Term Structure of Interest Rates

What is a Term Structure of Interest Rate? Recall that interest rates are the price of money (borrowing or lending) and, in equilibrium, interest rates equate the amount of borrowing to the amount of saving. A yield curve displaying the relationship between spot rates of zero-coupon securities and their term to maturity. The term structure of interest rates is the relationship between the yield to maturity and the time to maturity for pure discount bonds. Measure of the market’s expectations of future interest rates given the current market conditions.

relationship between yield and maturity is NOT constant sometimes short-term yields are highest, most of the time long-term yields are highest For example, the yield on a one-year bond might be 4% while the yield on a 30 day-bond is 6%.

The yields differ for several reasons, including differences in risk for short-term and long-term investments and differences in expectations about future interest rates.

Importance of Term Structure The term structure of interest rates is of fundamental importance in macroeconomics because monetary policy affects short-term interest rates, but investment depends on long-term interest rates. Theories about the term structure of interest rates thus become theories about the connection between monetary policy and investment.

Economist and investors believe that the shape of yield curve reflects the market’s future expectations for interest rates and the conditions for monetary policy. Forms the basis of valuation of all fixed income instruments.

The Yield Curve It is the plot of maturity vs. yield slope of curve indicates relationship between maturity and yield

Yield Curve (continued) The term structure of interest rates is the relation between different interest rates for different term-to-maturity loans. If we observe r1 = 8%, r2 = 9%, r3 = 9.5%, r4 = 9.75% and r5 = 9.875% then the current term structure of interest rates is represented by plotting these “spot rates” against their terms-to-maturity. The curve plotted through the above points is also called the “yield curve”

Definitions – Spot Rates The n-period current spot rate of interest denoted rn is the current interest rate (fixed today) for a loan (where the cash is borrowed now) to be repaid in n periods. Note: all spot rates are expressed in the form of an effective interest rate per year. In the example above, r1, r2, r3, r4, and r5, in the previous slide, are all current spot rates of interest. Spot rates are only determined from the prices of zero-coupon bonds and are thus applicable for discounting cash flows that occur in a single time period. This differs from the more broad concept of yield to maturity that is, in effect, an average rate used to discount all the cash flows of a level coupon bond.

Definitions – Forward Rates The one-period forward rate of interest denoted fn is the interest rate (fixed today) for a one period loan to be repaid at some future time period, n. I.e., the money is borrowed in period n-1 and repaid in period n. There fore it is the rate agreed upon today for a loan where the money is borrowed in one year and repaid the following year. Note: forward rates are also expressed as effective interest rates per year.

Definitions – Forward Rates (continued) To calculate a forward rate, the following equation is useful: 1 + fn = (1+rn)n / (1+rn-1)n-1 where fn is the one period forward rate for a loan repaid in period n (i.e., borrowed in period n-1 and repaid in period n) Calculate f2 given r1=8% and r2=9% Calculate f3 given r3=9.5%

Definitions – Future Spot Rates Current spot rates are observable today and can be contracted today. A future spot rate will be the rate for a loan obtained in the future and repaid in a later period. Unlike forward rates, future spot rates will not be fixed (or contracted) until the future time period when the loan begins (forward rates can be locked in today). Thus we do not currently know what will happen to future spot rates of interest. However, if we understand the theories of the term structure, we can make informed predictions or expectations about future spot rates. We denote our current expectation of the future spot rate as follows: E[n-trn] is the expected future spot rate of interest for a loan repaid in period n and borrowed in period n-t.

Future Spot Rates… Who Cares? You are considering locking in your mortgage rate for one year or for five years. You would use the longer term if you thought interest rates would be much higher in one year. I.e., if you expect future spot rates in one year to be much higher, you will choose the longer term mortgage right now … so, yes, it matters for your personal life. As a financial manager, you must decide whether your firm should borrow long term or short term. You would prefer to borrow short term as these rates are currently lower, however, you are concerned about what rates you will face when you refinance your loan. I.e., you are concerned about future spot rates at the time you refinance and your expectations will affect your current decision to finance long or short term… so, yes, it matters for the corporation.

Notation Review rn spot rate for a loan negotiated today and repaid in period n fn forward rate for a one-period loan repaid in period n n-tfn forward rate for a t-period loan repaid in period n and borrowed in period n-t E[n-trn] our current expectation for the future spot rate of interest for a t-period loan repaid in period n and borrowed (and contracted) in period n-t.

Nature of Yield Curves Yield curves can take different forms: Upward Sloping Downward Sloping Flat Humped

upward sloping maturity yield yields rise with maturity (common)

downward sloping (inverted) maturity yield yield falls with maturity (rare)

flat maturity yield yields similar for all maturities

humped maturity yield intermediate yields are highest

Factors influencing the Yield Curve Monetary policy Fiscal policy Inflation Economic growth Attractiveness of Debt Markets

Discussion Identify factors that influence Yield Curve (interest rate) in Zimbabwe and explain how they influence the Yield Curve. Why are interest rates high in Zimbabwe? Explain why there is no inflow of interest bearing investment in Zimbabwe while there are high interest as compared to other countries in the region.

