Understanding Interest Rates PDF

1 UNDERSTANDING INTEREST RATES Organization of the chapter Types of Credit Market Instruments - 2 Present Value Calculations & Yields – 2 – 8 Yield to Maturity - 3 Current Yield - 4 Holding Period Returns - 5 Interest Rates and Returns - 6...

1 UNDERSTANDING INTEREST RATES Organization of the chapter Types of Credit Market Instruments - 2 Present Value Calculations & Yields – 2 – 8 Yield to Maturity - 3 Current Yield - 4 Holding Period Returns - 5 Interest Rates and Returns - 6 Yield Calculations for Money Market Instruments - 7 The Behavior of Interest Rates – 8 – 21 Theory of Portfolio Choice - 8 Demand and Supply in the Bond Market - 9 Risks associated with bonds - 15 The Liquidity Preference Framework 17 The Risk and Term Structure of Interest Rate – 21 – 28 Risk Structure of Interest Rates - 21 Term Structure of Interest Rates - 23  2 UNDERSTANDING INTEREST RATES I. TYPES OF CREDIT MARKET INSTRUMENTS Different instruments have very different streams of cash payments to the holder, called cash flows, with different timings. In terms of the timing of cash flow payments, there are four basic types of credit market instruments: 1. Simple loan – where a sum is lent with a maturity date at which the borrower must pay the principal amount with an additional interest payment. Interest rates could be simple or compound interest rates. 2. Fixed payment loan or fully amortized loan – here the lender provides the borrower with an amount of funds which must be repaid by making the same payment every period, consisting of a part of principal and interest for a set number of years. Auto loans or mortgages are frequently of the fixed payment type. 3. Coupon Bond – a coupon bond pays the owner of the bond a fixed interest payment, called coupon payment, every year until the maturity date when a specified final amount, called face value or par value or simply par, is repaid. 4. Discount Bond – a discount or zero-coupon bond is bought at a price below its face value and the face value is repaid at the maturity date. Unlike a coupon bond, a discount bond generally does not make any interest payments. II. PRESENT VALUE CALCULATIONS & YIELDS Different instruments are compared using the concept of present value or present discounted value. It is the discounted value of future cash flows or income. For instance, if Rs 100 is lent out with a maturity period of 1 year and interest payment of Rs 10, then after one year the total payment would be Rs 110. If this money is further lent out then after one year total payment would be 110  (1+0.10) = 121. Alternatively, with compound interest rate of 10 percent Rs 100 would become Rs 121 following the formula. 100  (1+0.10) (1+0.10) = 100  (1+0.10)2 = 121 Generalising, we obtain that after n years the payment would be 100  (1+0.10)n. Now working backward, the present value of a future payment, say 121, can be calculated as 100 = which canbe again generalised for n years as100 = where CF is cash (. ) (. ) flow in n period where 100 is the present value (PV).  3 Yield to Maturity This is the most important and accurate measure of interest rates that equates the present value of cash flows from a debt instrument with its value today. In order to explain the concept let us take up examples of calculations of yield to maturity for the four types of credit market instruments. For a simple loan of Rs 100 if the payment in one year’s time would be Rs 110, then the yield to maturity is calculated from 100 = which would give i = 0.10 or 10 percent. ( ) For a fixed payment loan, since there is more than one cash flow payment, today’s value of the loan is equated with the sum of the present values of all cash flow payments. Suppose the loan amount is Rs 1000 and the borrower makes monthly payment of Rs 125 for 24 months. Then i ( ) is calculated from 1000 = + + + ⋯+ = as i = 0.116 or ( ) ( ) ( ) ( ) 12 percent per month. The yield to maturity for a coupon bond is also calculated following the same strategy as the one used for fixed payment loan. However, beside present values of the coupon payments, present value of the final payment of the face value is also included. Thus if the price of a coupon bond is Rs 900 and the face value is Rs 1000 with a coupon rate of 10 percent (i.e. annual coupon payment of Rs 100) and eight years to maturity the yield to maturity can be calculated from the equation 900 = + + ⋯+ + to obtain i = 0.12 or12 ( ) ( ) ( ) percent. It should be noted that first, when the coupon bond is priced at its face value then the yield to maturity equals the coupon rate. Second, the price of a coupon bond is negatively related to the yield to maturity. For instance, if the price increases to Rs1000, then i would decrease and become equal to the coupon rate of 10 percent. And third, the yield to maturity is greater than the coupon rate when the bond price is lower than the face value. A special case of a coupon bond is a perpetual bond with no maturity and no payment of face value, called a consol or perpetuity. They were introduced by the British Treasury during the Napoleonic Wars and are still traded in Britain, though quite rare in other places. The yield to maturity for a consol is calculated from the formula, 𝑃 = where P is the present value, C is the perpetual coupon payment and i is the yield to maturity.  4 The calculation for yield to maturity for a discount bond is similar to that for a simple loan. If a discount bond is purchased at Rs 900 with a face value of Rs 1000 for one year then i would be calculated from 900 = or i = = 11 percent. Current Yield Current yield is a commonly used, easy-to-compute measure of the proceeds the bondholder receives for making a loan. It is the yearly coupon payment divided by the price: 𝑦𝑒𝑎𝑟𝑙𝑦 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 = 𝑝𝑟𝑖𝑐𝑒 𝑝𝑎𝑖𝑑 Looking at this expression, we can see that the current yield measures that part of the return from buying the bond that arises solely from the coupon payments. It ignores the capital gain or loss that arises when the price at which the bond is purchased differs from its face value. So if the price is below par, the current yield will be below the yield to maturity. Let’s consider a one-year 5 percent coupon bond and assume that it is selling for $99. The current yield is easy to calculate as 5 = 0.0505 99 or 5.05 percent. The yield to maturity for this bond is the solution to 5 100 + = 99 1+𝑖 1+𝑖 which is 6.06 percent. The yield to maturity is higher because, if you buy the bond for $99, one year later you get not only the $5 coupon payment but also a guaranteed $1 capital gain for a total of $6. We can repeat these calculations for a case in which the bond is selling for $101. Then the current yield is 5 = 0.0495 101 or 4.95 percent, and the yield to maturity is 5 100 + = 101 1+𝑖 1+𝑖  5 or 3.96 percent. Putting all this together, we see the relationship between the current yield and the coupon rate. Again, it comes from the fact that current yield moves in the opposite direction from the price: it falls when the