In this unit we begin our study of the interest rate. We start with the components of the interest-rate and build them up to construct the interest rates investors apply to an individual bond.
Up to this point we have taken the interest-rate (discount rate) as given. That is to say, in personal finance questions you were told that your investments earned a certain percentage APR and in the case of bonds you were given the YTM. Now we want to see where those inputs to the TVM are coming from.
Even before this course you were probably familiar with the notion of an interest rate from your transactions with a bank, either through having a bank account or getting a bank loan. We see then that the interest rate is related to a borrowing/lending relationship.
We know from the unit on bonds that the bond market is one place the borrowing/lending relationship exists: the issuer of the bond is borrowing and the buyer of the bond is lending. The resulting price on the bond reflects the interest-rate implicitly agreed-upon by the buyer and seller of the bond.
We will use the bond market to examine the components of the interest-rate by exploiting the inverse relationship between bond prices and interest rates.
We know that the price of any asset is the present value of the cash flows expected to be received from owning the asset. In the case of a bond we have:
When interest rates are higher (larger R) bond prices are lower and vice-versa. Therefore, anything that affects bond prices also affects interest rates. Let’s turn now to what affects bond prices.
Like the price of anything, the price of bonds is determined by the supply and demand of bonds. We can use a simple supply and demand graph to illustrate this.
We can now explore the factors that affect the price of bonds (and thus interest rates) in terms of what affects the supply and demand for bonds.
Supply of Bonds
You recall from your knowledge of bonds that bonds are issued (sold) by firms and governments as a means of borrowing from the investing public. So why do firms and governments need to borrow? Let’s consider firms first.
Firms consist of assets (plant & equipment) that produce the products they sell. These assets need to be maintained. There are times when these assets need to be maintained and the company does not have enough money currently available for the maintenance, so it must borrow the funds from investors. Think about what might happen if the transmission on your car breaks. You may not have enough money in your bank account to pay for the repair. So what do you do? You borrow the money by putting the expense on your credit card. The same sort of situation can happen to companies, except they may borrow to create funds for maintaining their plant and equipment.
A second reason firms may borrow is to expand their current facilities to take advantage of a better business environment. Firm sales are often affected by the general business environment such as recessions and expansions. When the economy expands, customers want to buy more of the firm’s products and the firm may not have enough production capacity to meet the increased demand. Under such circumstances, firms may borrow from the investing public to add onto their existing factories. Also, when the economy is good, firms may introduce new products for consumers and new production facilities may be needed. In both of these situations, firms need to fund their expansion and one way they do this is by borrowing from investors using bonds.
To summarize, firms borrow to
Maintain current plant & equipment
Add to plant & equipment to exploit a favorable economic environment.
Based on these two influences on firms, let’s use our bond supply and demand framework to make a prediction about interest rates.
Question 1: The economy is booming and firms cannot meet existing consumer demand with their current factories. Consequently, the firms decide to build more factories. Based on this scenario, what would you expect interest rates to do?
Answer 1: Firms would need to fund the building of the new factories. The firms would sell more bonds (increase the supply of bonds). As seen in figure 2, the supply of bonds would increase and the price of bonds would decrease. Since bond prices and interest rates move in opposite directions, interest rates would increase.
Demand for Bonds
The demand for bonds comes from investors’ willingness to save. So to get a better understanding of the demand for bonds, we have to look at why investors save. Investors save for a combination of three motivations:
Investors save for a combination of three motivations:
(1) Delayed consumption
(2) Precaution (for a “rainy day”)
The first reason investors save is for “delayed consumption.” Examples of this type of motivation would include saving-up to buy a Christmas gift, or more long term, saving now to pay for your retirement years from now. Investors’ willingness to delay consumption is call their “time-preference” of consumption and it is measured by how much additional future consumption would have to be promised in order for the consumer to not consume now. For example, if I gave you a choice of getting $100 today (which represents a certain amount of purchasing ability today) or waiting one year, virtually everyone would take the $100 today vs. $100 in a year. If you tell me that the minimum future amount you would take to wait a year is $110 (10% more for waiting) then you have revealed to me that your time preference is 10% per year.
The more impatient you are, the greater your time preference will be. The time preference concept applies to whole cultures as well as individuals. The US culture is very impatient so this makes the time preference of the US high relative to other cultures. From this we can observe a pattern regarding delayed consumption.
