Interest Rates and the Pricing of Bonds



Interest Rates and the Pricing of Bonds

Types of Loans

We will examine four types of loan agreements; simple loan, fixed payment loan, coupon bond, and a discount bond.

Simple Loan

A simple loan is when the borrow repays the amount borrowed plus interest when the loan matures. For example, suppose one borrows 100 at the interest rate of 10%, and the loan matures in one year. Then in one year the borrower pays the lender the principle of 100, plus an interest payment of 10%(100) = 10, for a total payment of 110.

This is called a simple loan because there is only one payment when the loan matures, and it includes both the principle and interest.

Fixed Payment Loan

A fixed payment loan is when the borrower makes periodic payments (e.g. monthly payments) on a loan until the loan is repaid. The fixed payment includes both an interest and principle component. House and car loans are fixed payment loans. For example, if you borrow 6000 to buy a car, you may negotiate with the bank to repay the loan over 36 months, paying the bank 200 per month. At the end of the 36 months you would have repaid the bank 7200, implying interest and principle were repaid.

How can one find the fixed payment? To do so we use the present value formula. Recall,

[pic]

In this case we know the interest rate i and we know the present value of the loan, which is simply the amount borrowed. Finally, since this is a fixed payment loan (meaning every payment in the future is the same amount), it must be [pic]. We will refer to these constant fixed payments as FP.

Now suppose the borrowed amount (i.e. the PV of the loan) is 10000. Also suppose the interest rate is 1%, so that 1+i = 1.05. Furthermore assume that the loan is repaid in 4 years. Then from the present value formula we have

[pic]

Now we must simply solve the equation for FP.

[pic]

Solving for FP we have

[pic]

Coupon and Discount Bonds

Recall that a coupon bond has a face value repaid to the bondholder when the bond matures. The coupon bond also makes regular interest payments to the bondholder. This is called the coupon payment. The coupon payment is equal to the face value times the coupon rate, as stated on the bond. That is,

[pic]

Recall a discount bond promises to pay the bondholder the face value when the bond matures, but there are no coupon payments of interest. Hence a discount bond is sometimes called a “zero-coupon” bond.

Determining the Price of a Discount Bond

The market price of a bond is simply the present value of all its future payments. To demonstrate this we begin with a discount bond, since this has only one future payment.

Note the present value of a discount bond maturing in n years is given by

[pic]

where FV is the face value and P is the price (or present value).

As an example, suppose the face value is 5000, the interest rate is 7%, and the bond matures in 10 years. Then the price of the bond is given by

[pic]

That is, a discount bond paying 5000 in ten years when the current market interest rate is 7% is worth 2542 today.

Effects of changes in interest rate and years to maturity

Now let us consider how a change in the current market interest rate affects bond prices. Consider again the above example in which the face value is 5000 and the bond matures in 10 years. But now suppose the interest rate is 15%. Then the price of the bond is given by

[pic]

Thus we see that an increase (decrease) in the interest rate will cause a decrease (increase) in the price of a bond.

Now consider again when the bond has a face value of 5000 and the interest rate is 7%, but let the years to maturity be 20. Then the price of the bond is given by

[pic]

Thus we see that an increase in the years to maturity will lower the bond price.

Determining the Price of a Coupon Bond

As in the case of the discount bond the price of a bond is the present value at current market interest rates. That is,

[pic]

Where n is the years to maturity. Note that the interest rate i is the current market rate, not the coupon rate.

As an example, suppose the face value is 1000, the coupon rate is 7%, and the years to maturity is 3. In this case the coupon payment is 70 and the pricing formula is given by

[pic]

Now we will find the price of the coupon bond for alternative values of i, the market interest rate. We will consider three cases; when the market interest rate equals the coupon rate, when the market interest rate is less than the coupon rate, and when the market interest rate is more than the coupon rate.