bond’s price goes up and rises when the price goes down. So when the price equals the face value of the bond, the current yield and coupon rate are equal. When the price rises above the face value, the current yield falls below the coupon rate. And when the price falls below the face value, the current yield rises above the coupon rate. The table below summarizes the relationships among the price, coupon rate, current yield, and yield to maturity. We know that when the bond price is less than face value, the current yield and the yield to maturity are both higher than the coupon rate. But because the yield to maturity takes account of the capital gain the bondholder receives, while the current yield does not, the yield to maturity must be even higher than the current yield. When the price is above the face value, the yield to maturity is lower than the current yield, which is lower than the coupon rate. Holding Period Returns Most holders of long-term bonds plan to sell them well before they mature. And since the price of the bond may change between the time of the purchase and the time of sale, the return to buying a bond and selling it before it matures – the holding period return or realised return – can differ from the yield to maturity. Take an example in which you pay Rs 100 for a 10-year 6 percent coupon bond with a face value of Rs 100. You intend to hold the bond for one year and then sell off a 9-year bond. If the interest rate (and hence bond price) does not change then your return will be 6/100 = 0.06 or 6 percent. But if the interest rate changes then return also changes. For instance, if the interest rate changes from 6 to 5 percent then from the following equation the price of a 9-year bond with 5 percent yield rate can be calculated as 𝑃𝑉 = 1− + =107.11. (. ) (. )  6.. Therefore, one year holding period return = + = = 13.11 % Alternatively if there is an increase in the interest rate from 6 to 7 percent then bond price falls to 1−( ) +( ) = 93.48... This implies that the holding period return is -0.52 percent calculated from the same equation as above. To generalize these examples note that the one-year holding period return is the sum of yearly coupon payment divided by the price paid for the bond and the change in the price divided by the price paid. 𝑦𝑒𝑎𝑟𝑙𝑦 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒 𝑜𝑓 𝑏𝑜𝑛𝑑 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 = + 𝑝𝑟𝑖𝑐𝑒 𝑝𝑎𝑖𝑑 𝑝𝑟𝑖𝑐𝑒 𝑝𝑎𝑖𝑑 The first part on the right-hand side of this equation is the current yield. The second part is the capital gain. So, the holding period return is, Holding period return = Current yield + Capital gain Whenever the price of a bond changes, there is a capital gain or loss. The greater the price change, the more important a part of the holding period return the capital gain or loss becomes. The potential for interest-rate movements and changes in bond prices creates risk. The longer the term of the bond, the greater those price movements and the associated risk can be. Interest Rates and Returns Return or rate of return is the defined as the payments to the owner plus the change in its value expressed as a fraction of the purchase price. For instance, if a Rs 1000-face value coupon bond is bought at Rs 1000 at a coupon rate of 10 percent for one year and then sold for Rs 1200 then the rate of return from the bond would be Rs 300 (Rs 100 + Rs 200) expressed as a fraction of Rs 1000, i.e. 30 percent. Whereas for this bond the yield to maturity or interest rate is 10 percent. A generalised formula for calculation of rate of return is 𝑅 = where R is the rate of return, Pt is the price of the bond at period t, or purchase price, Pt+1 is the selling price at period t+1 and C is total coupon payment. The formula for rate of return can be segmented into two parts: the first part (C/Pt)  100 is the current yield and the second term is called the rate of capital gain and is measured by 𝑔 =  7. Essentially, return from an asset would differ substantially from the interest rate if the price of the bond experiences wide fluctuations causing capital gain or loss. It has been observed that the following relation holds between returns, yield to maturity, bond prices and date of maturity.  When the holding period is short, returns are closest to yield to maturity.  A rise in yield is associated with a fall in bond prices, resulting in capital loss.  The more distant a bond’s maturity, the greater the size of the percentage price change associated with an interest rate change.  The more distant a bond’s maturity, the lower the rate of return that occurs as a result of increase in the interest rate. Since the prices of long maturity bonds reacts more dramatically to changes in interest rates, the prices and returns for long-term bonds are more volatile than those for short-term bonds. The riskiness of an asset’s returns associated with interest rate changes are termed as interest rate risk. Yield Calculations for Money Market Instruments In India the RBI calculates the YTM for the Treasury Bills following the formula, Where FV is the face value which is ₹ 100 for Treasury Bills and t is the maturity period which are 91 days, 182 days and 364 days. P, the price of the T-bill is determined through the auction process and announced along with the corresponding YTM in the auction results. For uniform price auction, the YTM will be same for all investors while for the multiple price auction, they will be different for different investors depending on what price an individual investor pays. The YTM formula has a multiplicative term, 365/t which annualizes the yields of the short term instruments. That’s why these are called annualised YTMs. For a 91-day T-bill, annualised YTM measures the yield received by an investor over a period of 365 days if the investor could have invested in the same instrument with same yield for 365 days. Alternatively, If 91-day represents a quarter, then the annualised YTM measures the total yield that the investor would by investing in the same instrument in all four quarters of a year. The importance of annualised yield can be explained with an example. Suppose, Mr A borrows a sum of ₹ 100 for a month at a monthly interest rate of 10 percent. This implies that he pays only ₹ 10 for a month and settles the loan after a month. However, if this interest rate is annualised then it would imply  8 that annually he would pay 10  12 = 120 for the ₹ 100 loan if he holds the loan for 12 month. Thus, the annualised interest rate is 120 percent which is massive amount. Thus apparently, annualised rates magnify the rates associated with short term instruments. In order to attract investors, issuers of credit instruments would prefer to quote annualised interest rates. On the other hand, when financial organizations offer short term loans, they prefer to quote non- annualised or per month interest rates to attract borrowers. Because, as the example shows above, the per month interest rate of 10 percent would imply 30 percent interest for a quarter, 60 percent interest