Transcript
BOND YIELDS AND PRICES
Bond Yields:
Bond yields and interest rates are the same concept. Therefore, we begin our discussion of bond yields with a brief consideration of interest rates. Interest rates measure the price paid by a borrower to a lender for the use of resources over some time period—that is, interest rates are the price for loan able funds. The price differs from case to case, based on the demand and supply for these funds, resulting in a wide variety of interest rates. The spread between the lowest and highest rates at any point in time could be as much 10 to 15 percentage points. In bond parlance, this would be equivalent to 1,000 to 1,500 basis points, since 1 percentage point of a bond yield consists of 100 basis points.
It is convenient to focus on the one interest rate that provides the foundation for other rates. This rate is referred to as the short-term riskless rate (designated RF in this 'text) and is typically proxied by the rate on Treasury bills. All other rates differ from RF because of two factors:
Maturity differentials
Risk premiums
The Basic Components of Interest Rates:
Explaining interest rates is a complex task that involves substantial economics reasoning and study. Such a task is not feasible in this text. However, we can analyze the basic determinants of nominal (current) interest rates with an eye toward recognizing the factors that affect such rates and cause them to fluctuate. The bond investor who understands the foundations of market rates can then rely on expert help for more details, and be in a better position to interpret and evaluate such help.
The basic foundation of market interest rates is the opportunity cost of foregoing consumption; representing the rate that must be offered to individuals to persuade them to save rather than consume. This rate is sometimes called the-real risk-free rate of interest because it is not affected by price changes or risk factors. We will refer to it simply as the real rate and designate it RR in this discussion.
Thus, for short-term risk-free securities, such as three-month Treasury bills, the nominal interest rate is a function of the real rate of interest and the expected inflationary premium. This is expressed as, which is an approximation:
RF ?RR + EI
Where;
RF = short term Treasury bill rate
RR = the real risk-free rate of interest
El = the expected rate of inflation over the term of the instrument
Measuring Bond Yields:
Several measures of the yield on a bond are used by investors. It is very important for bond investors to understand which yield measure is being discussed, and what the underlying assumptions of any particular measure are. To illustrate these measures, we will use as an
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example a three-year, IC-percent coupon, AAA-rated corporate bond, with interest payments occurring exactly six months from now, one year from now, and so forth.
Current Yield:
The ratio of the coupon interest to the current market price is the current yield, and, this is the measure reported daily In the Wall Street Journal for those corporate bonds shown under the sections "New York Exchange Bonds" and "AMEX Bonds." The current yield is clearly superior to simply citing the coupon rate on a bond, because it uses the current market price as opposed to the face amount of a bond (almost always $1,000). However, current yield is not a true measure of the return to a bond purchaser, because it does not account for the difference between the bond's purchase price and its eventual redemption at par value.
Yield to Maturity:
The rate of return on bonds most often quoted for investors is the yield to maturity (YTM), a promised rate of return that will occur only under certain assumptions. It is the compound rate of return, an investor will receive from a bond purchased at the current market price if:
The bond is held to maturity, and
The coupons received while the bond is held are reinvested at the calculated yield to maturity.
Barring default, investors will actually cam this promised rate if, and only if, these two Conditions are met. As we shall see, however, the likelihood of the second condition actually being met is extremely small.
The yield to maturity, is the periodic interest rate that equates the present value of the expected future cash flows (both coupons and maturity value) to be received on the bond to the initial investment in the bond, which is its current price. This means that the yield to maturity is the internal rate of return (IRR) on-the bond investment, similar to the IRR used in capital budgeting analysis.
Yield to Call:
Most corporate bonds, as well as some government bonds, are callable by the issuers, typically after some deferred call period. For bonds likely to be called the yield to maturity calculation is unrealistic. A better calculation is the yield to call. The end of the deferred call period, when a bond can first be called, is often used for the yield to call calculation.
This is particularly appropriate for bonds selling at a premium (i.e. high-coupon bonds with market prices above par value).