Case 1: Market Interest Rate = Coupon Rate

In this case i = 7%. Hence the pricing formula becomes

[pic]

Hence, when the when the market interest rate equals the coupon rate the price of the bond is equal to its face value.

Case 2: Market Interest Rate < Coupon Rate

In this case let i = 3%. Hence the pricing formula becomes

[pic]

Hence, when the when the market interest rate is less than the coupon rate the price of the bond is greater than its face value.

Case 3: Market Interest Rate > Coupon Rate

In this case let i = 10%. Hence the pricing formula becomes

[pic]

Hence, when the when the market interest rate is more than the coupon rate the price of the bond is less than its face value.

Yield to Maturity

In the above analysis we assumed we knew the market interest rate, the coupon payments, the face value, the years to maturity, and the market interest rate. However, in the real world we observe the price of the bond and infer from it what the implied market interest rate is. This implied interest rate is called the yield to maturity.

For example, suppose a discount bond has a face value of 10000, matures in 2 years, and has a current price of 9000. Then by the present value formula we have

[pic]

We must simply solve the equation for i to find the yield to maturity. That is,

[pic]

So the yield to maturity on a two year discount bond is 5.4%.

In general, the yield to maturity on a discount bond is given by

[pic]

The yield to maturity on a coupon bond is found in a similar way. Suppose the coupon payment is 100, the face value is 1000, and there are 5 years to maturity. Also suppose the price is 800. Then by the present value formula we have

[pic]

From this equation one must solve for i to find the yield to maturity. However, because the interest rate is raised to different exponents there is no simple formula, though there are methods to find the yield to maturity from the above equation.

Nominal vs. Real Interest Rates

When discussing interest rates it is important to understand the difference between nominal and real interest rates.

• The nominal interest rate refers to the percentage increase in money one receives/pays on a loan.

• The real interest rate refers to the percentage increase in purchasing power one has after making a loan.

Of the two, the interest rate that is agreed to on a loan or bond is the nominal interest rate. However, it is clear that it is the real interest rate that is important to borrowers and lenders.

One may wonder why the two interest rates are different. The two interest rates will differ if there is a change in the price of goods/services.

For example, suppose you make a loan today of RO100 at the nominal interest rate of 10%. This implies in one year you will receive RO110. Now also suppose that today the price of a good is RO5. Hence the real value of today’s loan of RO100 is 20 goods. That is, at the price of RO5, RO100 will buy 20 goods (this is found by finding RO100/RO5).

Now suppose the price in one year is still RO5. Then the RO110 received in one year will buy RO110/RO5 = 22 goods. Hence one could say you loaned 20 goods today and received 22 goods in the future. So the purchasing power of your money increased by 10%, the same as the nominal interest rate. Thus we can conclude the following:

If the price of goods is constant over time (i.e. inflation is zero) then the real interest rate will equal the nominal interest rate.

Now suppose the price in one year rises by 10% to RO5.5. In this case the RO110 received in one year will buy RO110/RO5.5 = 20 goods. Hence one could say you loaned 20 goods today and received 20 goods in the future. So the purchasing power of your money increased by 0%. In fact, 10% increase in money combined with a 10% rise in prices, cause the purchasing power to remain constant. Thus we can conclude the following:

If the price of goods rises over time (i.e. inflation is positive) then the real interest rate will be less than the nominal interest rate. In fact, the real interest rate will be equal to the nominal interest rate minus the rate of inflation.

Letting r be the real interest rate, i be the nominal interest rate, and π be the rate of inflation, we have the following formula

[pic]

Inflation Risk

The fact that the real interest rate (which is the one borrowers and lenders care about) depends on inflation implies that if inflation changes unpredictably the real interest rate will change unpredictably; that is, inflation risk creates risk in the real interest rate.