for six months and so on. However, there is one problem with the annualised rate. It does not take into consideration the compounded return. For example, as sated above, the annualised yield (AY) of a quarterly instrument is AY = HPY+HPY +HPY +HPY= 4 HPY. However, an investor can always reinvest the principal plus the plus the yield in the second quarter and repeat the same thing in the third and the fourth quarter. Consequently, the investor would receive some additional return coming out of the investment of the yields generated by the previous quarter’s investments. The yield generated through this process is known as Effective Annual Yield (EAY) and is obtained as EAY = [(1+HPY)(1+HPY)(1+HPY)(1+HPY)] – 1 = (1+HPY)4 – 1. Further generalizing, this can be written as EAY = (1+HPY)365/t – 1. III. THE BEHAVIOR OF INTEREST RATES Here we explain what determines the level of interest rate and movements in it. For that purpose we begin with the assumption that there is only one asset, one type of bond, and one interest rate in the economy. Later on there will be discussion on why different assets have different interest rates or why interest rates on different assets differ from each other. Now, determination of interest rates depends on the interaction between supply and demand of assets. Taking up the demand side first, the Theory of Portfolio Choice is discussed below. Theory of Portfolio Choice The Theory of Portfolio Choice relates the quantity demanded of an asset with four determining factors, such as wealth, expected returns, risk and liquidity. When there is an increase in the wealth possessed by an individual it is obvious that s/he will have more resources to purchase assets and therefore, quantity demanded of assets increases.  9 Return on an asset is measured as the monetary gain from holding that asset. Expected return is essentially (weighted or non-weighted) average return from an asset. If expected return from an asset rises relative to alternative assets (or expected returns from alternative assets fall) then the demand for that asset increases. The more the returns from an asset are uncertain the greater the risk associated. A risk-averse person would prefer a less risky asset. Usually, people are risk-averse in their financial decisions. Therefore, they prefer to hold less risky asset. When the riskiness of an asset increases relative to alternative assets, the demand for that asset decreases. Liquidity is defined as the ease of converting an asset into cash and it is a desirable property of an asset. Therefore, as the liquidity of asset increases relative to alternative assets, the greater is the quantity demanded of that asset. Summarizing the above mentioned observations the Theory of Portfolio Choice states that holding all other factors constant  The quantity demanded of an asset is positively related to wealth.  The quantity demanded of an asset is positively related to its expected return relative to alternative assets.  The quantity demanded of an asset is negatively related to the risk of its returns relative to alternative assets.  The quantity demanded of an asset is positively related to its liquidity relative to alternative assets. Demand and Supply in the Bond Market Some investors are supplying bonds, while others are demanding them. The demand curve for a bond relates the bond price with the total value of the bond demanded in the market. As the price falls, the reward for holding the bond rises, so the demand goes up. That is, the lower the price potential bondholders must pay for a fixed-dollar payment on a future date, the more likely they are to buy a bond. Let us consider a discount or zero coupon bond whose expected return would be equal to the interest rate. At a purchase price of Rs 950 and face value Rs 1000, the return is 5.3 percent. Suppose at that price bonds worth of Rs 100 million is the demand. We know that bond price and interest rate have a negative relationship. Therefore, when bond price falls to Rs 900, interest rate or return from the bond increases to 11.1 percent. Hence, demand for the bond increases to say Rs 200 million. Thus, we can observe a negative  10 relationship between price of bond and the total value of bond demanded, ceteris paribus. The demand curve is downward sloping. The bond supply curve is the relationship between the price and the quantity of bonds people are willing to sell, all other things being equal. The higher the price of a bond, the larger the quantity supplied for two reasons. From investors’ point of view, the higher the price, the more tempting it is to sell a bond they currently hold. From the point of view of companies seeking finance for new projects, the higher the price at which they can sell bonds, the better. Taking our example of a $100 one-year zero-coupon bond, the quantity supplied will be higher at $95 per bond than it will be at $90 per bond, all other things being equal. On the other hand, when there is fall in the bond price and increase in the interest rate, borrowing becomes more expensive for the corporations. Therefore, there will be a decrease in the supply of bonds. Alternatively, when the price rises and interest rate falls, borrowing becomes cheaper and the supply of bonds increases in the market. Therefore, the supply curve for bonds is upward sloping, ceteris paribus. The equilibrium takes place where the market clears, that is demand is met with supply. If at any given price there is an excess supply of bonds, then the price is reduced. As price decreases, demand increases and the gap between demand and supply narrows down. This process continues until the equilibrium is reached. Similarly, if there is an excess demand that puts an upwards pressure on the price which stimulates supply, and the demand supply gap reduces till the equilibrium point is reached. In the figure below equilibrium is reached at the point C where bond price is Rs 850 and quantity of bonds sold is worth Rs 300 million  11 Shifts in demand for bonds – the demand curve will shift upward or downward under the influence of the five factors that affect asset demand: Wealth – at a given bond price if wealth increases, say during a period of expansion, then demand for bond increases and the demand curve would shift to the right. Similarly, during recession, the demand shrinks; so there will be leftward shift in the demand curve. Expected inflation - Changes in expected inflation alter investors’ willingness to purchase bonds promising fixed-dollar payments. A decline in expected inflation means that the payments promised by the bond’s issuer have a higher value than borrowers originally thought, so the bond will become more attractive. This fall in expected inflation shifts the bond demand curve to the right, increasing demand at each price and lowering the yield, as shown in the Figure 2. In short, the higher real return on the bond increases the willingness of would-be lenders to buy it at any given price. Note that the decline in expected inflation has reduced the nominal interest rate that