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Chapter 34 Measuring yield.


Yield measures the annual (or semi-annual, or some other time period) rate of return on an investment in fixed-income securities (securities that offer a stated rate of return). In evaluating fixed-income investments such as corporate or government bonds, the investor is faced with a variety of measures of investment return or yield: current yield; yield-to-maturity; yield-to-call; after-tax yield; taxable equivalent yield; and realized compound yield.

This chapter presents the basics of bond prices and describes these measures of yield. It also explains advanced topics such as duration and convexity. Finally, this chapter examines the special categories of U.S. Treasury bills (short-term government securities) and convertible bonds.

Fixed-Income Terminology

The terminology associated with fixed-income investing provides a foundation for understanding the differences between seemingly similar yields and ultimately for managing fixed-income investments (i.e., bonds).

Bonds represent debt. The issuer of the bond is borrowing funds. The original purchaser of the bond is effectively making a loan to the issuer. The principal of the bond is the amount the issuer borrows and promises to repay. The principal is also referred to as the face value or par value of the bond. Bonds are usually issued in $1,000 increments and have a specified maturity date on which the issuer promises to repay the principal.

The coupon rate is the interest rate the issuer promises to pay each period on the borrowed funds. The term "coupon" comes from the fact that bonds were formerly issued in "bearer"-form (where the interest and principal were payable to the bearer, or holder, regardless of the person it was originally issued to). With the original bearer-form bonds, the individual or agent in possession of the bond would literally clip a coupon off of the bond and send it in to receive each interest payment. The coupon would note the amount of a specific interest payment and date the payment was due. There was generally one coupon for each interest payment due.

The coupon rate is also known as the stated rate. Most bonds pay interest semi-annually; a small number of bonds pay interest annually or quarterly. Bonds that do not have/pay a coupon rate of interest are called zero-coupon bonds.

Example 1: Palmetto Enterprises, Inc. issues bonds with a total face value of $10,000,000 on July 1, 2006. The bonds pay a coupon rate of 3.5% semi-annually and mature in 5 years on June 30, 2011. Purchasers of each $1,000 bond will receive cash interest payments of $35 every six months, on December 31 and June 30. The company will pay each $1,000 bondholder $35 in interest on each of these dates and promises to repay the $1,000 at maturity.

Bond Pricing

While a bond has a specified face value, the bond cannot necessarily be bought or sold at this price. Circumstances change, and investors will demand different interest rates at different times. Only if the bond offers a coupon rate of return equal to the return then required by investors will it sell for face value. Example 1 presented a bond issued in face value increments of $1,000 and paying interest of 3.5% semi-annually. If a bond investor requires a 3.5% semi-annual rate of return for an investment at this risk level, then $1,000 is a fair price.

But what if a new investor considering a purchase of the bond requires a semi-annual rate of 4% for this bond's level of risk? Since the bond is only paying 3.5% semi-annually, the investor would purchase the bond only if it were offered at a discount from face value.

As with practically any other investment, the current value (selling price) of a bond can be determined by taking the present value of the future cash flows expected to be received:

Equation 34-1

[V.sub.0] [C.sub.1]/[(1 + r).sup.1] + [C.sub.2]/[(1 + r).sup.2] + [C.sub.3]/[(1 + r).sup.3] + ... + [C.sub.n]/[(1 + r).sup.n]

Here, [V.sub.0] is the current value, C is the cash flow for each period (1, 2, 3 ...), r is the required rate of return, and n is the number of periods. Dividing by [(1 + r).sup.n] is the mathematical formula for taking the present value of each cash flow. For example, if an investor expects to receive $1 one year from today and the desired rate of return of 10% (r = 0.10), the present value would be $0.9091 ($1.0/1.1). If the investor expects to receive $1 in two years, the present value would be $0.8264 ($1/1.1 (2)). Rather than using the formula to determine the present value, a present value table for a single sum for each cash flow (see Appendix A), a present value table for an ordinary annuity for the series of cash flows (see Appendix C), or a financial calculator can be used.

In the case of a bond, the cash flows include interest to be received each semi-annual period and the face value to be received at maturity. This can be expressed as:

Equation 34-2

[V.sub.0] [C.sub.1]/[(1 + r).sup.1] + [C.sub.2]/[(1 + r).sup.2] + [C.sub.3]/[(1 + r).sup.3] + ... + [C.sub.n]/[(1 + r).sup.n] + [Face.sub.n]/[(1 + r).sup.n]

In this version of the discounted cash flow equation, C represents each periodic coupon payment and Face represents the face or par value of the bond to be received at maturity (n).

For the Palmetto Enterprises bond the cash flows would be as follows:
  Semi-annual          Cash
    Period             Flow

1                      35.00
2                      35.00
3                      35.00
4                      35.00
5                      35.00
6                      35.00
7                      35.00
8                      35.00
9                      35.00
10                  1,035.00
Total               1,350.00

Note that since the bond matures in five years and there are semi-annual payments, there are ten semi-annual periods. In periods one through nine, the bond investor will receive coupon payments of $35.00 each period. In period ten, the bond investor receives the final interest payment, plus the face value of the bond.

Taking the present value of each cash flow (discounted at the investor's required rate of return, 4% semi-annually) and summing them would produce the following:
Semi-annual            Cash             Present
Period                 Flow              Value

1                      35.00             33.65
2                      35.00             32.36
3                      35.00             31.11
4                      35.00             29.92
5                      35.00             28.77
6                      35.00             27.66
7                      35.00             26.60
8                      35.00             25.57
9                      35.00             24.59
10                  1,035.00            699.21
Total               1,350.00            959.45

While the bondholder will receive nominal cash flows of $1,350, they are worth just $959.45 in today's dollars given a desired rate of return of 4%. Stated another way, if the bondholder invested $959.45 and achieved a semiannual rate of return of 4%, the bondholder would have an equivalent total return at the end of the time period as the promised cash flow stream shown above. The fair price of this bond that would satisfy the investor's required rate of return is therefore $959.45. Since this bond has a par value of $1,000.00, it is said to sell at a discount, or 95.945 percent of par. So whenever the market's desired or required rate of return exceeds the bond's coupon rate of interest, the bond will sell at a discount.