To understand the risk better, consider the following definitions:

• Let πe be the expected inflation rate

• Let πa be the actual inflation rate

• Let re be the expected real interest rate

• Let ra be the actual real interest rate

In this case we have

[pic]

And

[pic]

Hence the difference between the actual real interest rate and the expected real interest rate is given by

[pic]

Which implies

[pic]

This says that if the actual inflation rate is greater than expected inflation, then the actual real interest rate will be less than the expected real interest rate. Hence the more difficult it is to predict inflation, the more risk one is exposed to. For this reason, uncertainty in inflation can have a negative effect on bond markets.

Exchange rate risk on foreign bond holdings

We consider now the possibility of holding foreign bonds, which exposes the investor risk owing to changes in the exchange rate. To begin let us define [pic] as the foreign nominal interest rate from time t to t+1. Let us also define the amount of domestic money invested in the foreign bond as the amount [pic]. Now let [pic] be the exchange rate of the number of domestic currency for one unit of foreign currency. Hence the amount [pic]will convert to [pic] of foreign currency and will earn an interest rate of [pic]. Hence in one year’s time, at time t+1, the foreign bond will pay [pic] in foreign currency. This must now be converted back into domestic currency. At time t+1 the exchange rate is given by [pic]. Since this is the amount of domestic currency for one unit of foreign currency, the exchange rate to convert foreign currency into domestic currency is simply [pic]. Hence at time t+1 this foreign bond will pay in domestic currency the amount.

[pic].

Now to compute the return on this foreign bond, in domestic currency, we find

[pic]

Since [pic] is close to zero, we can neglect it, and we write the return as

[pic]

Notice the return on the foreign bond depends on the rate of change in the exchange rate.

Interest Rate Risk

The above explains that bondholders face a risk due to inflation. But there is another risk bondholders face. When considering one year rates of return on bonds, bondholders face a risk in their annual rate of return that derives from the fact that the current market interest rate can change. Recall that if the market interest rate rises (falls), the price of an existing bond will fall (rise). Since future interest rates cannot be perfectly predicted, this means the future price of one’s bond holdings cannot be perfectly predicted, and hence the risk.

To understand this better, let us consider the rate of return on a bond. The rate of return is given by the following:

[pic]

Where I is the interest payment, P0 is the price one pays for the bond, and P1 is the price of the bond after one year. If we split this into two parts we have

[pic]

The first terms is the interest payment divided by the purchase price of the bond. This is simply the interest rate, and is known at the time of the bond purchase.

The second term is the capital gain or loss on the bond. It is not known at the time the bond is purchased and thus represents a source of risk.

Now recall that the price of a bond falls with an increase in the interest rate. This is implies if one buys a bond and then the interest rate rises, the price will fall and P1 – P0 will be negative; that is, the bondholder will experience a capital loss.

To understand the effect better let us consider an example. Suppose P0 = 1000, i = 10%, and there are two years to maturity. Now suppose after one year the market interest rate rises to 20%. Using the present value formula, the new price of the bond (now with only one year to maturity) is given by

[pic]

Thus the rate of return on the bond is given by

[pic]

Hence the rate of return is 10% minus the capital loss experienced. Now note that the longer the time to maturity the greater is the price change in a bond following an interest rate increase.

To illustrate this consider again the example above in which P0 = 1000, i = 10%, and after one year the market interest rate rises to 20%. However, assume that at the time of purchase the bond has three years to maturity. This implies that after one year the bond has two years to maturity. So using the present value formula, the new price of the bond (now with only two year to maturity) is given by

[pic].

So notice that while the two year bond fell in price from 1000 to 917, the three year bond fell in price from 1000 to 846. The reason is simply due to the fact that denominator in the present value formula is being raised to a higher exponent.