Realized Compound Yield:
After the investment period for a bond is over, an investor can calculate the realized, compound yield (RCY). This rate measures the compound yield on the bond investment actually earned over the investment period, taking into account all intermediate cash flows and reinvestment rates. Defined in this mannerist cannot be determined until the investment is concluded and all of the cash flows are known. Thus, if you invest $ 1,000 in a bond for five years, reinvesting the coupons as they are received, you will have X dollars at the
conclusion of the five years, consisting of the coupons received, the amount earned from reinvesting the coupons, and the $1,000 par value of the bond payable at, maturity. You can then calculate your actual realized rate of return on the investment.
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The RCY for a bond can be calculated by dividing the total ending wealth (including the purchase price) at the bond's maturity by the amount invested; and raising the result to the 1/n power, where n is the number of compounding periods. Next, subtract 1.0 from the result. Finally, because of the semiannual basis for bonds, multiply by 2 to obtain the bond equivalent rate.
The semi-annual realized compound yield can be calculated using the following formula:
RCY = [total ending wealth / purchase price of bond] 1/ n – 1.0
BOND PRICES:
The Valuation Principle:
A security's intrinsic value, or estimated value, is the present value of the expected cash flows from that asset. Any security purchased is expected to provide one or more cash flows some time in the future. These cash flows could be periodic, such as interest or dividends, or simply a terminal price or redemption value, or a combination of these. Since these cash flows occur in the future, they must be discounted at an appropriate rate to determine their present value. The sum of these discounted cash flows is the estimated intrinsic, value of the asset. Calculating intrinsic value, therefore, requires the use of present value techniques.
n
Value t = 0 = ?cash flows / (1 + k)t i = 1
Where;
Value t = 0 = the estimated value of the asset now (time period 0) Cash flows = the future cash flows resulting from ownership of the asset
k = the appropriate discount rate or rate of return required by an investor for an investment of this type
n = number of periods over which the cash flows are expected
To solve and derive the “intrinsic value of a security”, it is necessary to determine the following:
The expected cash flows from the security. This includes the size and type of cash flows, such as dividends, Interest, face value expected to be received at maturity, or the expected price of the security at some point in the future.
The riming of the expected cash flows. Since the returns to be generated from security occur at various times in the future, they must be properly documented far discounting back to time period 0 (today). Money has a time value, and the timing of future cash flows significantly affects the value of the asset today.
The discount rate, required rate of return demanded by investors. The discount rate used will reflect the time value of the money and the risk of the security. It is an opportunity cost, representing the rate foregone by an investor in the next best alternative with comparable risk.
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Bond Valuation:
The price of a bond should equal the present value of its expected cash flows. The coupons and the principal repayment of $1,000 are known, and the present value, or price, can be determined by discounting these future payments from the issuer at an appropriate required yield, r, for the issue. To solve for the value of an option-free coupon bond.
n
P = ?ct / (1 + r)t + FV / (1 + r) n i = 1
Where;
P = the present Value or price of the bond today (time period 0) c = the semiannual coupons or interest payments
FV = the face value (or par value) of the bond
n = the number of semiannual periods until the bond matures r = the appropriate semiannual discount rate or market yield
In order to conform with the existing payment practice on bonds of paying interest annually rather than annually, the discount rate being used (r), the coupon (ct on the bond, and the number of periods are all on a semiannual basis.
For expositional purposes, we will illustrate the calculation of bond prices by referring to the present value tables at the end of the text; in actuality, a calculator or computer is used. The present value process for a typical coupon-bearing bond involves three steps, given the dollar coupon on the bond, the face value, and the current market yield applicable to a particular bond:
Using the present value of an annuity table, determine the present value of the coupons (interest payments).
Using the present value table, determine the present value of the maturity (par) value of the bond; for our purposes, the maturity value will always be $1,000.
Add the present values determined in steps 1 and 2 together.
BOND PRICE CHANGES:
Bond Price Changes Over Time:
We now know how to calculate, the price of a bond, using the cash flows to be received and the YTM as the discount rate. Assume that we calculate the price of a 20-year bond issued five years ago and determine that it is $910. The bond still has 15 years to maturity. What can we say about its price over the next 15 years?