The reverse is also true: If the investor required a rate of return of only 3% semi-annually for a bond of this risk level, an investor would be willing to pay more than the $1,000 par value. The fair price would be determined as above, using a 3% discount rate:
Semi-annual            Cash             Present
Period                 Flow              Value

1                      35.00             33.98
2                      35.00             32.99
3                      35.00             32.03
4                      35.00             31.10
5                      35.00             30.19
6                      35.00             29.31
7                      35.00             28.46
8                      35.00             27.63
9                      35.00             26.82
10                  1,035.00            770.14
Total               1,350.00          1,042.65

The bond would need to sell at a premium, or 104.265 percent of par. At this price, the bondholder would expect to earn 3% semi-annually. Note, however, that the investor receives the coupon rate of interest in cash in all cases.

Let's summarize the present value of the bond given the various desired rates of return:
Desired Semi-annual       Present Value
   Rate of Return            of Bond

4.0%                         $ 959.45
3.5%                        $1,000.00
3.0%                        $1,042.65

This exercise illustrates the important principle that bond prices and returns are inversely related; all else being equal, the higher the desired rate of return, the lower the value of the bond to the investor. This makes sense, as a rational investor that wishes to achieve a certain rate of return, and will contractually receive precisely the stated rate of interest from the bond, can only achieve the desired return by purchasing the underlying bond at a discount or premium - such that the total return, at the end of the period, will be adjusted by the gain or loss on the face value of the bond to achieve this goal.

In practice, the required rate of return is not that of a single investor, but of all investors in aggregate--the market rate of interest. This is the rate of return that market participants require in order to purchase a particular bond or other offerings that are functionally equivalent. All investors will generally be purchasing a bond based upon the market rate of interest because, in the aggregate, any bond that is over-priced relative to the market rate of interest will not be sold until the seller lowers the price, while any bond that is under-priced will be quickly snatched up by market participants until the seller realizes the price can be raised. The market rate of interest is equivalent to the yield realized if the bond is purchased on the open market and is held to its maturity date - and is generally called the yield-to-maturity.

Assume now that the investor purchases the above bond for $1,042.65, which is the price available in the open market. Knowing the semi-annual interest payments, the face value, and the number of payments (as these are all stated quite clearly on the bond), Equation 34-2 can be solved for r, the required rate of return. This tells what rate of interest is required by the market. In this case, the required rate of return is 3% semi-annually. This is the expected rate of return from the investment or its yield-to-maturity.

More formally, the yield-to-maturity can be defined as the rate of return that will make the net present value (present value of future cash flows minus the initial investment) of investing in the bond zero - assuming:

* The bond is held to maturity;

* All promised interest and principal payments are received when due; and

* Coupon payments can be reinvested at the same rate of interest.
Equation 34-3

   Present Value of
   Future Cash Flows
 - Initial Investment
   Net Present Value

The mathematically-inclined reader will note that when the net present value is zero, the yield is equal to, and also known as, the internal rate of return.

For example, as shown above, if the semi-annual yield or internal rate of return is 4%, both the price of a five-year $1,000 bond with a coupon rate of 3.5% and the present value of the future cash flows is $959.45, and the net present value is zero ($959.45 - $959.45 = $0). And, if the semi-annual yield or internal rate of return is 3%, both the price of a five-year $1,000 bond with a coupon rate of 3.5% and the present value of the future cash flows is $1,042.65, and the net present value is zero ($1,042.65 - $1,042.65 = $0).

A much more complex and important question is, "How do investors arrive at the desired or required rate of return?" In the next section, the factors that underlie the market's required rate of return are examined.

Market Interest Rates

The market rate of return is determined by a variety of factors at any given time. There are two primary components to the required rate of return: the risk-free rate and the risk premium.

A risk-free asset is one whose future return can be predicted with near certainty. The typical example is a 3-month Treasury bill, which is virtually risk-free. As a component of the required rate of return, the risk-free rate of return is that return that would be expected if an investment had virtually no risk. The risk-free rate is the amount necessary to properly compensate the lender for the use of his cash before risk is taken into consideration. Expected inflation is a major factor in determining this risk-free rate.

Short of treasury bills though, most investments in bonds (such as corporate bonds) entail at least some degree of risk (such as risk of default). Therefore, a charge must be imposed, in addition to the risk-free rate, to compensate the lender for the risk taken. The additional return that must be paid to compensate investors for such risk is called the risk premium.

One way to assess the risk premium for a bond is to examine the bond's yield spread. The spread is the difference between the bond's yield to maturity and that of a risk-free bond (e.g., a government bond) with otherwise similar underlying characteristics (such as maturity). The spread is measured in basis points, where 100 basis points equals 1%. So if a 20-year U.S. governmental bond were offering a yield of 5.00% and the spread on a AAA-rated corporate bond of the same maturity is 50 basis points, the corporate bond would have a required return (yield to maturity) of 5.50%. Figure 34.1 presents an example of spreads for industrial bonds by maturity and credit rating. Note that for a 10-year industrial bonds rated AAA (the highest rating), a spread of 30 basis points (0.30%) is required, whereas for a CCC-rated bond, (a much lower bond rating, indicating the investor is taking significant risk) a 1,375 basis point (13.75%) spread is required. Since the bonds are being compared to similar-maturity government bonds, and government bonds are generally considered to be risk-free, the spreads represent the risk premium.

Bond Yields

Having reviewed the factors that determine the required return on a bond investment, one must then be sure that the price of the bond reflects those factors. While this should be straightforward, an array of interest rates is quoted by market participants to communicate the price of a single bond. This section is designed to clarify the meaning and distinctions among: coupon rate, yield-to-maturity, after-tax yield, taxable equivalent yield, current yield, realized compound yield, and yield-to-call.

