Explains why 50% of banks were insolvent on a PV basis in 1982.

Bank strategies to manage Interest Rate Risk, Duation Mismatch:

Decrease D of Assets          Increase D of LIAB

IMMUNIZATION: To immunize (protect) against inerest rate risk, Match DA with DL, or more accurately: PVA * DA = PVL * DL In that case, the Bank (or firm) would be completely protected (immunized) against changes in interest rates. Changes in interest rates would NOT affect the Value of the Bank. Immunization is a risk management strategy.