Given the fall in price to 846, the rate of return on the bond is given by

[pic]

Given that the longer the time to maturity the greater is the price change in a bond following an interest rate increase. This implies that bonds of longer maturities will have a greater capital loss when the interest rate increases. The following table shows the results for bonds of different maturities. In all cases, the initial interest rate is 10%, the market interest rate rises to 20%, and the initial price is 1000.

|Years to Maturity when |P1 |Initial Interest Rate |Capital Loss |Rate of Return |

|Purchased | | | | |

|2 |917 |10% |-8.3% |1.7% |

|5 |741 |10% |-25.9 |-15.9 |

|10 |597 |10% |-40.3 |-30.3 |

|20 |516 |10% |-48.4 |-38.4 |

|30 |503 |10% |-49.7 |-39.7 |

As one can see, the longer the time to maturity, the greater the risk associated with an interest rate change.

Term Structure of Interest Rates

As explained above, the yield to maturity is the implied interest rate from a bond with a particular maturity date. The term structure compares the interest rate of bonds of different maturities. Often this is expressed graphically, as below, in what is known as the Yield Curve.

[pic]

In this graph we see that a 1 year bond has a current yield of 4%, a 5 year bond has a current yield of 6%, and a 10 year bond has a current yield of 8%.

Three Facts about Yield Curves

1. Yields on bonds of different maturities move together. That is, when yields on short-term bonds rise, we also observe that yields on long-term bonds rise.

2. When the yields on short-term bonds are low, the yield curve is more upward sloping.

3. Yield curves are usually upward sloping, as in the graph above.

Expectations Hypothesis

We now attempt to explain the above three facts regarding yield curves by a theory called the Expectations Hypothesis.

Before we begin let us introduce some notation. Let

• [pic]be the annual interest rate on a 1-year bond at time t

• [pic]be the annual interest rate on a 2-year bond at time t

• [pic]be the annual interest rate on a n-year bond at time t

• [pic]be the expected annual interest rate on a 1-year bond at time t+j

The basic idea of the expectations hypothesis is that to see bonds of different maturities as substitutes. For example, suppose one wants to invest in bonds for two years. There are two methods of doing this.

Method 1

Invest in one 2-year bond. Such a bond will pay an annual interest rate of [pic]. Hence one receives this interest rate two times. So the total interest received with this method is 2[pic].

Method 2

Invest in two 1-year bonds. That is, at time t invest in a one year bond that pays [pic]. Then at time t+1 reinvest in another 1-year bond, which pays an expected interest rate of [pic]. Hence the total interest paid with this method is [pic].

Since these two methods are substitutes we expect their interest rates to be the same. That is, we expect

[pic]

Or, dividing by 2 we have

[pic]

This is our equation of the expectations hypothesis. It links the annual interest rate on a long-term bond (i.e. 2-year bond) to the annual interest rates on 1-year bonds, both current and expected future. In fact, a 2-year bond’s annual interest rate is just the average of current and expected future annual interest rates on 1-year bonds over the next two years.

As an example, suppose [pic]and [pic], then

[pic]

Now what is true for the two-year bond is true for three-year bonds, four-year bonds, etc. That is, an n-year bond’s annual interest rate is just the average of current and expected future annual interest rates on 1-year bonds over the next n years. Hence we have

[pic]

[pic]

[pic]

.

.

.

[pic]

Explaining the Three Facts of Yield Curves

Fact 1: Yields on bonds of different maturities move together. That is, when yields on short-term bonds rise, we also observe that yields on long-term bonds rise.

By the construction of equations above for yields of bonds of different maturities it is obvious that if today’s yield on a 1-year (i.e. [pic]) bond rises, then the yields on all bonds of different maturities will rise. That is, [pic]is in the numerator of [pic], [pic],…,[pic] thus as [pic]rises all the others must also rise.

For example, before we had [pic]and [pic], then

[pic]

Now if we let [pic], then [pic]

Thus we conclude that the expectations hypothesis explains this first fact about yield curves.

Fact 2: When the yields on short-term bonds are low, the yield curve is more upward sloping.