(1) Delayed consumption:
Less patient –> greater time preference –> lower savings –> lower bond demand –> price of bond decreases –> interest rate increases
Now let’s look at the second reason investors save: (2) Precaution (for a “rainy day”). Everyone has heard the expression that you should save some money for a “rainy day.” No one can predict the future perfectly, so it is considered prudent planning to have an “emergency” fund of cash saved so that when bad events happen (your car breaks down, your furnace stops working) you will have money available to pay for the emergency. The amount you decide to set aside depends on how much you dislike risk, a characteristic we call risk aversion.
What happens to interest rates when people save more for a “rainy day?” Again we can use our bond supply and demand graph to address this question.
If investors set aside more money as a precaution to bad events, then investors will buy more bonds as a means of saving. The demand for bonds will increase, the price of bonds will increase and (because bond prices and interest rates move in opposite directions) interest rates will decrease.
2) Precaution (for a “rainy day”):
Investors save more for a “rainy day” –> demand for bonds increases –> price of bonds increases –> interest rates decrease
The last reason that we believe motivates investors to save is speculation. Here the idea is that on occasion certain financial “deals” arise and you want to have money available to take advantage of the situation. As a simple example, say that your favorite home entertainment store runs a one day sale where everything is half-off. You would want to have cash on hand to take advantage of such a situation. In the investment world, you would like to have some cash in case the stock market takes a temporary drop in price so that you can by when the time is right. We call money set aside for this purpose “speculative” savings.
We can summarize that
Investors save more for speculation –> demand for bonds increases –> price of bonds increases –> interest rates decrease.
Now that we see WHY investors save, we next turn our attention to why investors get compensated for investing.
Compensation for Investing
When you invest (save) today you are doing two things that deserve compensation in the form of return:
Investors get compensated for
(1) Delaying Consumption
(2) Bearing Risk
Once again, we see that to get investors to wait to consume, the investor has to expect to be able to consume more at some future date. Additional future compensation is required even when the additional future compensation is guaranteed. This additional compensation is riskless, so we conclude that the riskless interest rate is affected by investors’ willingness to delay consumption. But we know that the willingness to delay consumption is given by the investors’ time preference of consumption. We therefore can conclude:
The riskless interest rate is affected by investors’ time preference of consumption.
The more patient the investor –> lower the time preference of consumption –> lower the riskless interest rate
The second reason investors get compensated for investing is from bearing risk. All else equal, investors do not like to have their income fluctuate, even if the trend in income is increasing. We call this dislike in income fluctuations risk aversion. The more you dislike your income to fluctuate the more risk averse you are, and the more you would have to earn on average from an investment to be willing to own it.
There are a variety of factors that cause an asset’s return to fluctuate. We call these factors “risk factors.” In order for an investor to be willing to own an asset with a risk factor, the investor must expect to get additional compensation. We call the additional compensation associated with risk a “risk premium.” Each type of risk has its own associated risk premium.
The return to an investor can thus be represented by the following relation.
Expected Return = Risk-Free Interest Return + Risk Premiums
All assets share the risk-free component of return. The risk premiums associated with a particular asset vary depending on the asset. We now consider different risk premiums.
Different Risk Premiums
We will consider three types of risk that investors face when they purchase a bond. These are
First let’s consider default risk. When we talk about default risk, we mean that the party that sold (issued) the bond fails to make a coupon and/or the face value payment to the bond owner on the day agreed upon by the bond’s terms. The likelihood that the issuer defaults on a particular bond it has issued depends on who the bond issuer is. Bonds issued by the US Federal Treasury are considered to have the least likelihood of a default. In fact, the likelihood is considered so low that we consider US Treasury bonds to be default free. Bonds from other issuers, such as local governments or corporations contain default risk. One way we indicate the amount of default risk a bond contains is from the bond’s rating. The “credit quality” of a bond is measured by its “rating.” Three firms provide ratings for bonds: Standard & Poor’s, Moody’s and Fitch Inc. Each bond is given a rating related to the likelihood the issuer of the bond will not be able to make the coupon payments and/or face value on the date indicated by the bond’s terms. A bond’s rating is in many ways analogous to your individual credit score. When a bond has the potential to default, investors will pay less for that bond than they would for a bond with the same cash-flows, but with no chance of default. As a consequence, the YTM on bonds that have the possibility of default is higher than a no-default bond. We can see the consequences of this from the following figure.
Here we compare 4 four risky bonds (left-hand side) with four bonds that are default free (right-hand side). In the case of the bonds with default, bonds 1, 3 and 4 end-up NOT defaulting so the return of those three bonds is high. Bond 2 on the other hand defaulted and the investor earned nothing. The four bonds together earned an average return given by the dashed line.