The coupon rate of interest determines the periodic cash interest payment that will be made by the issuer to the bondholder(s). The yield-to-maturity is the effective rate that the bondholder expects to receive based upon the actual selling price of the bond (which may not necessarily be the same rate the issuer pays in interest and principal repayments). Only if the bond is sold at par value will the coupon rate equal the yield-to-maturity. If the bond is callable (redeemable by the issuer prior to maturity at a predetermined price), then it is also useful for the bondholder to compute the expected yield-to-call.

The yield-to-call is the internal rate of return obtained by solving the bond present value equation in Equation 34-2 and using the call price for the principal payment and the number of periods until call for the time to maturity.

Equation 34-4

[V.sub.0] = [C.sub.1]/[(1 + r).sup.1] + [C.sub.2]/[(1 + r).sup.2] + [C.sub.3]/[(1 + r).sup.3] + ... + [C.sub.n]/[(1 + r).sup.n] + [Call Price.sub.n]/[(1 + r).sup.1]

As before, [V.sub.0] is the selling price of the bond. Now n is the number of periods to the call date and Call Price is the price to be received by the bondholder if the bond is called (the call price usually includes a small "call premium," or slight excess above the face value). Call prices are normally expressed as a percentage of face value. For example, a $1,000 par value bond callable at 102 would have a call price of 102% of the $1,000 par, or $1,020. Using the example, the bond is initially offered at par value of $1,000.00 and is callable at the end of six periods (three years) at 102. [V.sub.0] is $1,000, Call Price is $1,020, each coupon payment is $35 and n is 6. Solving for r results in a value of 3.803% semi-annually [r is the discount rate at which the present value of the cash flow to the call date equals the bond purchase price of $1,000, see below]. This is the bond's yield-to-call at issue.
  Semi-annual          Cash             Present
    Period             Flow              Value

1                      35.00             33.72
2                      35.00             32.48
3                      35.00             31.29
4                      35.00             30.15
5                      35.00             29.04
6                   1,055.00            843.32

Total               1,230.00          1,000.00

It was noted earlier that the rate of return actually realized on a bond investment held to maturity could differ from the expected yield-to-maturity (or yield-to-call). This is due to reinvestment risk. Let's reconsider the case of the $1,000 bond from Example 1. If it is sold at par value, then the expected yield-to-maturity is 3.5% semi-annually. If interest rates subsequently fall to 3% before any coupon payments are received, the investor must reinvest those amounts at only 3%. The bondholder's wealth at the end of the period is $1,401.24. This is comprised of the accumulated coupon payments reinvested at 3% semi-annually and the principal that is received at maturity. The realized compound yield can be found by solving an internal rate of return problem. Here, one uses $1,000 as the initial investment (Present Value (1)) $1,401.24 as the wealth at the end of 10 periods (Future Value, see below), and 10 is the number of periods; solving for the rate of return results in a realized compound yield of 3.43% (see below). This is lower than the expected yield-to-maturity of 3.5% since the interest payments were reinvested at a lower rate.
  Semi-annual          Cash             Present
    Period             Flow              Value

1                      35.00             45.67
2                      35.00             44.34
3                      35.00             43.05
4                      35.00             41.79
5                      35.00             40.57
6                      35.00             39.39
7                      35.00             38.25
8                      35.00             37.13
9                      35.00             36.05
10                  1,035.00          1,035.00
Total               1,350.00          1,401.24

The future value of each payment is calculated by multiplying the cash flow by [(1 + i).sup.n], where n is the remaining number of periods until maturity and i is the reinvestment rate. Thus, the future value for the first period equals $45.67 [$35 x [(1 + 3%).sup.9]]. The realized compound yield, r, is then calculated as follows: [(FV / PP).sup.1/p] - 1, where FV equals the future value of payments, PP equals the purchase price of the bond, and p equals the number of periods. Thus, r equals 3.43% [[(1,401.24 / 1,000).sup.1/10] - 1].

Since most bonds (other than municipal bonds) are subject to federal income tax, it is also useful to express all of these yields on an after-tax basis:

Equation 34-5

After Tax Yield = Before Tax Yield x (1 - Tax Rate)

If the expected semi-annual yield-to-maturity is 3.5% and the tax rate is 25%, the after-tax yield would be 2.625% on a semi-annual basis. In the case of a nontaxable bond (municipal bond), the process can be reversed to determine the taxable equivalent yield of a tax-free bond:

Equation 34-6

Taxable Equivalent Yield = Tax Free Yield/(1 - Tax Rate)

If a municipal bond has a tax-free yield-to-maturity of 3% and the tax rate is 25%, then the taxable equivalent yield would be 4.0%. In other words, a tax-free bond yielding 3% is equivalent to a taxable bond yielding 4.0% when the investor's tax rate is 25%.

Both to minimize confusion and because bond math is customarily performed using semi-annual rates, all yields have been expressed in semi-annual form. However, although the math is done on a semi-annual basis, bonds are frequently discussed in terms of annual rates. The coupon rate can be converted to an annual rate by simply multiplying by two. So, the 3.5% semi-annual rate bond pays a coupon of 7.0% annually. However, calculating the actual annualized yield is slightly more complicated, since the interest must be compounded (interest is earned during the second six months on the interest payment received after the first six months). A compound annualized yield can be obtained as follows:

Equation 34-7

Annualized Yield = [([(1 + Non Annualized Yield).sup.n] - 1).sup.*] 100

In this equation, n is the number of compounding periods and Non-Annualized Yield is the periodic yield (such as semi-annual yield) and is expressed in decimal format. So a 3.5% (0.035) semi-annual yield-to-maturity converts to an annualized yield of 7.1225% (a little higher than doubling due to the compounding of interest).

The current yield is an approximation as to how much the bondholder is earning from his investment on a current basis - the yield based upon the interest payments received relative to the exact current price of the bond. It is measured as the annual coupon interest payment divided by the price of the bond. If the $1,000 bond is sold at par, the current yield would be ($35 x 2) / $1000 or 7.0%. When a bond is sold at par, the annual coupon rate, expected yield-to-maturity, and current yield are all equal. For the earlier bond example, if the current market rate of interest is 3% semi-annually (6.09% effective annual rate) and the bond sells at $1,042.65, the current yield would be 6.71% ($70 / $1,042.65). Note that for a bond sold at a premium, the current yield is less than the coupon rate (7%) because the coupon rate essentially assumes the bond is priced at par, and more than the expected yield-to-maturity (6.1%) because of the lower rate that will be earned on reinvested funds If the required semi-annual return were 4% (8.16% effective annual rate), then the bond would sell for $959.45. The current yield would be 7.30%. For a discount bond, the current yield is more than the coupon rate (7%) but less than the expected yield-to-maturity (8.16%) (the bond discount boosts the total return as the bond is purchased for $959.45 but will mature at $1,000).

