H14. Bond Prices and Yields.
Bond: a security that is issued in connection with a
borrowing arrangement. It is a loan made by an investor
to a firm or government. A borrower obtaining a bank
loan gets money up front, makes interest payments each
term, and then repays a principle when the loan is due. A
bond is the opposite: the investor pays the bond price and
then receives regular interest payments plus principle
when the bond matures. To value a security, we discount its expected cash flows by the
appropriate discount rate.
Zero-coupon bonds: bonds that make no coupon payments.
The yield to maturity is a measure of the average rate of return to an investor who purchases
the bond for the asked price and holds it until its maturity date.
If a bond is purchased between coupon payments, the buyer must pay the seller for accrued
interest, the prorated share of the upcoming semiannual coupon. The formula of the amount of
accrued interest between two dates is:
!""#!$ !"#$"% !"#$%&' !"#$ !"#$% !"#$ !"#$"% !"#$%&'
Accrued interest = ! (!"#$%&&'%()
× !"#$ !"#"$%&'() !"#$"% !"#$%&'(
The invoice price, sale price, equals the stated price of the bond + the accrued interest.
Some corporate bonds are issued with call provisions allowing the issuer to repurchase the
bond at a specific call price before maturity. Callable bonds are bonds that the issuer can
decide to buy back at a certain time, but these bonds come with a period of call protection in
which the bonds are not callable. The option to call the bond is valuable to the firm, allowing
it to buy back the bonds and refinance at lower interest rates when market rates fall. Callable
the bonds are issued with higher coupons and promised yields to maturity than non-callable
bonds.
Puttable bonds: while the callable bonds give the issuer the option to extend or retire the
bond at the call date, the extendable or put bond gives this option to the bondholder. If the
bond’s coupon rate exceeds current market yields, for instance, the bondholder will choose to
extend the bond’s life.
Preferred stock: promises to pay a specified stream of dividends. If a corporation fails to pay
dividend, the dividends accumulate and the common stockholders may not receive any
dividends until the preferred stockholders have been paid in full.
Price-yield relationship.
A fundamental property of an option-free bond is that its price
changes in the opposite direction from the change in the yield.
Hence, when interest rates increase, bond prices decline and
vice-versa. Simple reason: the price of a bond is the present
value of its cash flows. At a higher interest rate, the present
value of the payments to be received by the bondholder is
lower. An increase in the interest rate results in a price decline
that is smaller than the price gain resulting from a decrease of
equal magnitude in the interest rate. Thus, progressive increases in the interest rate result in
progressively smaller reductions in the bond price.
One key factor determines the sensitivity of bond prices to market yield, namely, the maturity
of the bond. As a general rule, keeping all other factors the same, the longer the maturity of
the bond, the greater the sensitivity of the price to fluctuations in the interest rate.
, There are three methods usually used to measure the return an investor would make on an
investment in bonds:
!""#$% !"#$"% !"#$%&'
1. Current yield: !"#$% !"#$
. However, this is not a good measure because:
- It only takes into account the interest income earned on the bond.
- It doesn’t include any capital gains or losses on your investment.
2. Yield to Maturity. The yield to maturity is the standard measure of the total rate of return.
The YTM is defined as the interest rate that makes the present value of a bond’s cash flows
equal to its market price. The interest rate is often interpreted as a measure of the average rate
of return that will be earned on a bond if it is bought now and held until maturity. Limitations:
- It assumes that you hold the bond until maturity.
- It assumes that all coupons are reinvested at the YTM.
- It assumes the term structure is flat.
3. Rate of return. This is also called the Holding-Period Return (HPR). When the yield to
maturity is unchanged over the period, the rate of return on the bond will equal that yield.
However, when yields fluctuate, so will a bond’s rate of return. An increase in the bond’s
yield acts to reduce its price, which reduces the HPR. In this event, the HPR is likely to be
less than the initial yield to maturity. Conversely, a decline in the yield will result in a HPR
greater than the initial yield. Limitation:
- It can only be computed at result.
𝐶𝑜𝑢𝑝𝑜𝑛 𝑖𝑛𝑐𝑜𝑚𝑒 + (𝑃! − 𝑃! )
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑡𝑢𝑟𝑛 =
𝑃!
Par value bonds: P = FV ! c = ytm
Premium bonds: P > FV ! c > ytm
Discount bonds: P < FV ! c < ytm
Treasury ‘strips’: zero-coupon bonds created by selling each coupon or principal payment
from a whole Treasury bond as a separate cash flow. A bond dealer who purchases a Treasury
(zero) coupon bond may ask the Treasury to break down the cash flows to be paid by the bond
into series of independent securities, where each security is a claim to one of the payments of
the original bond. Treasury stripping suggest exactly how to value a coupon bond.
Bond markets
" US Treasury Bonds. This is one of the largest and most liquid bond markets.
T-bills: maturity up to 1 year; Treasury notes: maturity 1 to 10 years; Treasury bonds: maturity 10 to 30 years.
Bonds can be purchased directly from the US Treasury, but also a very active secondary
market.
" Corporate bonds. Like governments, corporations also issue bonds with variety of
maturities. Commercial Paper is short-term lending by firm (less than 270 days). Some
corporate bonds may have special features that government bonds typically don’t have:
• Callable bonds: firms can repurchase before maturity.
• Convertible bonds: can be converted into shares of common equity stock.
• Floating-rate bonds: bond’s coupon is adjustable to current market rates instead of fixed
over time.