investors require in order to make a loan. Expected returns relative to alternative assets - for bonds with maturity greater than one year, expected returns decreases with increase in interest rate. Thus, if people think that interest rates on long-term bond would increase in future than what they expected, then the expected return  12 would fall and hence, the quantity demanded of bonds. There will be a leftward shift in the demand curve. Alternatively, lower expected interest rates in future increase expected return, and a consequent increase in the demand for long term bonds which shifts the demand curve to the right. Risk relative to alternative assets and - an increase in the riskiness of bonds lowers the demand for the same and hence, the demand curve shifts to the left. By contrast, it would shift to the right if the riskiness of a bond is lowered. Liquidity relative to alternative assets–since liquidity is a desirable property of any asset, increased liquidity of bonds results in an increased demand for bonds and the demand curve shifts to the right. Similarly, increased liquidity of alternative assets lowers the demand for bonds and shifts the demand curve to the left. Fig. 1: Equilibrium in the Bond Market after a Demand Shift (Pic courtesy: Cecchetti & Schoenholtz, 2017)  13 Figure 2. Factors affecting Bond Demand (Pic courtesy: Mishkin, 2019) Shifts in supply of bonds–the factors that cause shifts in the supply curve are identified as three factors: expected profitability of investment opportunities or in general business conditions, expected inflation and changes in government borrowing or government budget deficit. Expected profitability of investment opportunities – During business cycle expansions, when general business conditions are good, investment opportunities abound, it prompts firms to increase their borrowing. As the amount of debt in the economy rises, the quantity of bonds outstanding with a given risk goes up (Figure 4). So as business conditions improve, the bond supply curve shifts to the right, forcing bond prices down and interest rates up. This connection between general business conditions and the supply of bonds also helps explain how weak economic growth can lead to rising bond prices and lower interest rates for bonds with unchanged risk.  14 Expected inflation – Bond issuers care about the real cost of borrowing—the cost of the loan taking inflation into account. At a given nominal interest rate, higher expected inflation means a lower real interest rate. And at a lower real interest rate, fewer real resources are required to make the payments promised by a bond. So when expected inflation rises, the cost of borrowing falls and the desire to borrow at every nominal interest rate rises; the bond supply curve shifts to the right. Higher expected inflation increases the bond supply, reducing bond prices and raising the nominal interest rate. Government budget deficit – government issues bonds to partially finance its budget deficit, the difference between government’s expenditure and revenues. Both changes in tax policy and adjustments in spending can affect a government’s need to borrow. Regardless of the reason, any increase in the government’s borrowing needs increases the quantity of bonds outstanding, shifting the bond supply curve to the right. The result is an increase in quantity of the bonds supplied at every price. Because the demand curve stays where it is (remember, we’re holding everything else constant), the increase in supply drives the price down. The added supply of U.S. government bonds has reduced prices, raising interest rates. On the other hand, government surpluses decrease the supply of bonds and shift the supply curve to the left. Fig. 3: Equilibrium in the Bond Market after a Supply Shift (Pic courtesy: Cecchetti & Schoenholtz, 2017)  15 The following table summarizes the factors that increase the quantity of bonds supplied at every price, shifting the bond supply curve to the right. Before moving on to shifts in the demand for bonds, we should mention that there is one other factor that shifts the bond supply: changes in corporate taxation. Because such changes in the tax code require government legislation, they don’t occur very often. But when they do, they can affect the economywide supply of bonds. Corporations pay taxes on their profits, just as individuals pay taxes on their income, so they are concerned with after-tax profits. Fig. 4. Shifts in Bond Supply (Pic courtesy: Cecchetti & Schoenholtz, 2017) Risks associated with bonds Bondholders face three major risks. Default risk is the chance that the bond’s issuer may fail to make the promised payment. Inflation risk means an investor can’t be sure of what the real value of the payments will be, even if they are made. And interest-rate risk arises from a bondholder’s investment horizon, which may be shorter than the maturity of the bond. If, for example, the interest rate were to rise between the time the bond is purchased and the time it is sold, the investor would suffer a capital loss.  16 Remember that risk arises from the fact that an investment has many possible payoffs during the horizon for which it is held. So in looking at the risk a bondholder faces, we need to ask what the possible payoffs are and how likely each one is to occur. As we do, we will be looking at the impact of risk on the bond’s return relative to the risk-free rate. That is, we will try to figure out how certain risks affect the premium investors require over the risk-free return. Once again, risk requires compensation. Default Risk - There is no guarantee that a bond issuer will make the promised payments. While we usually ignore default risk in thinking about Treasury bonds issued by government, we cannot do so when discussing bonds issued by many other governments or by private corporations. When corporations or governments fail to meet their payments, then the possibility of default risk arises. Default risk is simply the risk associated with non-payment by the borrower or issuer of a bond. The higher the default risk, the higher is the probability that the bondholders will not receive the promised payments. Risk reduces the expected value of a given promise, lowering the price an investor is willing to pay and raising the yield. The higher the default risk, the higher the yield. Inflation risk - With few exceptions, bonds promise to make fixed-dollar payments. That is, a $100 face value, one-year bond at 5 percent is a promise to make a $105 payment in one year. If this promise is free of default risk, the bondholder can be sure of receiving the $105 payment. Still, there is a risk of inflation. Investors care about the purchasing power of the money, not the number of Rupees or dollars. In other words, bondholders are interested in the real interest rate, not just the nominal interest rate. And they don’t know what the inflation rate will be. Inflation reduces the purchasing power of money. And the risk associated with falling value of money, so that the real value of returns from a bond also decreases, is called inflation risk. Interest rate risk - To explain interest-rate risk, we’ll focus on a U.S. Treasury bond and assume that it is free of default risk and that we know how much inflation there will be, so there also is no inflation risk. Interest-rate risk arises from the fact that investors don’t know the holding period return of a long-term bond. Remember that when interest rates change, bond prices move; the longer the term of the bond, the larger the price change for a given change in the interest rate. Now think about what happens if you have a short investment horizon. If you buy a long-term bond, you will need to sell the bond before it matures, so you have to worry about what will happen if the interest rate changes.  