Figure 34.2 presents bond prices as presented in the Wall Street Journal.

The Goldman Sachs bond is selling at a discount (96.413%) to par value. As a result, the yield-to-maturity is higher than the promised coupon payment. This bond matures in 30 years and offers a spread or premium over ten-year U.S. Government bonds of 155 basis points. (Remember that a basis point is one hundredth of a percentage point, or 0.01%.) The Valero Energy bond on the other hand is offering a higher coupon rate than is required given its risk and it is therefore selling at a premium. The yield-to-maturity is 6.670% versus a coupon rate of 7.50%. Note that both bonds are long term and offer similar yields to maturity in spite of their differing coupon rates.


Interest Rate Volatility

As already seen, there is an inverse relationship between interest rates (required returns) and bond prices; when interest rates rise, bond market prices decline. Interestingly, some bonds are more sensitive to changes in interest rates than others. That is, a given change in interest rates results in a greater change in some bond prices than in others. This section will explore some of the factors that affect a bond's sensitivity to interest rates and thus its price volatility.

Figure 34.3 shows the relationship between the yield-to-maturity and the bond price for the sample bond ($1,000 par value, 3.5% semi-annual coupon, and maturing in 5 years).

This relationship between bond yields and bond prices shows that a 1% (relative) change in interest rates (as measured by YTM) produces some relative change in the price of the bond. (2) From the previous example and the above graph, it is known that if the required market rate of interest is 3.5% semi-annually, the bond price should be $1,000.00. If interest rates decline to 3% semi-annually, then the bond price will rise to $1,042.65. This is usually referred to as a 100 basis point change (50 basis points semi-annually = 100 basis points annually) in interest rates (100 basis points = an absolute 1.00%). The price of the bond rose by $42.65, which is 4.265% of the original price. So a 100 basis point (or 1% absolute) decline in annual interest rates (50 basis points or 0.5% semi-annually) caused the price of the bond to change by 4.265% and in the opposite direction (interest rate decline caused a price increase). This is a measure of the bond's volatility. By definition, a more volatile bond will show a greater change in price for a given change in interest rates.

As an investor in bonds, it is certainly important to understand the potential change in the price of the investment. This factor will be one major determinant in the selection of bonds in which to invest (as it is a major element of risk given that interest rates do in fact fluctuate). Furthermore, the expected volatility of a bond can potentially be used as part of an investment strategy.

Fortunately, a bond's volatility can be estimated using a concept known as duration. Duration is the weighted average time until receipt of all of a bond's cash flows. For the sample bond, the maturity is five years. However, since cash flows occur throughout the bond's life, the weighted average time is less than five years. The duration can be measured precisely by discounting each expected cash flow by the current market rate of interest (4% semi-annually here) and weighting each one by the period in which it occurs:
Semi-annual        Cash         Present        PV times
   Period          Flow          Value          Period

1                   35.00          33.82          33.82
2                   35.00          32.67          65.35
3                   35.00          31.57          94.70
4                   35.00          30.50         122.00
5                   35.00          29.47         147.35
6                   35.00          28.47         170.84
7                   35.00          27.51         192.57
8                   35.00          26.58         212.64
9                   35.00          25.68         231.13
10               1,035.00         733.73        7337.31
Total            1,350.00       1,000.00       8,607.69

The sum of the present value of each cash flow is $1,000.00 which is the bond's current price given a 3.5% required rate of return. The sum of the present value multiplied by the period in which each cash flow occurs is $8,607.69. Dividing this sum by the current price of the bond ($1,000) produces the duration of 8.6 periods ($8,607.69/$1,000), or the weighted average time until receipt of the bond's cash flows. Duration is normally expressed in years rather than periods, 4.3 years in this case.

This process can be expressed by the following formula for duration determined in periods:

Equation 34-8

Dur [n.summation over (t=1)] ([C.sub.t](t)/[(1 + i).sup.t])/[n.summation over (t=1)] ([C.sub.t])/([(1 + i).sup.t])

Where C is each cash flow for time period t. The numerator represents the sum of the present value of each cash flow multiplied by the time period. The denominator represents the sum of each present value (the price). This value may need to be adjusted if the period is not already expressed in terms of years.

Since duration is calculated based upon the current price and current yield-to-maturity, it will change as the price and yield-to-maturity change. The calculated duration is therefore valid only for the current price and yield. This version of duration is more formally known as Macaulay duration, named for Frederick Macaulay, who derived it in a paper published in 1938. As noted above, Macaulay duration is expressed in years, so the Macaulay duration for the example bond is 4.3.

Macaulay duration can also be approximated, avoiding the need to compute all of the present values. The approximation formula is:

Equation 34-9

Dur = 1 + y/Y - (1 + y) + T (c - y)/c [[(1 + y).sup.T] - 1] + y

Where y is the yield-to-maturity, c is the coupon rate, and T is the time to maturity. For the sample bond, using y = .035, c = .035, and T = 10 (all in periods) yields an approximation for duration of 8.6 periods or 4.3 years.

Take this one step further to measure the interest rate sensitivity of a bond. Divide Macaulay duration (in periods) by one plus the periodic (semi-annual in this case) yield-to-maturity.