When everything else is held constant, including market interest rates, bond prices that differ from the bond's face value (assumed to be $1,000) must change over time. Why? On a bond's specified maturity date, it must be worth its face value or maturity value. Therefore, over time, holding all other factors constant, a bond's price must converge to $1,000 on the maturity date because that is the amount the issuer will repay on the maturity date.
After bonds are issued, they sell at discounts (prices less than $1,000) and premiums (prices greater than $1,000) during their lifetimes. Therefore, a bond selling at a discount will experience a rise in price over time, holding all other factors constant, and a bond selling at
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a premium will experience a decline in price over time, holding all other factors constant, as the bond's remaining life approaches the maturity date.
Bond Price Changes As a Result of Interest Rate Changes:
Bond prices change because interest rates and required yields change. Understanding how bond prices change given a change in interest rates is critical to successful bond management. The basics of bond price movements as a result of interest rate changes have been known for many years. For example, over 40 years ago, Burton Malkiel derived five theorems about the relationship between, bond prices and yields. Using the bond valuation model, he showed the changes that occur in the price of a bond (i.e., its volatility), given a change in yields, as a result of bond variables such as time to maturity and coupon. We will use Malkiel's bond theorems to illustrate how bond prices change as a result of changes in interest rates.
Bond Prices Move Inversely to Interest Rates:
Investors must always keep in mind the fundamental fact about the relationship between bond prices and bond yields. Bond prices, move inversely to market yields. When the level of required yields demanded by investors on new issue changes, the required yields on all bonds already outstanding will also change. For-these yields to change, the prices of these bonds must change. This inverse relationship is the basis for understanding, valuing, and managing bonds.
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Effects of Maturity:
The effect of a change in yields on bond prices depends on the maturity of the bond. An important principle is that for a given change in market yields, changes in bond prices are directly related to time to maturity: Therefore, as interest rates change, the prices of longer term bonds will change more than the prices of shorter term bonds, everything else being equal.
The Effects of Coupon:
In addition to the maturity effect, the change in the price of a bond as a result of a change in interest rates depends on the coupon rate of the bond. We can state this principle as (other, things equal): Bond price fluctuations (volatility) and bond coupon rates are inversely related. Note that we are talking, about percentage-price fluctuations; this relationship does not necessarily hold if we measure volatility in terms of dollar price changes rather than percentage-price changes..
BOND YIELDS AND PRICES Contd…
MEASURING BOND PRICE VOLATILITY:
Duration:
In managing a bond portfolio, perhaps the most important consideration is the effects of yield changes on the prices and rates of return for different bonds. The problem is that a given change in interest rates can result in very different percentage-price changes for the various bonds that investors hold. We saw earlier that both maturity and coupon affect bond price changes for a given change in yields. One of the problems, however, is that we examined the effects of these two variables separately.
Duration is a measure of a bond's lifetime that accounts for the entire' pattern of cash flows over the life of the bond
Duration measures the weighted average maturity of a (non-callable) bond's cash flows on a present value basis. We can also say that duration is the weighted average of the times until each payment (coupon or principal repayment) from the bond is received.
Calculating Duration:
To calculate duration, it is necessary to calculate a weighted time period, because duration is stated in years, the time periods at which the cash flows are received are expressed in terms of years and denoted by t in this discussion. When all, of these t's have been weighted and summed, the result is the duration, stated in years.
The present values of the cash flows, as a percentage of the bond's current market price, serve as the weighting factors to apply to the time periods. Each weighting factor shows the relative importance of each cash flow to the bond's total present value, which is simply its current market price. The sum of these weighting factors will be 1.0, indicating that all cash flows have been accounted for. The sum of all the discounted cash flows from the bond will equal the bond's price. The equation for duration is shown as:
n
Macaulay Duration = D = ?PV (CFt) / market price * t i = 1
Where;
t = the time period at which the cash flow is expected to be received n = the number of periods to maturity
PV (CFt) = present value of the cash flow in period t, discounted at the yield to maturity. Market price = the bond's current price or present value of all the cash flows
Understanding Duration:
How is duration related to the key bond variables previously analyzed? The calculation of duration depends on three factors:
• The final maturity of the bond
• The coupon payments
• The yield to maturity
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Duration expands with time to maturity but at a decreasing rate (holding the size of coupon payments and the yield to maturity constant particularly beyond 15 years time to maturity). Even between 5 and 10 years time to maturity, duration is expanding at a significantly lower rate than in the case of a time to maturity of up to
years, where it expands rapidly. Note that for all coupon-paying bonds, duration is always less than maturity. For a zero-coupon bond, duration is equal -to time to maturity.