Theories of the term structure These are theories which explain relationship between yield and maturity what does the yield curve tell us?

Theories of the term structure PURE EXPECTATION THEORY Liquidity-Preference Hypothesis Segmented Markets Theory Preferred Habitat Theory

The Pure Expectations Theory What does the theory say? The Pure-Expectations Hypothesis states that expected future spot rates of interest are equal to the forward rates that can be calculated today (from observed spot rates). The forward rates exclusively represent expected future rates. In other words, the forward rates are unbiased predictors for making expectations of future spot rates. Thus the entire term structure at a given time reflects the market’s current expectations of future short term rates.

Assumption bond buyers do not have any preference about maturity i.e. bonds of different maturities are perfect substitutes LT = long-term ST = short-term

then investors care only about expected return if assumption is true, then investors care only about expected return if expect better return from ST bonds, only hold ST bonds if expect better return from LT bonds, only hold LT bonds

but investors hold both ST and LT bonds so, must EXPECT similar return: LT yields = average of the expected ST yields Under expectancy theory, the slope of yield curve tells us direction of expected future ST rates

why? if expect ST rates are to RISE, then average of ST rates will be > current ST rate so LT rates > ST rates so yield curve SLOPES UP

ST rates expected to rise maturity yield

if expect ST rates to FALL, then average of ST rates will be < current ST rate so LT rates < ST rates so yield curve slopes DOWN

ST rates expected to fall maturity yield

if expect ST rates to STAY THE SAME, then average of ST rates will be =current ST rate so LT rates = ST rates so yield curve is FLAT

ST rates expected to stay the same maturity yield

Pure-Expectations Hypothesis (Mathematical demonstration) Consider that given expectations for inflation over the next year, investors require 4% for a one year loan. Suppose investors currently expect inflation for the next year (the second year) to be higher so that they expect to require 6% for a one year loan (starting one year from now).

Then, the Pure-Expectations Hypothesis, is consistent with the current 2-year spot rate defined as follows: (1+r2)2=(1+r1)(1+E[1r2]) = (1.04)x(1.06) so r2=4.995238% Restated, if we observe r1=4% and r2=4.995238%, then, under the Pure-Expectations Hypothesis, we would have E[1r2] to be 6% (which is equal to f2).

Is this theory true? not quite. FACT: yield curve usually slopes up but expectations theory would predict this only when ST rates are expected to rise 50% of the time

What went wrong? back to assumption: bonds of different maturities are perfect substitutes but this is not likely long term bonds have greater price volatility short term bonds have reinvestment risk assumption is too strict so implication is not quite correct

Drawbacks in the Theory 1. Slope of the yield curve is due to two things: Expectation about future rates Size of liquidity premium 2. Difficult to interpret the yield curve

Liquidity-Preference Hypothesis Empirical evidence seems to suggest that investors have relatively short time horizons for bond investments. Thus, since they are risk averse, they will require a premium to invest in longer term bonds. The Liquidity-Preference Hypothesis states that longer term loans have a liquidity premium built into their interest rates and thus calculated forward rates will incorporate the liquidity premium and will overstate the expected future one-period spot rates.

Liquidity Theory Assumptions Assume: bonds of different maturities are imperfect substitutes, and investors PREFER ST bonds

so if true, investors hold ST bonds UNLESS LT bonds offer higher yield as incentive higher yield = liquidity premium IF LT bond yields have a liquidity premium, then usually LT yields > ST yields or yield curve slopes up.

Reconsider investors’ expectations for inflation and future spot rates Reconsider investors’ expectations for inflation and future spot rates. Suppose over the next year, investors require 4% for a one year loan and expect to require 6% for a one year loan (starting one year from now). Under the Liquidity-Preference Hypothesis, the current 2-year spot rate will be defined as follows: (1+r2)2=(1+r1)(1+E[1r2]) + LP2 (LP2 = liquidity premium: assumed to be 0.25% for a 2 year loan) (1+r2)2 = (1.04)x(1.06) + 0.0025, so r2=5.11422%

Problem How do we interpret yield curve? slope due to 2 things: (1) expectations about future ST rates (2) size of liquidity premium do not know size of liquidity premium.

if liquidity premium is small, then ST rates are expected to rise yield curve maturity yield small liquidity premium if liquidity premium is small, then ST rates are expected to rise

if liquidity premium is larger, yield curve maturity yield large liquidity premium if liquidity premium is larger, then ST rates are expected to stay the same

Interpret the yield curve using liquidity premium theory expect short-term i to ---- in the future steeply upward sloping rise slightly upward sloping be unchanged flat decline moderately downward sloping decline sharply You can figure out what the market is predicting about future short-term interest rates by looking at the slope of the yield curve.

Comparing with the expectations theory, liquidity premium theory implies a more steeply upward sloped yield curve. Gap = liquidity premium

Segmented Markets Theory This theory sees two separate maturity habitats or segments one long and the other short. Each segment has a schedule of supply (lenders) and demand (borrowers) for loanable funds. The point at which the demand and supply intersect determines the prevailing rate for that sector.