Equation 34-10

D = Macaulay Duration/1 + y

Where D is modified duration and y is the yield-to-maturity per period (the annual yield-to-maturity divided by the number of compounding periods per year). For the previous sample bond, 4.3/1.035 equals 4.15. Modified duration measures the approximate relative change in the bond price for every 100 basis point (1%) change in annual interest rates. Since there is an inverse relationship in bond yields and prices, the change (A) in the price of a bond can be expressed as:

Equation 34-11

[DELTA]P/P = - D ([DELTA]y) or [DELTA]P/P = Macaulay Duration ([DELTA]y)/1 + y

Where [DELTA]y is the change in annual yield in decimal form. So for the bond at hand, a decline in semi-annual interest rates from 3.5% to 3% is a move of 50 basis points semi-annually or 100 basis points annually. In decimal form this is 0.01 (1%). A decrease of 100 basis points on an annual basis should cause the bond price to change by approximately -4.15*(-.01) = +0.0415 or +4.15%. This is close to the observed price increase of the sample bond above, from $1,000.00 to $1,042.65 (which was 4.265%).

Note that duration provides only an approximation of the percentage change in price for a given (absolute) change in interest rates. Duration, as formulated here, measures the slope of a straight-line tangent to the curve presented in Figure 34.3 at a semi-annual yield of 3.5% and a price of $1,000.00. Note also that, in fact, the relationship shown in Figure 34.3 is not a straight line but a curve. As one moves along any particular tangent line (where the slope of the line represents duration) from the intersection/starting point (i.e., a more substantial interest rate change), the curvilinear relationship 'curves' away and the distance between the tangent line and the actual relationship curve increases. This represents the "error" of the estimate under the duration formula (the tangent line) from the actual effect of an interest rate change (the relationship curve). As a result of the curvature of the relationship, duration is a fairly accurate predictor for very small changes in interest rates, but not for larger changes - even the 100 basis point change presented here partially revealed the inaccuracy.

As a further check of understanding duration, consider the special case of zero-coupon bonds. Investors in "Zeroes" (as this type of bond is often labeled) receive no periodic interest payments. (3) Instead, the interest accrues from the purchase date and is received along with the principal at maturity. Since the desired rate of return is always greater than zero, zero-coupon bonds will always sell at a discount.

Example 2: TRR Financial Enterprises is raising debt to fund a business acquisition. In order to preserve cash flow while integrating this new business, TRR decides to issue zero-coupon bonds with a total face value of $5,000,000. Each $1,000 bond matures in 5 years. Given the risk level of TRR and current market conditions, investors desire a 4% semi-annual rate of return on their investment.

Pricing a zero-coupon bond is easier than pricing a coupon bond, since there is only one cash flow involved. The investor will receive $1,000 at maturity in five years. The present value of $1,000 over 10 semi-annual periods at a 4% semi-annual rate is $675.56 - the price that investors would buy this bond when initially issued. This is 67.556% of par value and a substantial discount to $1,000.

How would an investor calculate the duration of a zero-coupon bond? In this and only this special case, one can see that the duration is exactly equal to the remaining life of the bond. For a 5-year zero-coupon bond that is currently priced at $675.56 and is expected to return $1,000 at maturity, the duration is calculated as follows:

Present value of the (single) cash flow = $675.56

Time to maturity = 5.

So, according to Equation 34-8, duration equals 5 x (675.56) / (675.56) = 5. Therefore, the duration of a zero-coupon bond is identical to its remaining life. This is logical, as duration is the average weighted time to receive the cash flows of a bond - if the only cash flow occurs 5 years from now, then the average weighted time is the average of one number - the time to maturity, or 5 years.

Across all bonds, a higher duration equates to higher bond price volatility. Since volatility is directly related to duration, one can infer that zero-coupon bonds are more volatile than other bonds of the same maturity, all else being equal. Further, bonds with a longer maturity (hence a longer duration) are more volatile than bonds of shorter maturity.

Because the relationship between interest rates and bond prices is curvilinear, as noted earlier, one can modify the measure of duration to predict bond volatility more precisely. Recall from Figure 34.3 and the discussion above that duration measures the slope of a line tangent to the curve at a particular point. However, the relationship between price and yield itself is curvilinear--this phenomenon is called convexity. Convexity is a measure of the curvature of the relationship between bond yields and prices. Specifically, convexity can help explain the movement in bond prices that is not accounted for by duration. Convexity is determined by taking the duration computation one step further and multiplying (PV times t) by (t+1):
Semi-annual       Cash         Present       PV times     (PV times t)
   Period         Flow          Value         Period      times (t+1)

1                   35.00         33.82          33.82          67.63
2                   35.00         32.67          65.35         196.04
3                   35.00         31.57          94.70         378.82
4                   35.00         30.50         122.00         610.01
5                   35.00         29.47         147.35         884.07
6                   35.00         28.47         170.84       1,195.85
7                   35.00         27.51         192.57       1,540.54
8                   35.00         26.58         212.64       1,913.72
9                   35.00         25.68         231.13       2,311.25
10               1,035.00        733.73       7,337.31      80,710.41
Total            1,350.00      1,000.00       8,607.69      89,808.33

The total of the last column is divided by the price of the bond times the number of compounding periods squared times (one plus y) squared.

Convexity for this bond is therefore: 89,808.33 / ((1,000.00 * 4) (1.035)2) or 20.9593.

Incorporating these factors together, the price change in a bond using both duration and convexity can be measured as:

Equation 28-12

[DELTA]P/P = - D ([DELTA] y) + 1/2 Convexity [([DELTA] y).sup.2]

For the bond, the change in price for a 100 basis point decline in interest rates can be measured as (-4.15(-.01) + 0.5 x 20.9593 x (0.0001) = 0.0425 or 4.25%. Recall that the bond in question actually increased in value 4.265%. Although the expected change did not fully anticipate the actual change, it was substantially closer than the initial estimate of 4.15% based upon modified duration alone. Note that convexity is always a good thing for the bond investor, regardless of whether interest rates rise or fall. If interest rates fall, convexity augments the increase in the price of the bond. If interest rates rise, convexity dampens the decline in the price. This is evident in the original example: a 100 basis point decline in rates caused the bond price to increase to $1,042.65, or 4.265%; a 100 basis point increase in rates caused the bond price to decline to $959.45, or (only) 4.055%. It is important to remember that due to convexity the price change for a rise in interest rates is not equal to the (opposite-direction) price change for an equivalent decline in interest rates. Therefore, when evaluating the potential consequences of interest rate changes, both effects should be evaluated separately - particularly for large interest rate changes, where the impact of the curvilinear nature of the relationship (i.e., convexity) is larger.

