Fin 410 Chp 10

Percentage Change in Bond Price formula

- Modified Duration * Change in YTM

longer a bond's maturity..

..the longer is its duration.

higher a bond's coupon..

..the shorter is its duration

Bond price Formula

= (C/YTM)*(1-(1/(1+(YTM/2)^2m))+(FV/(1+(YTM/2)^2m))

Pct Change in Bond Price formula

= -Duration*(Change in YTM/(1+(YTM/2))

Yield Value of 32 formula

= 1/(32*Dollar Value of 01)

Current Yield Formula

= Annual Coupon / Bond Price

Modified Duration Forumla

= Macauley Duration / (1+(YTM/2))

Dollar Value of 01 formula

= Modified DurationxBond Pricex.0001

Current Yield

Annual Coupon / Bond Price

Coupon Rate formula

Annual Coupon / Par Value

Malkiels 1st theory

Bond prices and bond yields move in opposite directions. As a bond's yield increases, its price decreases.

Premium Bond

Bonds with a price greater than par value are said to be selling at a premium. The yield to maturity of a premium bond is less than its coupon rate.

Discount Bonds Def

Bonds with a price less than par value are said to be selling at a discount. The yield to maturity of a discount bond is greater than its coupon rate.

suppose a $1,000 par value bond paying an $80 annual coupon has a price of $1,032.25. The current yield is

Book * Annual Coupon / Bond Price 80/1032.25 = 7.75%

suppose a $1,000 par value bond pays semiannual coupons of $40. The annual coupon is

Book * Annual Coupon / Par Value 80/1000 = 8%

a bond has a $1,000 face value, 20 years to maturity, an 8 percent coupon rate, and a yield of 9 percent. What's the price?

Book* = (C/YTM)*(1-(1/(1+(YTM/2)^2m))+(FV/(1+(YTM/2)^2m)) (80/.09) * (1-(1/(1+(.09/2)^2*20)) + (1000/(1+(.09/2)^2*20)) = 907.99

A bond has a modified duration of seven years. Suppose its yield increases from 8 percent to 8.5 percent. What happens to its price?

Book* = - Modified Duration * Change in YTM = -7 * (.085-.08) = -3.5%

A bond has a Macaulay duration of 11 years, and its yield increases from 8 percent to 8.5 percent. What will happen to the price of the bond?

Book* = -Duration*(Change in YTM/(1+(YTM/2)) -11 * ((.085-.08)/(1 + (.08/2)) = -5.29

suppose a bond has a Macaulay duration of six years, and its yield decreases from 10 percent to 9.5 percent. The resulting percentage change in the price of the bond is calculated as follows:

Book* = -Duration*(Change in YTM/(1+(YTM/2)) -6 * ((.095-.10) / (1+(.10/2)) = 2.86%

A bond has a Macaulay duration of 8.5 years and a yield to maturity of 9 percent. What is its modified duration?

Book* = Macauley Duration / (1+(YTM/2)) = 8.5 / (1+(.09/2)) = 8.134

Suppose a bond has eight years to maturity, a price of 110, and a coupon rate of 8 percent. What is its yield?

Book* TVM...N=8/I=/PV=-110/PMT=.08*100/FV=100

Now consider two bonds, both with a 9 percent coupon rate and the same yield to maturity of 11 percent, but with different maturities of 5 and 10 years. Which has the higher price? Verify your answer by calculating the prices.

Both are at discount Coupon < YTM. Shorter maturity will have higher price...880.50 & 924.62

Both are at Premium coupon (9%) > YTM (7%). Longer maturity will have higher price...so 10 years

Consider two bonds, both with a 9 percent coupon rate and the same yield to maturity of 7 percent, but with different maturities of 5 and 10 years. Which has the higher price? Verify your answer by calculating the prices

Bond A is 7% Coupon > 5% YTM = Premium Bond B is 7% Coupon < 9% YTM = Discount

Consider two bonds, both with eight years to maturity and a 7 percent coupon. One bond (A) has a yield to maturity of 5 percent while the other (B) has a yield to maturity of 9 percent. Which of these bonds is selling at a premium and which is selling at a discount? Verify your answer by calculating each bond's price

Malkiels 2nd Theory

For a given change in a bond's yield to maturity, the longer the term to maturity of the bond, the greater will be the magnitude of the change in the bond's price.