In simple terms it assumes that: bonds of different maturities are NOT substitutes at all if assumption is true, separate markets for ST and LT bonds slope of yield curves tells us nothing about future ST rates

MST recognizes that there are institutional restrictions on the asset side and hedging pressures on the liability side which allow for very little substitutability between bonds of different maturities. Some of these restrictions result from government regulation, company policy, Securities and Exchange Commission regulations, goals and objectives, and fiscal and operational considerations.

Drawbacks unrealistic to assume NO substitution between ST and LT bonds unrealistic to assume NO substitution

The Market Segmentation and the Preferred Habitat Theory The Preferred Habitat Theory is an extension of the Market Segmentation Theory. The Preferred Habitat Theory allows market participants to trade outside their preferred habitat (maturity preferences) if adequately compensated for the additional risk by a risk premium. Investors will reallocate their security holding in response to the expected risk premium. This allows for humps or twists in the yield curve, but limits the discontinuities possible under Segmentation Theory.

Preferred Habitat Theory the preferred habitat theory is a modification by Modigliani and Sutch of Culbertson’s “market segmentation theory.” assume: bonds of different maturities are imperfect substitutes, and investor preference for ST bonds or LT bonds is not constant

liquidity premium could be positive or negative yield curve very difficult to interpret do not know size or sign of liquidity premium Implications rates of interest are an average of expected short-term rates plus an additional term premium some sequences of securities are expected to produce higher rates of return than others implied forward rates of interest are biased estimates of future spot rates arbitrage across maturities will not totally eliminate term premia changes in term structure shape may reflect changes in expectations of future rates or maturity preference

Link Between Theories and Term Structure of interest Rates A flat term structure means constant forward rates equal to today’s spot rates and thus … Expectations for the same future spot rates as today if you believe in the Pure-Expectations Hypothesis Expectations for declining future spot rates compared to today if you believe in the Liquidity-Preference Hypothesis

Interpreting the Link Between Theories and Term Structure (continued) A declining term structure means declining forward rates and thus … Expectations for similarly declining future spot rates under the Pure-Expectations Hypothesis Expectations for more sharply declining future spot rates under the Liquidity-Preference Hypothesis

Interpreting the Link Between Theories and Term Structure (continued) An increasing term structure means increasing forward rates and thus … Expectations for similarly increasing future spot rates under the Pure-Expectations Hypothesis Expectations for future spot rates that increase to a lesser degree or possibly remain flat or decrease (depending on the size of the Liquidity Premiums) under the Liquidity-Preference Hypothesis

Which Theory is Right? Day-to-day changes in the term structure are most consistent with the Preferred Habitat Theory. However, in the long-run, expectations of future interest rates and liquidity premiums are important components of the position and shape of the yield curve.

Economic implications of the Yield Curve Yield Curves are directly related to the level of economic activity (the Business Cycle) and profits of financial institutions: An ascending (upward sloping) yield curve indicates the market expectations of higher interest rates, higher periods of economic expansion and/or higher inflation levels. The yield curve starts sloping upwards at the beginning of the business cycle. Banks and financial institutions are expected to make better profits. The more the steeper the yield curve, the higher are the expected profits A descending (downward sloping) yield curve forecasts the opposite indications.

Interest-Rate and Yield-Curve Patterns Over the Business Cycle

Uses of the Yield Curve At any point in time, the slope of the yield curve can be used to assess the general expectations of borrowers and lenders about future interest rates! Investors can use the yield curve to identify under-priced securities for their portfolios. Issuers may use the yield curve to price their securities.

Default Risk It is the probability of the borrower not honoring the security contract. Losses may range from “interest a few days late” to a complete loss of principal. Risk averse investors want adequate compensation for expected default losses. Investors charge a Default Risk Premium for added risk of default. It is measured as: DRP = i - irf The DRP is the difference between the promised or nominal rate and the yield on a comparable (same term) risk-free securities. Investors are satisfied if the default risk premium is equal to the expected default loss.

Default Risk: Risk Premiums (May 2004)

Default Risk, cont. Default Risk Premiums Increase (Widen) in Periods of Recession and Decrease in Economic Expansion In good times, risky security prices are bid up; yields move nearer that of riskless securities. With increased economic pessimism, investors sell risky securities and buy “quality” widening the DRP. Credit Rating Agencies Measure and Grade Relative Default Risk Security Issuers Cash flow, level of debt, profitability, and variability of earnings are indicators of default riskness. As conditions change, rating agencies alter rating of businesses and governmental debtors.

Corporate Bond-Rating Systems

Tax Effects on Yields The Taxation of Security Gains and Income Affects the Yield Differences Among Securities The after-tax return, iat, is found by multiplying the pre-tax return by one minus the marginal tax rate. iat = ibt(1-t) Government and Municipal bond interest income is tax exempt. Coupon income and capital gains have been taxed differently in the past, but are now both taxed at the same rate as ordinary income for individuals.

To Buy a Municipal or a Corporate Bond?