The combination of duration and convexity allow us to predict movements in the bond's value for certain expected changes in interest rates.

Special Cases

U.S Treasury Bills are short-term securities issued for 13, 26, and 52 week periods. They are issued on a discount basis (non-interest paying zero-coupon bonds), in increments of $1,000 face value. The price quotes for Treasury bills are somewhat unusual compared to other fixed-income investments and warrants some explanation. Consider the following quote from the U.S. Treasury web site (
Term:                   182-Day
Issue Date:             09-28-2000
Maturity Date:          03-29-2001
Discount Rate:          5.985%
Investment Rate:        6.258%
Price Per $100:         96.974
CUSIP:                  912795FZ9

Treasury bill prices are quoted in terms of the Discount Rate, sometimes called a bank discount rate. The price is derived by computing the discount based upon a 360-day year and the maturity in days of the bill:

100 - ((182/360) x (5.985% x 100)) = 96.974

However, to evaluate the bill compared to other fixed-income investments, examine the annualized yield-to-maturity rather than the discount rate. The Treasury refers to this as the Investment Rate or equivalent coupon yield. To compute the yield-to-maturity on a Treasury bill, calculate the internal rate of return for a single period with a present value of 96.974 and a future value (par) of 100.000. The resulting rate is 3.1204%, which is the yield-to-maturity for a 182-day period. The annualized rate listed by the Treasury is 3.1204% x 365/182 or 6.258%. (4) This is not a compounded rate.

Convertible bonds allow the investor to exchange the bond for a pre-specified amount of another security, typically a common stock. (5) While the size of the U.S. market for convertibles is dwarfed by the markets for equities and nonconvertible bonds, it is a market that continues to grow in size and importance along with the growth in related derivative securities (for example, options).

Why do investors purchase convertible bonds? In short, convertibles offer much of the upside potential of an equity security and the downside protection of a bond. The upside portion of this interesting combination stems from the fact that the holder has the right to surrender the bond in exchange for a fixed number of shares of the common stock, which rise in value if the shares appreciate. The downside protection is a result of the combination of two factors. First, the bondholder need not acquire the stock if its price does not rise. Second, provided the issuer remains solvent, the investor can hold the bond to maturity and collect its face value at maturity or sell it to another investor in the interim period.

Convertible bonds pay the coupon rate of interest semi-annually on the face amount of the security. So a convertible bond with a $1,000 face value and a coupon rate of 5.5% will pay $27.50 (i.e., 1/2 x .055 x $1,000) twice each year. In addition, assume this convertible bond can be converted into 16 2/3 shares of common stock. In this case the conversion ratio is 16.667 and implies a conversion price of $60, which is simply the face value ($1,000) divided by the conversion ratio (16.667). To summarize, the convertible investor has the right to acquire 16.667 shares for each $1,000 bond by surrendering the bond for this pre-specified number of shares of the common stock.

In assessing the potential risk of a convertible investment, the prospective investor must break this hybrid investment vehicle into its two components and examine each in turn. Convertibles are called hybrid securities because they have both equity and fixed-income characteristics.

If the current price of the stock of the issuer is $50 and the investor could elect to surrender the bond today in exchange for 16 2/3 shares, then clearly the convertible could not be priced less than $833.33 (16 2/3 shares x $50 per share) in today's market because investors would simply buy and convert it, thereby reaping an immediate (arbitrage) profit. This amount, $833.33, is known as the convertible's conversion value.

One must also consider the fixed-income component alone in the process of assessing the downside risk of this investment. What price would the convertible bond move toward in the event that investors decide today that the common stock has little or no potential to exceed its conversion price? In effect, the convertible bond would then be priced as if it were a straight bond of the issuer. So, if a 20-year (40 coupon payment) straight $1,000 par value bond of this firm with a 5.5% annual coupon rate (payable $27.50 each semiannual period) were priced for a semi-annual yield-to-maturity of 3.5%, the convertible bond would then be priced to yield about the same rate of return.

Using the same approach to value bonds as earlier in this chapter, one could compute its expected market price as the present value of the future stream of interest and principal payments discounted at the appropriate rate (3.5% semi-annually in this case): $839.84. The value of the convertible as a bond alone is called its bond value. This indicates a second possible floor price to which the convertible could descend. Clearly, the convertible will not be priced in the market below its price as a straight bond because it also contains the conversion option, which has some non-zero market value.

The conclusion of the preceding two-stage analysis is that the minimum price of a convertible is the greater of its Bond Value and its Conversion Value. To complete this assessment of the downside risk of a convertible, its conversion premium is defined as the percentage difference between its current market price and its theoretical minimum value. Assuming the bond is currently selling for $1,050 its conversion premium would be $210.16 or 25.02% ($210.16/839.84). This is the premium the market is willing to pay for the potential increase in the stock price.


1. Frank J. Fabozzi, Ed., The Handbook of Fixed-Income Securities, 6th Edition (New York, NY: McGraw Hill, 2000).

2. Annette Thau, The Bond Book, 2nd Edition (New York, NY: McGraw Hill, 2001).

3. Marilyn Cohen and Nick Watson, The Bond Bible (New York, NY: New York Institute of Finance, 2000).


Question--What is the yield-to-maturity for a bond?

Answer--A bond's yield-to-maturity is the rate of return expected by a bond investor based on the current market price of the bond and promised future interest and principal payments.

Question--What is the relationship between investors' required rates of return and bond prices?

Answer--Bond prices are inversely related to investors' required rates of return: the higher the required rate of return the lower the bond price; the lower the required rate of return the higher the bond price. This relationship is a source of bond price volatility. As interest rates rise (fall), investors' required rates of return increase (decrease) and bond prices fall (rise).

Question--What is Duration?