Also from the construction of equations above for yields of bonds of different maturities it is clear that fact 2 can be explained. This fact implies that when the current yield on a 1-year bond is low, and all else is constant, there will be a bigger different between yields on 1-year bonds and long-term bonds. To see this return to our example where [pic]and [pic], and compare this to a case where [pic] falls to 1%. We then have

Case 1: [pic]

Case 2: [pic]

If we plot these two cases on the same graph we have the following:

[pic]

As one can see, the slope of the yield curve goes up when the short-term yield is lower.

Fact 3: Yield curves are usually upward sloping, as in the graph above.

An upward sloping yield curve implies [pic]. Let us just consider the statement that [pic]. Given that [pic], in order for [pic] it must be the case that [pic]. Hence according to the expectations hypothesis yield curves are only upward sloping if the expected future short-term yields are greater than the current short-term yields.

Or, said differently, yield curves are only upward sloping if one expects the short-term interest rates to rise in the future. However, while we expect the short-term interest rates to rise sometimes, we also expect them to fall just as often. Hence we should see downward sloping yield curves as much as upward sloping yield curves according to the expectations hypothesis.

Moreover, if we believe that short-term interest rates have a long-run natural level to which they go, then we would expect [pic]. This would imply that [pic], and the yield curve would be flat. Hence, we would expect flat yield curves if the expectations hypothesis is true. Thus we must conclude that the expectations hypothesis by itself cannot explain fact 3.

Amending the Expectations Hypothesis

How can we amend the expectations hypothesis to explain fact 3? Recall that long-term bonds are subject to more interest rate risk than short-term bonds (see page 9). What this means is that bonds of different maturities are not perfect substitutes for each other, as the expectations hypothesis assumes. In fact, since longer-term bonds have more risk people need a risk premium to be willing to hold such bonds. That is, they need the interest payments to be higher on a longer-term bond than a short term bond. And since the longer the time to maturity the greater the risk, it is also true that the longer the time to maturity the greater the risk premium. Thus, if we let kj be the risk premium on a bond of j years to maturity, then it must be that [pic]. Hence our equations for the yields of bonds of different maturities becomes

[pic]

[pic]

[pic]

.

.

.

[pic]

To see how this implies that yield curves will normally be upward sloping, consider the case in which [pic]. In this case we would have

[pic]

[pic]

[pic]

.

.

[pic]

But since [pic] it must be that [pic]; that is, the yield curve is upward sloping due to the risk premiums on longer-term bonds.

Coupon Bond Yields, Prices, and Interest Rate Risk

Coupon bonds, sometimes known as Straight Bonds, pay a coupon payment every six months and pay the face value at maturity. The coupon payment is given by

[pic]

Note it is divided by two because the coupon rate is an annual rate, but the payment is given twice a year.

Notice then one can write

[pic]

However because the bond price can differ from the face value when the current market interest rate differs from the coupon rate, we can also identify the current yield on a coupon bond as

[pic].

The current yield is expressing what the annual coupon payments translate to in terms of an interest rate given the purchase price of a bond. However, this does not reflect the face value of the bond returned when the bond matures. The yield to maturity (YTM) is the measure of the yield on the bond, given the bond price, that is the present value of all cash flows from the bond, including the face value when the bond matures. Bond prices must therefore adjust so that the future cash flows reflect the current market interest rate, or yield to maturity. For a coupon bond maturing in n years and paying coupon payments every six months, the bond price is given by

[pic]

Where YTM = yield to maturity (i.e. current market interest rate)

C = Annual coupon payment

n = years to maturity

FV = face value

Note that in the calculation of present value, YTM is divided by two since the payment is made twice per year, but then it is raised to the power of 2n because if there are n years to maturity, there are 2n periods to maturity, where each period is for six months.

As before the bond pricing formula shows that as the YTM rises (falls), the bond price will fall (rise). That is, the bond prices will move inversely with the YTM. This gives rise to the following terms:

Premium Bonds: Bond Price > Face Value…this occurs when the YTM < Coupon Rate

Discount Bonds[1]: Bond Price < Face Value…this occurs when the YTM > Coupon Rate

Par Bonds: Bond Price = Face Value…this occurs when the YTM = Coupon Rate

It should be noted that the size of the discount or premium is positively related to the time to maturity. The longer the time to maturity, the larger is the discount or premium if the YTM is different from the coupon rate. Or stated differently, the shorter the time to maturity the smaller is the discount or premium.

