Dr. Himanshu Joshi

Change in interest rates results in change in prices of bonds in reverse direction.

Interest rate increase = Price will fall to offer in-built capital gain to new investors

Interest rate decrease = Price will rise to offer in built capital loss to new investors.

Price Curve is convex, meaning that decreases in yield have bigger impact price than increase in yields of equal magnitude.(1)Bond prices and yield are inversely related, as yield increases, bond prices falls. And as yield decreases bond prices rise.

(2) An increase in a bond’s YTM results into a smaller price change (decrease) than a decrease in YTM of equal magnitude.

Compare Bond A and Bond B, which are identical in coupon payments but differ in maturity.

Bond B is more sensitive to interest rate having longer maturity.

Property (3) Prices of long term bonds tends to be more sensitive to interest rate changes than price of short term bonds.

Why? Impact of higher discount rate will be

greater as the rate applied to more distant cash flows.

Also Notice that Bond B has six times the maturity of Bond A, but it has less than six times sensitivity.

Although bond’s interest rate sensitivity seems to increase with maturity, it does so less than proportionally as bond maturity decreases.

Property 4. The sensitivity of bond prices to change in yields increases at a decreasing rate as maturity increases.

Now Compare Bond B and Bond C Bond B and Bond C are alike in all

respect except Coupon rate. Lower Coupon Bond C exhibits greater

sensitivity to changes in investment rates. This turns out to be a general property of Bond Prices.

Property 5. Interest Rate Risk in inversely proportional to the bond’s coupon rate.

Why?

Now Consider Bond C and Bond D Which are identical except for the

YTM at which the bond currently sell. Bond C with Higher YTM is less

sensitive than Bond D. Property 6. The Sensitivity of a

Bond’s price to a change in its yield to maturity is inversely related at which the bond is currently selling.

Why?

First five properties were described by Malkiel and are known as Malkiel bond pricing relationships.

The last (sixth) property was demonstrated by Homer and Liebowitz.

Maturity Coupon Rate YTM (at selling Price)

Excel Application…

To deal with the ambiguity of the maturity of a bond making many payments, we need a measure of the average maturity of the bond’s promised cash flows to serve as useful summary statistics of the effective maturity of a bond.

We would like use the measure as a guide to sensitivity of a bond to interest rate changes.

Since we have noted that price sensitivity tends to increase with time to maturity.

A measure of the effective maturity of a bond The weighted average of the times until each

payment is received, with the weights proportional to the present value of the payment

Duration is shorter than maturity for all bonds except zero coupon bonds

Duration is equal to maturity for zero coupon bonds

t tt

w CF y ice ( )1 Pr

twtDT

t

1

CF CashFlow for period tt

Numerator: present value of the cash flow occurring at time t,

while the denominator is the value of all the payments forthcoming from the bond.

These weights sums to 1.0, because of the cash flows discounted at the Yield to maturity equals the bond price

Excel…..

Price change is proportional to duration and not to maturity

D* = modified duration

(1 )

1

P yDx

P y

*P

D yP

Consider a 2-year maturity, 8% coupon bond making semiannual coupon payments and selling a price of $964.540. for a yield to maturity of 10%.

The duration of this bond is 1.8852 years. For comparison consider a zero coupon bond

with maturity and duration of 1.8852 years. As coupon bond makes payments

semiannually, it is best to treat one period as half year.

So the duration of each bond is 1.8852*2 =3.7704 (semi annual) with interest rate of 5%.

The modified duration = D/1+y = 3.7704/1.05 = 3.591.

Suppose semi-annual interest rate increases from 5% to 5.01%.

According to equation: ∆P/P = -D* ∆y = -3.591*0.01%= ( -0.3591%) Now Compute the price change each

bond direclty.

The zero coupon bond initially sells at : $1000/1.053.7704 = 831.9704 At higher yield: $1000/1.0513.7704 = 831.6717

So price falls by 831.9704 – 831.6717/831.9704 = 0.359%.

We can conclude that bonds with equal duration do in fact have equal interest rate sensitivity.

And the percentage change is the modified duration times the change in yield.

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

Duration of perpetuity = 1+y/y

At 10% yield, the duration of a perpetuity that pays $100 once a year forever is

1.10/.10 = 11 years. But at an 8% yield 1.08/0.08 = 13.5 years.

The relationship between bond prices and yields is not linear

Duration rule is a good approximation for only small changes in bond yields

n

tt

t tty

CF

yPConvexity

1

22

)()1()1(

1

Correction for Convexity:

21 [ ( ) ]2P

D y Convexity yP

Bond in the figure has a 30 year maturity, an 8% coupon, and sells at an initial yield to maturity of 8%.

Because coupon rate equals YTM, the bond sells at par value $1000.