17 Whenever there is a mismatch between your investment horizon and a bond’s maturity, there is interest-rate risk. Because the prices of long-term bonds can change dramatically, this can be an important source of risk. The lesson is that any move in interest rates changes the price of a bond. For investors with holding periods shorter than the maturity of the bond, the potential for a change in interest rates creates risk. The more likely interest rates are to change during the bondholder’s investment horizon, the larger the risk of holding a bond. The Liquidity Preference Framework John Maynard Keynes suggested an alternative approach for determination of interest rate through equilibrium in the demand and supply of money. It assumes that there are two types of assets, money and bonds. The total quantity of money and bonds supplied must equal the total quantity of money and bonds demanded; i.e. Bs + Ms = Bd + Md or Bs – Bd = Md – Ms This implies that if the money market is in equilibrium then the bond market will also be in equilibrium. Alternatively, it also indicates that if there is an excess supply for bonds then there must be an excess demand for money. That is increased demand for one asset has upset the demand for another asset resulting in an excess supply of the latter. However, assuming that there are only two assets the impact of alternative assets affecting interest rates through changes in expected returns, risk and liquidity is assumed away. The second assumption is that money defined as currency and checking account deposits do not pay any return, while bond, the only alternative asset does. Now if there is an increase in the interest rate, holding everything else constant, there will be an increase in the relative expected return from bonds, hence demand for bonds would increase and demand for money would fall. Therefore, interest rate and demand for money have a negative relationship. The reason we approach the determination of interest rates using both frameworks is that the bond supply and demand framework is easier to use when analyzing the effects caused by changes in expected inflation, whereas the liquidity preference framework is easier to use when analyzing the effects caused by changes in income, the price level, and the supply of money. Because the definition of money used by Keynes includes currency (which earns no interest) and checking account deposits (which in his time typically earned little or no interest), he assumed that money has a zero rate of return. Bonds, the only alternative asset to money in Keynes’s framework, have an expected return equal to the interest rate; even if they are not  18 equal, they are closely related. As this interest rate rises (holding everything else unchanged), the expected return on money falls relative to the expected return on bonds, causing a fall in the quantity of money demanded, as predicted by the theory of portfolio choice. We can also see that the quantity of money demanded and the interest rate should be negatively related by using the concept of opportunity cost, which is the amount of interest (expected return) sacrificed by not holding the alternative asset—in this case, a bond. As the interest rate on bonds, i, rises, the opportunity cost of holding money rises; thus, money is less desirable and the quantity of money demanded falls. Fig 5.: Supply and Demand for Money following Liquidity Preference Framework (Pic courtesy: Mishkin, 2019) The above diagram shows that if money supply is assumed to be fixed at Rs 300 billion (controlled by the monetary authority) then at the intersection of demand and supply the interest rate is 15 percent. If the interest rate is higher than the equilibrium rate then there is excess supply of money. That is people are holding idle money which they want to utilise to buy bonds, because the interest rate is high. Now as this demand pressure raises the bond price, interest rate falls and there will be a decreased demand for bond and increased demand for money. That’s how the interest rate keeps on slipping down the demand line until it reaches  19 the equilibrium. Similarly, for interest rate below the equilibrium level, there will be an excess demand for money. People will sell their assets for money. There will be a fall in bond price, a consequent increase in interest rate till the time it converges to the equilibrium rate. Shifts in the demand for money – Keynes identified two factors responsible for shifts in the demand curve, namely income and price level. In Keynes’s view, there were two reasons why income would affect the demand for money. First, as an economy expands and income rises, wealth increases and people want to hold more money as a store of value. Second, as the economy expands and income rises, people want to carry out more transactions using money as a medium of exchange, and so they also want to hold more money. The conclusion is that a higher level of income causes the demand for money at each interest rate to increase and the demand curve to shift to the right. Alternatively, increase in income would induce more spending which will increase demand for goods and services, induce further spending and thus this process would create an overall increased demand for money. Therefore, at each interest rate there will be an increased demand for money and hence, outward shifts in the demand curve. Second, Keynes took the view that people care about the amount of money they hold in real terms—that is, in terms of the goods and services it can buy. When the price level rises, the same nominal quantity of money is no longer as valuable; it cannot be used to purchase as many real goods or services. To restore their holdings of money in real terms to the former level, people will want to hold a greater nominal quantity of money, so a rise in the price level causes the demand for money at each interest rate to increase and the demand curve to shift to the right. It has been already assumed that the money supply is exogenous. Therefore, for both cases discussed above, an increase in demand for money would lead to a decrease demand for bonds and/or increased supply of bonds (as people liquidate their bond holdings) which will reduce the bond price and result in an increase in the equilibrium interest rate, ceteris paribus (shown in the Fig a and Fig b). If the central bank decides to increase the money supply then there will be a rightward shift in the money supply curve. Consequently, interest rate will increase for a given demand for money, ceteris paribus (shown in Fig. c).  