Answer--Duration measures the weighted average time to maturity for a bond. The duration of a zero-coupon bond is the same as its time to maturity. For bonds paying a coupon rate of interest, the duration will be less than the time to maturity. Duration is useful for assessing the sensitivity of bond prices to changes in interest rates. Prices of bonds with higher (lower) duration will be more (less) sensitive to interest rate changes. Duration is most frequently used to approximate the percentage change in the price of a bond given a 100-basis-point change in interest rates.

Question--Consider the following bond: $1,000 par value; 5 years to maturity; 5% annual coupon, paid semiannually. The market yield for a nonconvertible bond with similar characteristics is 5.5%. What should the price of this bond be? Assume the bond sells for this price. What is its compound yield to maturity? What is its initial current yield?

Answer--The price of the bond should be $978.40. The compound yield to maturity would be 2.75% semiannually; 5.58% annually [(1 + 2.75%)2 - 1]. The initial current yield would be 5.11% [$50 / $978.40].
Period             Cash         Present
                   Flow          Value

1                 25.00          24.33
2                 25.00          23.68
3                 25.00          23.05
4                 25.00          22.43
5                 25.00          21.83
6                 25.00          21.24
7                 25.00          20.68
8                 25.00          20.12
9                 25.00          19.58
10             1,025.00         781.46
Total                           978.40


(1.) On some financial calculators this initial investment is considered a cash outflow and must be entered as a negative number to solve for the internal rate of return.

(2.) Throughout the remainder of this discussion, it is important that the distinction between a relative and an absolute percentage change in rates be maintained. For example, when interest rates on a hypothetical bond move from 8% to 9% annually, this would constitute a 12.5% relative change in rates or a 1% absolute (or 100 basis point) change. From this point forward, the parenthetical word "relative" will be omitted when referring to the former type of change. To ensure further clarity, the word absolute or basis points will be explicitly used when referring to changes of the latter type.

(3.) Why would an investor in a fixed-income instrument purchase a bond that makes no periodic interest payments? One reason, which the reader might now suspect, is that the investor has thereby avoided reinvestment risk. The bond is purchased at a discount to its par value and, barring default, the investor receives the yield-to-maturity in the form of price increases (approaching par) as a rate of return until the bond matures.

(4.) The Investment Rate will always be higher than the Discount Rate. Given (with the bond example above) a discount amount (price difference) of $1,000 - $969.74 = $30.26 (accrued over the 182 day period): the Discount Rate is essentially the ratio of the discount amount to the par value, or $30.26 / $1,000; the Investment Rate is essentially the ratio of the discount amount to the purchase value, or $30.26 / $969.74. Since Treasury Bills are always purchased at a discount, this second ratio will always be lower than the first; the Discount Rate will always be lower than the Investment Rate.

(5.) While some convertible bonds can be exchanged for preferred stock or other instruments, this brief presentation of convertibles is confined to those that are most common--those that can be converted into the common stock of the bond issuer.

Year                Project 1      Project 2

0                  ($1,000.00)    ($1,000.00)
1                      400.00           0.00
2                      400.00           0.00
3                      400.00           0.00
4                      400.00           0.00
5                      400.00           0.00
6                      400.00           0.00
7                      400.00           0.00
8                    1,400.00       5,960.00
Annual Return           40%            25%

10% Reinvestment Rate

                              Project 1

          Reinvest Fund      Earnings On
           Before Cash        Reinvest            Cash
Year          Flow              Fund              Flow

0              N/A               N/A           ($1,000)
1              $0.00            $0.00               400
2             400.00            40.00               400
3             840.00            84.00               400
4           1,324.00           132.40               400
5           1,856.40           185.64               400
6           2,442.04           244.20               400
7           3,086.24           308.62               400
8           3,794.86           379.49             1,400

            Project 1         Project 2

           Fund After           Cash
Year        Cash Flow           Flow

0              N/A            ($1,000)
1            $400.00              0.00
2             840.00              0.00
3           1,324.00              0.00
4           1,856.40              0.00
5           2,442.04              0.00
6           3,086.24              0.00
7           3,794.86              0.00
8           5,574.35          5,960.00

Difference in terminal values $ 385.65

Figure 33.1

                           Present Value of Amount Paid @
Outflows       Paid         8%          10%        12.9%

Year 1       -$100,000   -$100,000   -$100,000   -$100,000

                           Present Value of Amount Paid @

Outflows                    20%

Year 1                   -$100,000   (Beginning of year)

                         Present Value of Amount Received @
 Inflows     Received       8%          10%        12.9%

Year 1       $ 10,000      $9,259      $9,091      $8,857
2              10,000       8,573       8,264       7,845
3             120,000      95,260      90,158      83,387

Total of Present
  Value of Inflows       $113,092    $107,513    $100,089

Net Present Value (NPV)   $13,092      $7,513         $89

                         Present Value of Amount Received @

 Inflows                    20%

Year 1                    $ 8,333    (End of year)
2                           6,944
3                          69,444

Total of Present
  Value of Inflows        $84,721

Net Present Value (NPV)  -$15,279

Figure 33.2

                                Present Value of Amount Paid @
 Outflows         Paid            8%            10%            20%

Year 1         -$100,000      -$100,000      -$100,000      -$100,000

Outflows                     Present Value of Amount Paid @

                                12.9%       (Beginning of year)

Year 1                        -$100,000

                               Present Value of Amount Received @
  Inflows       Received          8%            10%            20%

Year 1           $10,000         $9,259         $9,091         $8,333
2                 10,000          8,573          8,264          6,944
3                120,000         95,260         90,158         69,446

Total of Present
  Value of Inflows             $113,092       $107,513        $84,721

Net Present Value (NPV)         $13,092         $7,513       -$15,279

                             Present Value of Amount Received @

  Inflows                       12.9%

Year 1                           $8,857      (End of year)
2                                 7,845
3                                83,387

Total of Present
  Value of Inflows             $100,089

Net Present Value (NPV)             $89

Figure 33.3

                    Period 0         Period 1         Period 2

Cash Flow          (1,000.00)        4,700.00        (8,227.50)

Interest -5%          1.00000         1.05263           1.10803
PV CF              (1,000.00)        4,947.37        (9,116.34)
Cum. PV CF         (1,000.00)        3,947.37        (5,168.98)