Yield to maturity is inversely related to duration (holding coupon payments and maturity constant).
Coupon is inversely related to duration (holding maturity and yield to maturity constant). This is logical, because higher coupons lead to quicker recovery of the bond's value, resulting in a shorter duration, relative to lower coupons.
Why is duration important in bond analysis and management? First, it tells us the difference, between the effective lives of alternative bonds. Bonds A and B, with the same duration but different years to maturity, have more in common than bonds C and .D with the same maturity but different durations. For any particular bond, as maturity increases, the duration increases at a decreasing rate.
Estimating Price Changes Using Duration:
The real value of the duration mea-sure to bond investors is that it combines coupon and maturity, the two key variables that investors must consider in response to expected changes in interest rates, As noted earlier, duration is positively related to maturity and negatively related to coupon; However, bond-price changes are directly related to duration; that is, the percentage change in a bond's price, given a change in interest rates, is proportional to its duration. Therefore, duration can be used to measure interest rate exposure.
Convexity:
For very small changes in the required yield the approximation is quite close and at times could be exact. However, as the changes become larger the approximation becomes poorer. We refer to the curved nature of the price-yield relationship as the bond's convexity (the relationship is said to be convex because it opens upward). More formally, convexity is a term used to refer to the degree to which duration changes as the yield to maturity changes.
The degree of convexity is not the same for all bonds. Calculations of price changes should properly account for this convexity in order to improve the approximation of a bond's price change given some yield change.
Convexity is largest for low coupon bonds, long-maturity bonds, and low yields to maturity. If convexity is large, large changes in duration are implied, with corresponding inaccuracies in forecasts of price changes. Therefore, when dealing with securities that have high convexity, the convexity effect on price change must be considered.
Some Conclusions on Duration:
What does this analysis of price volatility mean to bond investors? The message is simple to obtain the maximum (minimum) price volatility from a bond; investors should choose bonds with the longest (shortest) duration. If an investor already owns a portfolio of bonds, he or she can act to increase the average modified duration of the portfolio if a decline in interest rates is expected and the investor is attempting to achieve the largest price appreciation possible. Fortunately, duration is additive, which means that a bond portfolio's
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modified duration is a (market value) weighted average of each individual bond's modified duration.
How popular is the duration concept in today's investment world? This concept has become widely known and referred to in the popular press. Investors can find duration numbers in a variety of sources, particularly with regard to bond funds.
Although duration is an important measure of bond risk, it is not necessarily always the most appropriate one. Duration measures volatility, which is important, but is only one aspect of the risk in bonds. If an investor considers volatility to be an acceptable proxy for risk, duration is the measure of risk to use along with the correction for convexity. Duration may not be a complete measure of bond risk, but it does reflect some of the impact of changes in interest rates.
Zero-Coupon Bonds:
Original issue discount bonds are less common than coupon bonds issued at par. These are bonds that are issued intentionally with low coupon rates that cause the bond to sell at a discount from par value. An extreme example of this type of bond is the zero-coupon bond, which carries no coupons and must provide all its return in the form of price appreciation.
Zeros provide only one cash flow to their owners, and that is on the maturity date of the bond.
What should happen to prices of zeros as time passes? On their maturity dates, zeros must sell for par value. Before maturity, however, they should sell at discounts from par, because of the time value of money. As time passes, price should approach par value. In fact, if the interest rate is constant, a zero's price will increase at exactly the rate of interest.
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