20 Fig a Fig. b Fig. c Fig. 7: Shifts in Demand and Supply of Bonds (Pic courtesy: Mishkin, 2019) An important criticism of the idea that an increase in the money supply lowers interest rates was raised by Milton Friedman, a Nobel laureate in economics. He acknowledged that the liquidity preference analysis was correct and called the result—that an increase in the money supply (everything else remaining equal) lowers interest rates—the liquidity effect. However, he viewed the liquidity effect as merely part of the story: An increase in the money supply might not leave “everything else equal” and will have other effects on the economy that may make interest rates rise. If these effects are substantial, it is entirely possible that when the money supply increases, interest rates might also increase.  Income Effect. An increasing money supply can cause national income and wealth to rise. Both the liquidity preference and bond supply and demand frameworks indicate that interest rates will then rise. Thus, the income effect of an increase in the money supply is a rise in interest rates in response to the higher level of income.  Price-Level Effect. An increase in the money supply can also cause the overall price level in the economy to rise. The liquidity preference framework predicts that this will lead to a rise in interest rates. Thus, the price-level effect from an increase in the money supply is a rise in interest rates in response to the rise in price level.  Expected-Inflation Effect. The higher inflation rate that can result from an increase in the money supply can also affect interest rates by influencing the expected inflation rate. Specifically, an increase in the money supply may lead people to expect a higher price level in the future—and hence the expected inflation rate will be higher. The bond supply and demand framework has shown us that this increase in expected inflation will lead to a higher level of interest rates. Therefore, the expected-inflation effect of an  21 increase in the money supply is a rise in interest rates in response to the rise in the expected inflation rate. IV. THE RISK AND TERM STRUCTURE OF INTEREST RATE In reality we have a large number of bonds and associated interest rates in the market where the interest rates differ from each other. The relationship between interest rates on bonds with same term to maturity is called the risk structure of interest rate though risk, liquidity and income tax rules, all play important role in determining the risk structure. The relationship between interest rates on bonds with different terms to maturity is called the term structure of interest rate. Risk Structure of Interest Rates The risk structure of an interest rate is explained by three factors: default risk, liquidity and the income tax treatment of a bond’s interest payments. Each one of these is explained below. Default risk – default risk or risk of default occurs when the issuer of a bond is unable to make interest payments or pay off the face value when the bond matures. Bonds having no default risk, like government bonds, are called default-free bonds. The spread between the interest rates on bonds with default risk and default-free bonds of same maturity are called the risk premium. Risk premium indicates how much additional interest people must earn to be willing to hold that risky bond. The impact of default risk on interest rates can be examined using the following diagram. To begin with suppose, a corporate bond has the same default risk as the govt. bond. Therefore, for the same term to maturity, the two bonds will have same interest rates and prices (Pc1 = PT1) and the risk premiums on corporate bonds will be zero.  22 Fig. 8: Determination of Risk Premium (Pic courtesy: Mishkin, 2019) Now, if the corporation begins to suffer large losses, the default risk on corporate bonds will increase, the asset will become riskier. Following the Theory of Portfolio Choice, the demand for riskier bond will decrease; the demand curve for the corporate bond would shift to the left (from Dc1to Dc2). The bond price will fall to Pc2 and interest rate will rise. On the other hand, the treasury bonds are now less risky relative to the corporate bond. Therefore, demand for this bond will increase as reflected in a rightward shift in the demand curve. The price of this bond will increase, and the interest will fall. Therefore, now there will be a positive risk premium (ic2 – iT2). So, a bond with default risk will always have a positive risk premium and the risk premium increases with increase in default risk. The information on the possibility of default is provided by credit-rating agencies, investment advisory firms that rate the quality of corporate and municipal bonds in terms of their probability of default. Top three international credit rating agencies are Moody’s Investor Service, Standard and Poor’s Corporation, and Fitch Ratings. There are five credit rating agencies in India: CRISIL (a Standard and Poor’s Company), Fitch Ratings India, ICRA, Credit Analysis and Research (CARE), Brickwork Ratings India, and SME Rating Agency of India (SMERA). There are separate ratings for short-term instruments, fixed deposits, corporate credit, and long-term and short-term structured finance instruments like derivatives, and so on. In essence, ratings are functions of both qualitative functions like financial ratios, balance sheet strength and qualitative factor such as competitive pressure and management competence.  23 Bonds with relatively low risk of default having a rating of BBB and above are called investment-grade securities. Bonds with ratings below BBB are called speculative-grade or junk bonds. Liquidity –treasury bonds are the most liquid of all long-term bonds because it is easiest to sell at a low cost. The lower liquidity of corporate bonds relative to treasury bonds increases the spread between the interest rates on these two bonds. Because with increased relative liquidity the demand for Treasury bonds will increase pushing its price up and interest rate down. On the other hand, when the relative liquidity of the corporate bond is less or lowered, its demand decreases (demand curve shifts to the left) resulting in a decrease in price and hence, increase in interest rate. Therefore, the risk premium on the corporate bond becomes positive. Income Tax Consideration– if payments on certain bonds are exempted from income taxes, then the expected returns from those bonds increase. This leads to an increase in the demand for those bonds which pushes up bonds’ prices. Consequently, the interest rate lowers. Therefore, these bonds tend to have a negative risk premium compared to other corporate bonds. In India there are not many tax-free bonds or securities. Most often pub sector companies, like Airport Authority of India (AAI), NHAI, Hudco, Rural Electrification Corporation (REC), Power Finance Corporation (PFC) issue tax-free bonds in India. Term Structure of Interest Rates Bonds with identical risk, liquidity and tax characteristics may have different interest rates because of different terms to maturity. A plot of the yields on bonds with different maturities but same risk, liquidity and tax considerations is called a yield curve. It describes the term structure of interest rates. There are three theories that attempt to explain the shapes and movements of yield curves: i) The Expectation Theory, ii) The Segmented Market Theory and iii) The Liquidity Premium Theory. It has been observed that  Interest rates on bonds of different maturities move together over time.  