Relationship Among Yield Measures

Given we have two yield measures (current yield and yield to maturity) it is important to understand their relationship to each other, and to the coupon rate. This relationship can be summarized as follows:

Premium Bonds: Coupon Rate > Current Yield > YTM

Discount Bonds: Coupon Rate < Current Yield < YTM

Par Bonds: Coupon Rate = Current Yield = YTM

Interest Rate Risk on Coupon Bonds

As we saw when looking at zero coupon bonds, changes in the market interest rate causes changes in bond prices, which then exposes investors to risk. This risk is termed interest rate risk. There are five statements, or theorems, that relate to some important relationships among bond prices, maturities, coupon rates, and yields. These are described by Burton Malkiel’s five bond price theorems (see page 326 of text). These five theorems are:

|Malkiel’s Theorems |

|1. Bond prices and bond yields move in opposite directions. As a |This just says that bond prices and the YTM move in opposite |

|bond’s yield |directions, as mentioned above. For example, in Table 1 below, as|

|increases, its price decreases. Conversely, as a bond’s yield |the YTM rises, all bond prices fall. |

|decreases, its price | |

|increases. | |

|2. For a given change in a bond’s yield to maturity, the longer |This establishes what was mentioned earlier when evaluating zero |

|the term to maturity of |coupon bonds. The longer the time to maturity, the more sensitive|

|the bond, the greater will be the magnitude of the change in the |is the bond price to a change in the YTM. For example, in table 1|

|bond’s price. |below we see the price difference for the 20 year bond is greater|

| |than that of the 10 year bond, which is greater than that of the |

| |5 year bond. |

|3. For a given change in a bond’s yield to maturity, the size of |We have established that the longer the time to maturity the |

|the change in the |greater then change in the bond price when the YTM changes. This |

|bond’s price increases at a diminishing rate as the bond’s term |theorem simply says that as YTM falls, the bond price increases a|

|to maturity |smaller amount as maturity is longer. So for example, in table 1 |

|lengthens. |below we see if the YTM rises by 2%, the price difference in the |

| |10 year bond is 68% more than the price difference in the 5 year |

| |bond, but the price difference in the 20 year bond is only 46% |

| |more than that of the 10 year bond. |

|4. For a given change in a bond’s yield to maturity, the absolute|This theorem says that the lower coupon rate bond are more |

|magnitude of the |sensitive to changes in the YTM than higher coupon rate bonds. |

|resulting change in the bond’s price is inversely related to the |For example, consider Table 2 below, and look at the case in |

|bond’s coupon rate. |which the yield rises from 8% to 10%. The 6% coupon bond falls in|

| |price from 802 to 657, which is a fall of 18%. The 8% bond falls |

| |in price from 1000 to 828, which is a fall of 17%. Hence the |

| |lower coupon rate bond is more responsive. |

|5. For a given absolute change in a bond’s yield to maturity, the|This just says that when looking at price changes caused by |

|magnitude of the price |changes in yield to maturity, the price increase coming from |

|increase caused by a decrease in yield is greater than the price |decreases in YTM are larger than the price decreases coming from |

|decrease caused by an |increases in YTM. For example, consider table 2 below. Look at |

|increase in yield. |the 8% coupon bond. Starting from a YTM of 8%, a 2% decrease in |

| |the YTM causes the bond price to rise by 231, but a 2% increase |

| |in YTM causes the bond price to fall by only 172. |

The table below illustrates the first three theorems (from page 327of the text).