In contrast, the four default free bonds sell at a high price (and thus have a lower YTM) than the bonds with default. Since these bonds are default-free, all four bonds earn their promised return. Note that the average return for the default-free bonds is lower than the average return for the bonds that can default. The difference is that the income from the default-free bonds is constant, while for the bonds that can default, the income fluctuates.
The second type of risk we look at is maturity risk.
We know that different bonds pay their cash over different investment horizons. We refer to the time to the last cash-flow the bond pays its investors as the bond’s maturity. If two bonds are identical in their coupon rates and face values, investors will pay less for longer maturity bonds because longer maturity bonds face more price risk, reinvestment risk and general uncertainty regarding whether promised cash-flows will be paid. This bias toward lower prices for longer maturity bonds results in longer maturity bonds tending to have higher YTMs.
The last type of risk we will consider for bonds is liquidity risk. Liquidity refers to the extent to which an asset can be converted to cash “quickly” at a price near its intrinsic value. Recall that the price of any asset is the present value of the cash-flows the asset is expected to pay its owners, where the discount rate used reflects the “true riskiness” of the cash-flows. The intrinsic value of an asset is the theoretical price that would result IF investors were able to estimate the true future cash-flows and the true appropriate discount rate. Assets that ARE liquid can be sold quickly at a price near the asset’s intrinsic value, whereas illiquid assets (assets that are not liquid) are sold at prices significantly less than intrinsic value when the asset must be sold on short notice. The table below shows examples of liquid and illiquid assets.
Illiquid (not liquid) Assets Liquid Assets House Cash Land Treasury Bonds Private Equity Stocks Traded on the NYSE
When an investor is thinking about buying an asset, he gathers information about the asset. This information gathering process takes resources (time and money) and we refer to these resources as information costs. Illiquid assets have high information costs because of the heterogeneous nature of these assets. For example, every house has unique characteristic that the buyer must evaluate to determine the proper price of a house. If a house has to be sold on a short notice, the buyer does not have time to make the proper evaluation and will therefore, “play it safe,” and pay a reduced price for the house. The same is true for land and private equity. Also, illiquid assets are characterized by high transactions costs. When buying a house there are significant broker and legal costs associated with the transactions. Illiquid assets are thus characterized as heterogeneous in nature, with high information and transaction costs.
In contrast, liquid assets are homogeneous with low transactions costs. It’s obvious that every dollar is just like every other dollar. Also, when the Treasury issues (sells) a bond the terms of the bond is exactly specified and new treasury bonds will have the same features. This similarity and consistency across Treasury bond features, lowers the information cost of Treasury securities. A similar feature exists with stocks that trade on the NYSE. To be listed on the NYSE, a firm must comply with well specified procedures for reporting its financial information. This lowers the cost to investors to learn about a company. One reason for wanting to be listed on the NYSE is that being listed increases the liquidity of a stock.
Each of these three types of risk commands additional expected return to induce an investor to buy an asset that has these risks. We call this additional expected return a “risk premium.” The expected return from any asset can thus be written as:
Expected Return = Risk free rate + risk premiums
The risk premiums can further be stated as:
Expected Return = Risk free rate + default premium + maturity premium + liquidity premium
These ideas are summarized in concept map 7.
Watch the video components of the interest rate for additional information about these premiums.
Click here for: In-Class Exercise 15: Components of the Interest Rate
What is the term structure of interest rates?
From the many bond examples you did in the bond unit, you know that bonds exist with different maturities. From the discussion on the components of the interest rate, you know that investors demand a different interest rate depending on the maturity of the bond (the maturity premium).
We now want to elaborate on the relationship between a bond’s maturity and the YTM on the bond. Here we formally introduce the idea of the term structure of interest rates.
The term structure of interest rates is the relationship between the YTM on a risk-free, pure discount bond, and the time to maturity of that bond.
Within this definition, we’ve introduced the additional concept of a pure discount bond.
Recall that the cash-flows of a typical bond consist of an annuity component (the coupons) and a lump sum component (the face value). With a typical bond, the investor earns return from current yield and capital gain (price appreciation). With a pure discount bond, the investor earns all of her return from capital gains. Examine the two cash-flows below to clarify the difference.
Using your knowledge of how to calculate the YTM, verify that both bonds have a YTM of 4%. The standard bond provides the 4% to the investor over the life of the bond from the coupon income AND capital gain/loss.
The pure discount bond’s price rises over the life of the bond until it reaches a price of $1,000 at maturity. The investor gains $209.69 in capital gains over the life of the bond.
Why is the term structure important?
Two important reasons the term structure is important are
The term structure of interest provides the foundation for determining the price of any bond.
The term structure of interest contains information regarding the market’s expectation of the future economy regarding economic growth and inflation
The term structure of interest provides the foundation for determining the price of any bond.