Yield curves are usually upward sloping, but they can be flat or downward sloping (often referred to as inverted yield curve) as well. For an upward sloping yield curve long term interest are above short-term interest rates and vice versa. For a flat yield curve, the two interest rates are same. However, they can take U or an inverted-U shape as well.  Yield curves almost always slope upward.  24 The first two statements are explained by the Expectation Theory. The Segmented Market Theory explains the third one while all three are explained by The Liquidity Premium Theory. Expectations Theory–The theory assumes that the expected returns or interest rates from different bonds are the determining factor regarding which one should be held and maturity does not play any role. If bonds with different maturities have same expected return then they will be perfect substitutes. The theory essentially says that the interest rates on long term bonds equal average interest rates on short term bonds. To see how the assumption that bonds with different maturities are perfect substitutes leads to the expectations theory, let us consider the following two investment strategies: 1. Purchase a one-year bond, and when it matures in one year, purchase another one-year bond. 2. Purchase a two-year bond and hold it until maturity. Because both strategies must have the same expected return, the interest rate on the two-year bond must equal the average of the two one-year interest rates. You will be willing to hold both the one and two-year bonds only if the expected return per year on the two-year bond equals this return. For instance, if the interest rate on a one-year bond is 9 percent and it is expected to be 10 percent in the next year, then the interest rate on a two-period bond should be 9.5 percent, the average interest rate on the two one-year bonds. Otherwise, people will not hold any amount of the bond which has lower expected interest rate. Generalizing this to n periods, one can say that the interest rate on an n-period bond, in must be equal to 𝒊𝟏 𝒊𝒆𝟏 𝟏 𝒊𝒆𝟏 𝟐 ⋯ 𝒊𝒆𝟏 (𝒏 𝟏) 𝒊𝒏 = (1) 𝒏 Where i1 is today’s interest on a one-period bond, ie1+1 is the expected interest on a one-period bond after 1 year and so on. Thus, the n-period interest rate equals the average of the one-period interest rates expected to occur over the n-period life of the bond. A simple numerical example might help clarify the expectations theory presented in the Equation (1). If the one-year interest rates over the next five years are expected to be 5%, 6%, 7%, 8%, and 9%, Equation (1) indicates that the interest rate on the two-year bond will be 𝟓% + 𝟔% = 𝟓. 𝟓% 𝟐  25 On the five-year bond, it will be 𝟓% + 𝟔% + 𝟕% + 𝟖% + 𝟗% = 𝟕% 𝟓 By doing similar calculations for the one-, three-, and four-year interest rates, you should be able to verify that the one- to five-year interest rates are 5.0%, 5.5%, 6.0%, 6.5%, and 7.0%, respectively. Thus we see that the rising trend in expected short-term interest rates produces an upward-sloping yield curve along which interest rates rise as maturity lengthens. The expectations theory is an elegant theory that explains why the term structure of interest rates (as represented by yield curves) changes at different times. When the yield curve is upward-sloping, the expectations theory suggests that short-term interest rates are expected to rise in the future, as we have seen in our numerical example. In this situation, in which the long-term rate is currently higher than the short-term rate, the average of future short-term rates is expected to be higher than the current short term rate, which can occur only if short-term interest rates are expected to rise. When the yield curve is inverted (slopes downward) , the average of future short-term interest rates is expected to be lower than the current short -term rate , implying that short -term interest rates are expected to fall, on average , in the future. Only when the yield curve is flat does the expectations theory suggest that short-term interest rates are not expected to change, on average, in the future. The expectations theory also explains the first statement, which states that interest rates on bonds with different maturities move together over time. Historically, short-term interest rates have had the characteristic that if they increase today, they will tend to be higher in the future. Hence a rise in short-term rates will raise people's expectations of future short-term rates. Because long-term rates are the average of expected future short-term rates, a rise in short-term rates will also raise long-term rates, causing short and long-term rates to move together. The expectations theory also explains the second statement that yield curves tend to have an upward slope when short-term interest rates are low and are inverted when short-term rates are high. When short-term rates are low, people generally expect them to rise to some normal level in the future, and the average of future expected short-term rates is high relative to the current short-term rate. Therefore, long-term interest rates will be substantially higher than current short -term rates, and the yield curve would then have an upward slope. Conversely, if short- term rates are high, people usually expect them to come back down. Long-term rates would  26 then drop below short-term rates because the average of expected future short-term rates would be lower than current short-term rates and the yield curve would slope downward and become inverted. However, the drawback of this theory is that it fails to explain the third observation or statement. Segmented Markets Theory - As the name suggests, the segmented markets theory of the term structure sees markets for different-maturity bonds as completely separate and segmented. The interest rate for each bond with a different maturity is then determined by the supply of and demand for that bond, with no effects from expected returns on other bonds with other maturities. The key assumption in the segmented markets theory is that bonds of different maturities are not substitutes at all, so the expected return from holding a bond of one maturity has no effect on the demand for a bond of another maturity. The argument for why