|TABLE 1 |

| | |

| |Time to Maturity for a $1000 bond with an 8% coupon rate |

|YTM |5 years |10 years |20 years |

|7% |1042 |1071 |1107 |

|9% |960 |935 |908 |

|Price difference |82 |136 |199 |

The table below illustrates the last two theorems(from page 327of the text).

|TABLE 2 |

| | |

| |Coupon Rates for a $1000 bond with 20 years to maturity |

|Yields |6% |8% |10% |

|6% |1000 |1231 |1462 |

|8% |802 |1000 |1198 |

|10% |657 |828 |1000 |

Duration of the Bond

What the above tells us is that not only does interest rate risk depend on the maturity of the bond (as it does with zero coupon bonds), but interest rate risk also depends on the coupon rate on the bond (theorem 4). That is, a bond’s price sensitivity to interest rate change depends on both the term and the coupon rate.

To capture the interest rate risk on a bond we then need a measure that reflects both the term of the bond and the coupon rate. Such a measure is called duration.

Duration is sometimes called the effective maturity. It measures how long until the bond repays the purchase price of the bond. Or stated differently, it measures in years the amount of time it takes to recover the money invested in the bond. It is important to remember that both the term of the bond and the coupon rate will determine the duration.

It turns out the duration of the bond can be used to easily estimate the sensitivity of a bond’s price to a change in the YTM. One measure of duration is called the Macauley Duration. Using this measure one can show

[pic]

Notice this provide a very convenient method of finding the percentage change in bond prices. Also note that since both the term of the bond and the coupon rate contribute to duration, the duration measure can be the same for bonds of different maturities, and thus the interest rate risk can be the same for bonds of different maturities.

An even simpler measure of duration is called the modified duration. It is given by

[pic]

Hence

[pic]

Calculating Duration

As stated above, duration measure the effective maturity; the number of years to recover the price paid for the bond.

A formula for calculating duration appears on page 331 of the text as equation (10.9). However a better understanding of duration can be achieved when looking a particular example. Suppose we have a 1000 face value bond that has a coupon rate of 8% that make semi-annual coupon payments of 40. Also suppose the current YTM is 8%, so the bond price equals the face value of 1000. Finally suppose there are three years to maturity. Then the calculations for the duration are as appears in Table 3 below.

|Years |Cash Flow |Discount Factor =1/1.08t |Present Value = cash |Years x Present Value |

| | | |flow*discount factor |/Bond Price |

|0.5 |40 |0.9615 |38.46 |.0192 years |

|1 |40 |0.9246 |36.99 |.0370 |

|1.5 |40 |0.8890 |35.56 |.0533 |

|2 |40 |0.8548 |34.19 |.0684 |

|2.5 |40 |0.8219 |32.88 |.0822 |

|3 |1040 |0.7903 |821.93 |2.4658 |

| | | |1000 |2.7259 year |

| | | |Bond price = PV |Bond duration |

PROPERTIES OF DURATION

Macaulay duration has a number of important properties. For coupon bonds, the basic properties

of Macaulay duration can be summarized as follows:

1. A zero coupon bond has a duration equal to its maturity.

2. All else the same, the longer a bond’s maturity, the longer is its duration.

3. All else the same, a bond’s duration increases at a decreasing rate as maturity

lengthens.

4. All else the same, the higher a bond’s coupon, the shorter is its duration.

5. All else the same, a higher yield to maturity implies a shorter duration, and a lower

yield to maturity implies a longer duration.

The graph below comes from page 333 of the text and shows the relationship between bond maturity and bond duration for different coupon rates on a 20 year bond.

[pic]

-----------------------

[1] We used the term discount bond before to refer to a zero coupon bond. A zero coupon bond only pays the face value when the bond matures. Hence the only way the bond pays a rate of return is if the bond sells at a discount from the face value. Hence, zero coupon bonds are also called discount bonds.

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Yield

Time to Maturity

1 5 10

8

6

4

4

Yield

Time to Maturity

1 2

7

5

1

Case 1

Case 2

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