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