bonds of different maturities are not substitutes is that investors have very strong preferences for bonds of one maturity but not for another, so they will be concerned with the expected returns only for bonds of the maturity they prefer. In the segmented markets theory, differing yield curve patterns are accounted for by supply and demand differences associated with bonds of different maturities. If, as seems sensible, investors have short desired holding periods and generally prefer bonds with shorter maturities that have less interest-rate risk, the segmented markets theory can explain the third statement that yield curves typically slope upward. Because in the typical situation the demand for long- term bonds is relatively lower than that for short term bonds, long-term bonds will have lower prices and higher interest rates, and hence the yield curve will typically slope upward. However, the drawback with segmented market theory is that it fails to explain the first two statements. Therefore, the third theory is brought in by combining these two theories. Liquidity Premium and Preferred Habitat Theories - The liquidity premium theory of the term structure states that the interest rate on a long-term bond will equal an average of short- term interest rates expected to occur over the life of the long-term bond plus a liquidity premium (also referred to as a term premium) that responds to supply and demand conditions for that bond. The liquidity premium theory's key assumption is that bonds of different maturities are substitutes, but not perfect substitutes. This means that the expected return on one bond does influence the expected return on a bond of a different maturity, but it allows investors to prefer one bond maturity over another. Investors tend to prefer shorter-term bonds because these bonds bear less interest-rate risk. For these reasons, investors must be offered a  27 positive liquidity premium to induce them to hold longer-term bonds. Such an outcome would modify the expectations theory by adding a positive liquidity premium to the equation that describes the relationship between long and short-term interest rates. The liquidity premium theory is thus written as 𝒊𝒆𝟏 𝟏 𝒊𝒆𝟏 𝟐 ⋯ 𝒊𝒆𝟏 (𝒏 𝟏) 𝒊𝒏 = +𝑙 (2) whereln is the liquidity premium for the n-period bond at time n, which is always positive and rises with the term to maturity or n. Closely related to the liquidity premium theory is the preferred habitat theory. It assumes that investors have a preference for bonds of one maturity over another, a particular bond maturity (preferred habitat) in which they prefer to invest. Because they prefer bonds of one maturity over another, they will be willing to buy bonds that do not have the preferred maturity (habitat) only if they earn a somewhat higher expected return. Because investors are likely to prefer the habitat of short-term bonds over that of longer-term bonds, they are willing to hold long-term bonds only if they have higher expected returns. This reasoning leads to the same equation implied by the liquidity premium theory with a term premium that typically rises with maturity. A simple numerical example, similar to the one we used for the expectations hypothesis, further clarifies the liquidity premium and preferred habitat theories given in Equation (2). Again suppose that the one-year interest rates over the next five years are expected to be 5%, 6%, 7%, 8%, and 9%, while investors’ preferences for holding short-term bonds means that the liquidity premiums for one- to five-year bonds are 0%, 0.25%, 0.5%, 0.75%, and 1.0%, respectively. Equation (2) then indicates that the interest rate on the two-year bond would be 𝟓% + 𝟔% + 𝟎. 𝟐𝟓% = 𝟓. 𝟕𝟓% 𝟐 On the five-year bond, it will be 𝟓% + 𝟔% + 𝟕% + 𝟖% + 𝟗% + 𝟏% = 𝟖% 𝟓 Let’s see if the liquidity premium and preferred habitat theories are consistent with all three empirical facts mentioned above. They explain fact l, which states that interest rates on different-maturity bonds move together over time: A rise in short-term interest rates indicates that short-term interest rates will, on average, be higher in the future, and the first term in the  28 equation then implies that long-term interest rates will rise along with them. They also explain why yield curves tend to have an especially steep upward slope when short-term interest rates are low and to be inverted when short-term rates are high. Because investors generally expect short-term interest rates to rise to some normal level when they are low, the average of future expected short-term rates will be high relative to the current short -term rate. With the additional boost of a positive liquidity premium, long-term interest rates will be substantially higher than current short-term rates, and the yield curve will then have a steep upward slope. Conversely, if short-term rates are high, people usually expect them to come back down. Long- term rates will then drop below short-term rates because the average of expected future short- term rates will be so far below current short-term rates that despite positive liquidity premiums, the yield curve will slope downward. The liquidity premium and preferred habitat theories explain the third statement, which states that yield curves typically slope upward, by recognizing that the liquidity premium rises with a bonds maturity because of investors' preferences for short-term bonds. Even if short-term interest rates are expected to stay the same on average in the future, long-term interest rates will be above short-term interest rates, and yield curves will typically slope upward. How can the liquidity premium and preferred habitat theories explain the occasional appearance of inverted yield curves if the liquidity premium is positive. It must be that at times short-term interest rates are expected to fall so much in the future that the average of the expected short-term rates is well below the current short-term rate. Even when the positive liquidity premium is added to this average, the resulting long-term rate will still be lower than the current short -term interest rate. As our discussion indicates, a particularly attractive feature of the liquidity premium and preferred habitat theories is that they tell you what the market is predicting about future short- term interest rates just from the slope of the yield curve. A steeply rising yield curve indicates that short-term interest rates are expected to rise in the future. A moderately steep yield curve indicates that short-term interest rates are not expected to rise or fall much in the future. A flat yield curve indicates that short-term rates are expected to fall moderately in the future. Finally, an inverted yield curve indicates that short term interest rates are expected to fall sharply in the future. References Mishkin, Frederick S. 2019. The Economics of Money, Banking, and Financial Marker, Twelfth Edition. Pearson Education, UK.  29 Cecchetti, Stephen G. & Schoenholtz, Kermit L. Money, Banking, and Financial Market, fifth edition. McGraw Hill, New York, 2017.