Modified duration for the bond is 11.26 years, and convexity is 212.4.

If the bond’s yield increases from 8% to 10%, bond price will fall to $811.46.

Change of 18. 85% The duration rule ∆P/P = -D* ∆y =-11.26 * .02 = -22.52%. Duration with Convexity: ∆P/P = -D* ∆y +1/2 * Convexity* (∆y)2

=11.26*.02 +1/2 * 212.4 * (.02)2

= -18.27%.

Passive bond management take bond prices as fairly set and seek to control only the risk of their fixed income portfolio.

Two broad class of passive strategies are followed in fixed income market:

(1)Indexing Strategy: attempt to replicate the performance of a given bond index.

(2)Immunization Techniques: designed to shield the overall financial status of the institution from exposure to interest rate fluctuations.

Indexing is to create a portfolio that mirrors the composition of an index that measures the broad market.

Three major bond indexes in US are:(1)Lehman aggregate bond index.(2) Solomon Smith Barney Broad

Investment Grade Index (BIG).(3) Merrill Lynch U.S Broad Market

Index

All are market weighted indexes of total return.All three include government, corporate, mortgage

backed, and Yankee bonds in their universe.All three indexes include only bonds with maturities

greater than 1 year.As time passes, and the maturity of a bond falls below 1

year, the bond is dropped from the index.These indexes include more than 5000 securities,

making it quite difficult to purchase each security in the index in proportion to its market value.

Many bonds are very thinly traded, meaning that identifying their owners and purchasing the securities at a fair price can be difficult.

Old bonds to be dropped and new bonds issued to be included to rebalance the portfolio.

In practice, it is deemed infeasible to precisely replicate the broad bond indexes. Instead a stratified sampling or cellular approach is taken.

First the bond market is stratified into several subclasses based on issuer, maturity, credit risk, coupon rate to form cells.

Bonds falling within each cell are then considered reasonably homogeneous.

Next the percentages of entire universe falling within each cell are computed and reported.

Finally, the portfolio manager establishes a bond portfolio with representation for each cell in the bond universe.

In contrast to indexing strategies many institutions try to insulate their portfolios from interest rate risk altogether.

The net worth of the firm or ability to meet future obligations fluctuate with interest rates.

Immunization refers to strategies used by such investors to shield their overall financial status from interest rate fluctuations.

Example: Pension Funds (Fixed Future obligation) and Banks (asset Liability maturity mismatch).

The lesson is that funds should match the interest rate exposure of assets and liabilities so that the value of assets will track the value of liabilities whether rate rises or falls.

The notion of immunization was introduced by F. M. Redington, an actuary for a life insurance company.

The idea behind immunization is that duration matched assets and liabilities let the assets portfolio meet the firm’s obligations despite interest rate movements.

Consider, an insurance company that issues a guaranteed investment contract, or GIC, for $10,000. if the GIC has the five year maturity and a guaranteed interest rate of 8%, the insurance company is obliged to pay: $10000*(1.08)5 = $14,693.28 in five years.

Suppose that insurance company chooses to fund its obligation with $10000 of 8% annual coupon bonds, selling at par, with 6 years to maturity and a guaranteed interest rate of 8%.

As long as interest rate stays at 8%, the company has fully funded its obligation. As the present value exactly equals the value of the bonds.

If interest rate stays at 8%, the accumulated funds from the bond will grow to exactly the $14,693.28 obligation.

If the interest rate rise, the fund will suffer a capital loss, impairing its ability to satisfy the obligation. However, at higher interest rate, reinvested coupons will grow faster, offsetting, the capital loss.

If portfolio maturity is chosen appropriately, price risk and re-investment rate risk will cancel out exactly.

When the portfolio duration is set equal to the investor’s horizon date, the accumulated value of the investment fund at the horizon date will be unaffected by interest rate fluctuations.

For a horizon equal to the portfolio’s duration, price risk and reinvestment risk exactly cancel out.

If the obligation was immunized, why is there any surplus in the fund?

Convexity. Coupon bond has greater convexity

than the obligation it funds.

Automatically immunize the portfolio from interest rate movement• Cash flow and obligation exactly offset each

other i.e. Zero-coupon bond

Not widely used because of constraints associated with bond choices

Sometimes it simply is not possible to do

Substitution swap: two identical substitute in same market with different yields. (Honda and Toyota)

Inter-market swap: when yield spread between two market is out of line (10 year baa rated Corporate bond and 10 year Treasury bonds spread is now 3%, historically 2%.)

Rate anticipation swap (if interest rate is expected to increase, then duration of portfolio should increase and vice -versa)

Pure yield pickup: if yield curve is slopping upward investor is bearing interest rate risk to earn term premium.)

Tax swap: if realization of capital loss is tax-advantageous.