You know that the price of any asset is the present value of the cash-flows the asset is expected to pay its owner. The cash-flows of a typical bond consist of an annuity (the coupons) and a lump sum (the face value). In this course we have used a single discount rate to discount all of the bond’s cash-flows regardless of when the cash-flows arrived. In more advanced courses (and in actual financial markets), each cash-flow of a bond is discounted at a different rate corresponding to the date of the cash-flow. These different rates come from the term structure of interest rates.
You can think of using a single discount rates like we’ve been doing as an average of all the different discount rates.
The term structure of interest contains information regarding the market’s expectation of the future economy regarding economic growth and inflation.
We will see shortly in the section on models of the term structure that using the “expectations model” of the term structure, we can get an idea of what the market thinks future interest rates will be.
How is the term structure estimated for the whole economy?
When we examine the term structure of interest rates for the whole economy, we select a risk-free, pure discount bond from available bonds at each maturity and calculate its YTM. An example of this process is shown in the table below.
The table below is an illustration of how you would determine the terms structure of interest rates. This table only shows 4 maturity dates, but in general, one would calculate longer maturities as well.
From the table we have 4 bonds with maturities 1, 2, 3, and 4 years. The face value of each bond is determined by whoever issued the bond. Based on the supply and demand of each bond at each maturity, the price of each bond is determined by market forces with the prices shown in the table.
With the price, maturity and face value data, the YTM of each bond can be determined (make sure you can verify the YTM of each bond).
The term structure is often displayed graphically. A term structure of interest rates graph is just a plot of the information in the table as illustrated below.
In this example, the term structure is “upward sloping” because the YTM at each maturity is higher than the YTM for earlier maturities.
In the next section we discuss models of the terms structure to answer the question “why does a particular term structure graph have the shape it does?”
Term Structure Models: The shape of the term structure
As we have discussed, the term structure of interest rates is the relation between the YTM (on a risk-free, pure discount bond) and the time to maturity. The graphs below indicate some different possibilities that can or have occurred with actual interest rates and maturities.
We would like to know what are investors doing (which bonds they buy and for how much) that would result in each different type of terms structure pattern. Here we will present three models that aim to give us insight into why the term structure takes on the shape it does. The three term structure models we will consider are
• The liquidity preference model • The market segmentation (preferred habitat) model • Expectations hypothesis model
The liquidity preference model
The first theory of the term structure is based on liquidity. Once again we build on what you already know. From your knowledge of the components of the interest rates, you know that, all else equal, investors prefer more liquidity to less liquidity. You also know that short term (short maturity) bonds are more liquid than longer maturity bonds. As a consequence, all else equal, investors will pay more for short term bonds than long term bonds based on their liquidity. Let’s see what this implies regarding the shape of the term structure.
Assume that based on their high liquidity, investors will pay $990 for the 1-year bond. This results in the YTM on the 1-year bond being 1.01% so we have
Next, we know that the 2-year bond is not as liquid as the 1-year bond, so it has to sell at a lower price, let’s say it sells for $966. This would imply the YTM for the 2-year bond is 1.74%. Continuing in this manner, we know that the 3-year bond is less liquid than the 2-year bond, so it has to sell for less than the 2-year bond. Let’s say investors are willing to pay $942 for the 3- year bond. This would imply a YTM for the 3-year bond of 2.01%.
Finally, the 4-year bond has the lowest liquidity so it also has the lowest price of the four bonds, let’s say $913. With a price of $913, the YTM for the 4-year bond is 2.30%. we now see the full term structure for the 4 years.
The liquidity preference model has led to a strictly upward sloping terms structure and this will always be the case. Since we assume that liquidity decreases with the maturity length, longer bonds always have less liquidity and therefore a lower price and finally a higher YTM. The liquidity preference model therefore, cannot explain term structure shapes such as “hump shaped” of “inverted.”
We conclude that there is some validity to the liquidity preference model, but it cannot be the whole story.
The market segmentation (preferred habitat) model
The market segmentation model is the second of our term structure models. It is based on two assumptions:
The market segmentation model assumes that there are different types of investors, and issuers of bonds, whose businesses and preferences lead them to issue and buy bonds of a particular maturity. (this is the preferred part)
Investors and issuers for one maturity do NOT trade bonds of a different maturity (this is the segmented part)
Different types of investors, and issuers of bonds
Different industries and investors may have a preference for a particular maturity of bonds. For example, if you are managing a pension fund you might have a preference for long-term bonds since your clients will not need their money for a long time into the future. In contrast, if you are managing a money market fund, you would have a preference for short term securities that you could easily turn into cash for your clients.