Malkiels 4th Theory

For a given change in a bond's yield to maturity, the resulting percentage change in the bond's price is inversely related to the bond's coupon rate.

Discount Bonds

If Coupon rate < YTM then Price < Face (par value)

Par Bonds

If Coupon rate = YTM then Price = Face (par value)

Premium Bonds

If Coupon rate > YTM then Price > Face (par value)

Price Quoted

If you buy a bond between coupon payment dates, the price you pay will usually be more than

decrease bond prices (price risk) but increase the future value of reinvested coupons (reinvestment rate risk).

Interest rate increases act to

realized yield will almost always differ from a promised yield

Interest rates fluctuate, causing bond prices to rise or fall.

Dollar Value of 01

Measures the change in bond price from a one basis point change in yield

Yield Value of 32nd

Measures the change in yield that would lead to a 1/32nd change in the bond price

What is the price of a straight bond with: $1,000 face value, coupon rate of 8%, YTM of 7%, and a maturity of 20 years?

Notes* Bond Price Formula = 1,106.78

Suppose a bond has a Macaulay Duration of 11 years and a current yield to maturity of 8%. If the yield to maturity increases to 8.50%, what is the resulting percentage change in the price of the bond

Notes* Pct Change in Bond -5.29%

Solution to reinvestment rate risk

Purchase zero coupon bonds

Clean Price

Quoted Price given without accrued interest

Price Risk

Risk that bond prices will decrease

U.S Treasury Bonds

Straight bonds

Target Date

The date the payment is due

Coupon Rate and Current yield

Two basic yield measures for a bond

Callable at Par

When a call price is equal to face value

Dedicated Portfolio

a bond portfolio created to prepare for a future cash payment, e.g. pension funds.

Dynamic Immunization

a periodic rebalancing of a dedicated bond portfolio for the purpose of maintaining a duration that matches the target maturity date

Duration

a way for bondholders to measure the sensitivity of a bond price to changes in bond yields

yield to call

a yield measure that assumes a bond issue will be called at its earliest possible call date.

Duration Matching

can immunize a dedicated portfolio

Coupon Bond

duration = a weighted average of individual maturities of all the bond's separate cash flows, where the weights are proportionate to the present values of each cash flow

Callable Bond

gives the issuer the option to buy back the bond at a specified call price anytime after an initial call protection period

..a shorter duration...

higher yield to maturity implies..

Yield to Call

is a yield measure that assumes a bond will be called at its earliest possible call date

Straight Bond

is an IOU that obligates the issuer of the bond to pay the holder of the bond

Yield to Maturity

is the discount rate that equates today's bond price with the present value of the future cash flows of the bond.

Interest Rate Risk

is the possibility that changes in interest rates will result in losses in the bond's value...faced by bondholders

Immunization

is the term for constructing a dedicated portfolio such that the uncertainty surrounding the target date value is minimized

when the coupon rate and YTM are held constant - for Premium

longer the term to maturity, the greater the premium over par value.

a longer duration...

lower yield to maturity implies..

Dirty Price

price the buyer actually pays with accrued interest

when the coupon rate and YTM are held constant - for Discount

the longer the term to maturity, the greater the discount from par value

The price of a bond is found by adding together

the present value of the bond's coupon payments and the present value of the bond's face value

Reinvestment Rate Risk

the uncertainty about the value of the portfolio on the target date

Macauleys Duration

values are stated in years and are often described as a bond's effective maturity

Coupon Rate and Yield

whether a bond sells at a premium or discount depends on the relation between