Bond: a security that is issued in connection with a
borrowing arrangement. It is a loan made by an investor
to a firm or government. A borrower obtaining a bank
loan gets money up front, makes interest payments each
term, and then repays a principle when the loan is due. A
bond is the opposite: the investor pays the bond price and
then receives regular interest payments plus principle
when the bond matures. To value a security, we discount its expected cash flows by the
appropriate discount rate.
Zero-coupon bonds: bonds that make no coupon payments.
The yield to maturity is a measure of the average rate of return to an investor who purchases
the bond for the asked price and holds it until its maturity date.
If a bond is purchased between coupon payments, the buyer must pay the seller for accrued
interest, the prorated share of the upcoming semiannual coupon. The formula of the amount of
accrued interest between two dates is:
!""#!$ !"#$"% !"#$%&' !"#$ !"#$% !"#$ !"#$"% !"#$%&'
Accrued interest = ! (!"#$%&&'%()
× !"#$ !"#"$%&'() !"#$"% !"#$%&'(
The invoice price, sale price, equals the stated price of the bond + the accrued interest.
Some corporate bonds are issued with call provisions allowing the issuer to repurchase the
bond at a specific call price before maturity. Callable bonds are bonds that the issuer can
decide to buy back at a certain time, but these bonds come with a period of call protection in
which the bonds are not callable. The option to call the bond is valuable to the firm, allowing
it to buy back the bonds and refinance at lower interest rates when market rates fall. Callable
the bonds are issued with higher coupons and promised yields to maturity than non-callable
bonds.
Puttable bonds: while the callable bonds give the issuer the option to extend or retire the
bond at the call date, the extendable or put bond gives this option to the bondholder. If the
bond’s coupon rate exceeds current market yields, for instance, the bondholder will choose to
extend the bond’s life.
Preferred stock: promises to pay a specified stream of dividends. If a corporation fails to pay
dividend, the dividends accumulate and the common stockholders may not receive any
dividends until the preferred stockholders have been paid in full.
Price-yield relationship.
A fundamental property of an option-free bond is that its price
changes in the opposite direction from the change in the yield.
Hence, when interest rates increase, bond prices decline and
vice-versa. Simple reason: the price of a bond is the present
value of its cash flows. At a higher interest rate, the present
value of the payments to be received by the bondholder is
lower. An increase in the interest rate results in a price decline
that is smaller than the price gain resulting from a decrease of
equal magnitude in the interest rate. Thus, progressive increases in the interest rate result in
progressively smaller reductions in the bond price.
One key factor determines the sensitivity of bond prices to market yield, namely, the maturity
of the bond. As a general rule, keeping all other factors the same, the longer the maturity of
the bond, the greater the sensitivity of the price to fluctuations in the interest rate.
, There are three methods usually used to measure the return an investor would make on an
investment in bonds:
!""#$% !"#$"% !"#$%&'
1. Current yield: !"#$% !"#$
. However, this is not a good measure because:
- It only takes into account the interest income earned on the bond.
- It doesn’t include any capital gains or losses on your investment.
2. Yield to Maturity. The yield to maturity is the standard measure of the total rate of return.
The YTM is defined as the interest rate that makes the present value of a bond’s cash flows
equal to its market price. The interest rate is often interpreted as a measure of the average rate
of return that will be earned on a bond if it is bought now and held until maturity. Limitations:
- It assumes that you hold the bond until maturity.
- It assumes that all coupons are reinvested at the YTM.
- It assumes the term structure is flat.
3. Rate of return. This is also called the Holding-Period Return (HPR). When the yield to
maturity is unchanged over the period, the rate of return on the bond will equal that yield.
However, when yields fluctuate, so will a bond’s rate of return. An increase in the bond’s
yield acts to reduce its price, which reduces the HPR. In this event, the HPR is likely to be
less than the initial yield to maturity. Conversely, a decline in the yield will result in a HPR
greater than the initial yield. Limitation:
- It can only be computed at result.
𝐶𝑜𝑢𝑝𝑜𝑛 𝑖𝑛𝑐𝑜𝑚𝑒 + (𝑃! − 𝑃! )
𝑅𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑡𝑢𝑟𝑛 =
𝑃!
Par value bonds: P = FV ! c = ytm
Premium bonds: P > FV ! c > ytm
Discount bonds: P < FV ! c < ytm
Treasury ‘strips’: zero-coupon bonds created by selling each coupon or principal payment
from a whole Treasury bond as a separate cash flow. A bond dealer who purchases a Treasury
(zero) coupon bond may ask the Treasury to break down the cash flows to be paid by the bond
into series of independent securities, where each security is a claim to one of the payments of
the original bond. Treasury stripping suggest exactly how to value a coupon bond.
Bond markets
" US Treasury Bonds. This is one of the largest and most liquid bond markets.
T-bills: maturity up to 1 year; Treasury notes: maturity 1 to 10 years; Treasury bonds: maturity 10 to 30 years.
Bonds can be purchased directly from the US Treasury, but also a very active secondary
market.
" Corporate bonds. Like governments, corporations also issue bonds with variety of
maturities. Commercial Paper is short-term lending by firm (less than 270 days). Some
corporate bonds may have special features that government bonds typically don’t have:
• Callable bonds: firms can repurchase before maturity.
• Convertible bonds: can be converted into shares of common equity stock.
• Floating-rate bonds: bond’s coupon is adjustable to current market rates instead of fixed
over time.