Issuers of bond may also have a preference for bonds of a certain maturity. When interest rates are low, a corporation that is raising large sums of money may want to lock-in the low rate for a long time period by issuing a long maturity bond. In contrast, a company may issue a short-term bond to meet temporary shortfalls in cash.
These preferences for different maturities can essentially create a different bond market for each maturity length. This is shown in the figure below.
Here we have 3 different markets for bonds of maturity 1 year, 5 years and 10 years. Market supply and demand result in the prices shown for each bond. From the prices, we can calculate the YTM for each bond. This is shown in the table. The resulting term structure is shown to be “humped shape.” Which is just one possibility.
Under different market conditions, investors may generate the following for the different bond maturities. In this case we have a strictly upward sloping term structure.
With the segmented market model, we can get any kind of shape for the term structure (a technical exception is that the term structure cannot slope very steeply downward).
The limitation with the segmented market model is that it is unrealistic to believe that investors and bond issuers will not pay attention to what is happening in the markets at every maturity. If interest rates are much lower at one particular maturity, bond issuers will issue more bonds with that maturity. The increase supply of bonds at that maturity will decrease the prices of those bonds, and thus increase those bonds’ YTM. Therefore, there will be feedback between the prices and YTM of bonds across the various maturities. This feedback violates one of the assumptions of the segmented market model and therefore limits the value of its conclusions.
Expectations hypothesis model
The expectations model of the terms structure states that the current term structure of interest rates reflects the market’s current expectation about what future interest rates will be.
The expectations model follows from the fact that investors form expectations about future interest rates and then invest in current bonds with the idea of gaining the maximum return they can expect with the investment strategy they have chosen. This leads to the idea that “any investment strategy that spans the same amount of time, is expected to earn the same amount over the investment horizon.”
Let’s give an example to illustrate this rather abstract notion.
Assume you are given the term structure shown in the following table.
Assume that you would like to invest $1,000 for two years and you can only invest in 1, 2 and 3 year bonds. What are the various ways that you can invest your money for a two year investment horizon?
The most direct way to have a two-year investment is to buy today’s 2 year bond and hold it for 2 years.
Let’s summarize that investment.
- Buy today’s 2 year bond and hold it for two years You will have $1000(1.0175)^2=$1035.31 (this is just future value)
For every $1 you invest in this 2 year bond you will have $1.0353 at the end of two years.
But what if you expected interest rates to go up next year? You would not want to lock-on a two year rate. In this case you could invest the $1,000 in today’s 1 year bond, hold it for one year and then take the $1,125 you would have and buy the “1 year bond that exists in 1 year.” Since this bond doesn’t exist yet, you don’t know what interest rate it will pay. For comparison purposes, assume the 1 year bond in 1 year pays a rate equal to R. With that assumption, we can write an expression for how much you would have from this strategy.
- Invest in today’s one year bond and then, in one year, roll the proceeds into the 1 year bond that exists in 1 year.
You will earn $1,000x(1+0.0125)x(1 +R)
For every $1 you invest in the one year bond and then roll the proceeds into the 1year bond that exists in 1 year, you will have $1,000x(1+0.0125)x(1 +R) at the end of two years. Note that “R” here is the 1 year interest rate, expected 1 year from now.
The expectation hypothesis says that in equilibrium, the two investment strategies should be expected to earn the same amount, which implies: \[\$1,000(1.0175)^2=(1.0125)^1(1+R)^1 \; \rightarrow \; R=0.0225 \; \rightarrow \; R=2.25\% \]
The market expects the one year interest rate, one year from now to be 2.25%. This is shown in the extended table below. Be careful to remember that as of today, only the bonds and interest rates in the shaded area exists. The 2.25% is just what the market expects. The actual 1 year interest rate in 1 year may be higher or lower than 2.25%. Predictions are rarely perfect.
We can apply the principle of “any investment strategy that spans the same amount of time, is expected to earn the same amount over the investment horizon” to find out outher interest rates the market expects.
Notice that the pattern in the term structure of today’s rates (downward arrow) is the same as the pattern in the expected interest rates (horizontal arrow). The current interest rates increase with maturity, which is mirrored in the expected future rates.
The expectations hypothesis would say that the current term structure is upward sloping BECAUSE the market expects future interest rates to rise.
Watch and study the material in video below to improve your understanding of the expectations hypothesis and how to calculate the expected future interest rates.
Expectations Hypothesis Problem Examples
Click here for: Worked Problems 13: Expectation Hypothesis Calcualtions
Click here for: In-Class Exercise 16: Term Structure - Expectations Calculations