Interest 10%          1.00000         0.90909           0.82645
PV CF              (1,000.00)        4,272.73        (6,799.59)
Cum. PV CF         (1,000.00)        3,272.73        (3,526.86)

Interest 25%          1.00000         0.80000           0.64000
PV CF              (1,000.00)        3,760.00        (5,265.60)
Cum. PV CF         (1,000.00)        2,760.00        (2,505.60)

Interest 40%          1.00000         0.71429           0.51020
PV CF              (1,000.00)        3,357.14        (4,197.70)
Cum. PV CF         (1,000.00)        2,357.14        (1,840.56)

                    Period 3         Period 4

Cash Flow            6,356.75       (1,828.78)

Interest -5%          1.16635          1.22774
PV CF                7,414.22       (2,245.23)
Cum. PV CF           2,245.23             0.00

Interest 10%          0.75131          0.68301
PV CF                4,775.92       (1,249.06)
Cum. PV CF           1,249.06             0.00

Interest 25%          0.51200          0.40960
PV CF                3,254.66         (749.06)
Cum. PV CF             749.06             0.00

Interest 40%          0.36443          0.26031
PV CF                2,316.60         (476.04)
Cum. PV CF             476.04             0.00

Figure 33.9

              10%                  Project B
 Year       Factor       Undiscounted     Discounted

0           1.00000        ($1,000)        ($1,000)
1           0.90909             500             455
2           0.82645             300             248
3           0.75131             200             150
4           0.68301             700             478
5           0.62092               0               0
UPI                           1.700
DPI                                           1.331

                   Project C                        Project D

 Year    Undiscounted     Discounted     Undiscounted     Discounted

0          ($1,000)        ($1,000)        ($1,500)        ($1,500)
1               200             182             700             636
2               300             248             515             426
3               500             376             285             214
4               700             478             700             478
5                 0               0             375             233
UPI           1.700                           1.717
DPI                           1.284                           1.325

Figure 33.10

                                   Project B
           Discount          Cash
Year        Factor           Flow         Discounted

0                $0        $(1,000)        $(1,000)
1           0.90909             500             455
2           0.82645             300             248
3           0.75131             200             150
4           0.68301             700             478
5           0.62092               0               0
UPI                           1.700
DPI                                           1.331

                    Project D                      Project D-B

             Cash                            Cash
Year         Flow         Discounted         Flow         Discounted

0          $(1,500)        $(1,500)         $(500)          $(500)
1               700             636            200             182
2               515             426            215             178
3               285             214             85              64
4               700             478              0               0
5               375             233            375             233
UPI           1.717                          1.750
DPI                           1.325                          1.314

Figure 33.11

Initial                            PROJECT B
 Outlay                $1,000 (Outlay B) + $500 Side Fund

            Side Fund       Earnings                     Side Fund
             Begin of       on Side                        End of
  Year      Year Bal.       Fund 10%      Cash Flow      Year Bal.

1              $500.00         $50.00        $500.00      $1,050.00
2             1,050.00         105.00         300.00       1,455.00
3             1,455.00         145.50         200.00       1,800.50
4             1,800.50         180.05         700.00       2,680.55
5             2,680.55         268.05           0.00       2,948.61

Terminal Values                                           $2,948.61
Difference in terminal values

Initial                             PROJECT D
 Outlay                         $1,500 (Outlay D)

            Side Fund       Earnings                     Side Fund
             Begin of       on Side                        at end
  Year      Year Bal.       Fund 10%      Cash Flow       of Year

1                $0.00          $0.00        $700.00        $700.00
2               700.00          70.00         515.00       1,285.00
3             1,285.00         128.50         285.00       1,698.50
4             1,698.50         169.85         700.00       2,568.35
5             2,568.35         256.84         375.00       3,200.19

Terminal Values                                           $3,200.19
Difference in terminal values                               $251.58

Figure 34.1

Rating         1 yr     2 yr     3 yr     5 yr

Aaa/AAA         5        10       15       22
Aa1/AA+         10       15       20       32
Aa2/AA          15       25       30       37
Aa3/AA-         20       30       35       45
A1/A+           30       40       45       58
A2/A            40       50       57       65
A3/A-           50       65       79       85
Baa1/BBB+       60       75       90       97
Baa2/BBB        65       80       88       95
Baa3/BBB-       75       90      105      112
Ba1/BB+         85      100      115      124
Ba2/BB         290      290      265      240
Ba3/BB-        320      395      420      370
B1/B+          500      525      600      425
B2/B           525      550      600      500
B3/B-          725      800      775      800
Caa/CCC       1500     1600     1550     1400

Rating         7 yr    10 yr    30 yr

Aaa/AAA         27       30       55
Aa1/AA+         37       40       60
Aa2/AA          44       50       65
Aa3/AA-         53       55       70
A1/A+           62       65       79
A2/A            71       75       90
A3/A-           82       88      108
Baa1/BBB+      100      107      127
Baa2/BBB       126      149      175
Baa3/BBB-      116      121      146
Ba1/BB+        130      133      168
Ba2/BB         265      210      235
Ba3/BB-        320      290      300
B1/B+          425      375      450
B2/B           450      450      725
B3/B-          750      775      850
Caa/CCC       1300     1375     1500


Figure 34.2

Company (ticker)              Coupon        Maturity        Price

Goldman Sachs Group (GS)       6.450      May 01, 2036      96.413
Valero Energy Corp (VLO)       7.500     Apr. 15, 2032     110.130

                              Last       EST                EST $ Vol
Company (ticker)              Yield    Spread      UST       (000's)

Goldman Sachs Group (GS)      6.729      155       10        57,055
Valero Energy Corp (VLO)      6.670      150       30        50,000
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Title Annotation:Techniques of Investment Planning
Publication:Tools & Techniques of Investment Planning, 2nd ed.
Geographic Code:1U2NY
Date:Jan 1, 2006
Previous Article:Chapter 33 Measuring investment return.
Next Article:Chapter 35 Security valuation.

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