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**THE STRUCTURE OF INTEREST RATES**

CHAPTER 6 THE STRUCTURE OF INTEREST RATES

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**Factors that Influence Yields /Rate of Return Differences**

Term to Maturity Default Risk Tax Treatment Marketability Options on Debt Securities: Call, Put or Convertibility option Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2008 John Wiley & Sons, Inc.**

Term Structure Relationship between yield and the maturity on securities that differ only in length of time to maturity. A yield curve is a graphical representation of the term structure; it shows the relationship between maturity and a security's yield at a point in time. The yield curve may be ascending (normal), flat, or descending (inverted). Several theories explain the shape of the yield curve. Copyright© 2008 John Wiley & Sons, Inc.

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**Yield Curves in the 2000s - Exhibit 6.1**

Copyright© 2008 John Wiley & Sons, Inc.

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**The Expectations Theory**

The shape of the yield curve is determined only by expectations of future interest rate movements, and changes in these expectations lead to changes in the shape of the yield curve . Ascending: future interest rates are expected to increase. Descending: future interest rates are expected to decrease. Flat :interest rates are expected to be stable in the future. Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2006 John Wiley & Sons, Inc.**

Assumptions of the Theory : 1. Investors are profit maximizers. 2. Investors are risk neutral ( indifferent between holding a long term security and holding a series of short -term securities). 3. Markets is assumed to be very efficient with excellent information and minimal trading costs. Long-term interest rates are geometric averages of current short term interest rate and expected short-term forward rates. Copyright© 2006 John Wiley & Sons, Inc.

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**Term Structure Formula**

Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2006 John Wiley & Sons, Inc.**

Alternatively: tRn = 1/n (tR1 + t+1f1 + t+2f1 + …….. + t+n-1f1) Copyright© 2006 John Wiley & Sons, Inc.

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**Dr. Hisham Handal Abdelbaki - FIN 221 - Chapter 6**

Example: Given the following Current one – year rate = 3%, Expected 1 – year rate, a year from now = 4% Expected 1 – year rate, 2 years from now = 5% What is the yield on a 3 – year security? Solution: 1 + tR3 = [(1 + 3%)(1+ 4%) (1+ 5%)]1/3 tR3 = [(1 + 3%)(1+ 4%) (1+ 5%)]1/ = 3.997%=4% Or tRn = 1/n (tR1 + t+1f1 + t+2f1 + …….. + t+n-1f1) tR3 = 1/ 3 (3% + 4% + 5%) = 4% Dr. Hisham Handal Abdelbaki - FIN Chapter 6

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**Copyright© 2006 John Wiley & Sons, Inc.**

A commercial bank made a 3 – year term loan at 10 %. The bank’s economics department forecasts that 1 year interest rate 1 year from now will be 10% and 1 year interest rate 2 years from now will be 14%, respectively. The current 1 – year rate is 8%. Given that the bank’s forecasts are reliable, has the bank set the 3 – year rate correctly? Answer = 10.6% Copyright© 2006 John Wiley & Sons, Inc.

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**An Implied One Year Forward Rate**

Copyright© 2008 John Wiley & Sons, Inc.

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**Finding a One-Year Implied Forward Rate**

Using the following term structure of interest rates, find the one-year implied forward rate for year two. 1-year Treasury note 1.95% 2-year Treasury note 2.39% 3-year Treasury note 2.71% Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2006 John Wiley & Sons, Inc.**

Calculate the one – year forward rate three – year from now if three and four year spot rates are 5.5% and 5.8% respectively? Answer = 6.7% Copyright© 2006 John Wiley & Sons, Inc.

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**Liquidity Premium Theory**

Unlike the expectation theory that assumes that investors are indifferent between the purchasing long term or short term securities. Liquidity Premium theory believes that long-term securities have greater risk and investors require greater premiums to give up liquidity. Long-term security prices are more sensitive to interest rates (have more price risk). Long-term securities have less marketability. The liquidity premium explains why the yield curve slopes upward most of the time. Liquidity premiums change over time. Copyright© 2008 John Wiley & Sons, Inc.

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**Market Segmentation Theory**

Investors are risk averse, who have maturity preferences by investors may affect security prices (yields), explaining variations in yields by time. The shape of the yield curve is determined by the supply and demand of securities at each maturity. Commercial banks prefer short term securities and they determine the short term yield curve, pension funds and life insurance companies prefer long term securities and they determine the long term yield curve. If market participants do not trade outside their maturity preferences, then discontinuities and spikes are possible in the yield curve. Copyright© 2008 John Wiley & Sons, Inc.

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**Preferred Habitat Theory**

The Preferred Habitat Theory (PH) is an extension of the Market Segmentation Theory. PH allows market participants to trade outside of their preferred maturity if adequately compensated for the additional risk. Therefore, investors are assumed to be not completely risk averse. PH allows for humps or twists in the yield curve, but limits the discontinuities possible under Segmentation Theory. PH is consistent with a smooth yield curve. Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2008 John Wiley & Sons, Inc.**

Which Theory is Right? According to the available evidence, it is difficult to conclude that one theory is totally able to explain fully the shape of the yield curve(term structure of interest rates). 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. Copyright© 2008 John Wiley & Sons, Inc.

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**Yield Curves and the Business Cycle**

Interest rates are directly related to the level of economic activity. An ascending yield curve notes the market expectations of economic expansion and/or inflation. A descending yield curve forecasts lower rates possibly related to slower economic growth or lower inflation rates. Security markets respond to updated new information and expectations and reflect their reactions in security prices and yields. Copyright© 2008 John Wiley & Sons, Inc.

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**Yield-Curve Patterns Over the Business Cycle**

Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2008 John Wiley & Sons, Inc.**

Uses of the Yield Curve The slope of the yield curve can be used to assess the market’s expectations about future interest rates!. Issuers may use the yield curve to price their securities. Investors use the yield curve for a strategy known as riding the yield curve. If the yield is expected to increase in the future, investors will prefer to buy short term bonds, while issuers will prefer to issue long term bonds. If the yield is expected to decrease in the future, investors will prefer to buy long term bonds, while issuers will prefer to issue short term bonds. Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2008 John Wiley & Sons, Inc.**

Default Risk It is the probability of the borrower not paying the promised amount of interest or principal at the agreed time. Risk averse investors want adequate compensation for expected default losses. Investors charge a default risk premium (above riskless or less risky securities) for added risk assumed. Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2008 John Wiley & Sons, Inc.**

Default Risk, cont. DRP = i - irf The default risk premium (DRP) is the difference between the promised or nominal rate and the yield on a comparable (same term) riskless security (Treasury security). Investors are satisfied if the default risk premium is equal to the expected default loss. The larger the default risk premium, the higher the probability of default and the higher the security market yield. Copyright© 2008 John Wiley & Sons, Inc.

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**As credit rating declines, the default risk premium increases**

Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2008 John Wiley & Sons, Inc.**

Default Risk, cont. Default risk premiums increase (widen) in periods of recession and decrease in economic expansion. In good times, investors tend to buy bonds with low credit ratings because they seek the highest yielding investments and during expansion, there is little chance of default. With increased economic recession, investors sell risky securities and buy “high credit rating ones, thus widening the DRP. Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2008 John Wiley & Sons, Inc.**

Credit Ratings Credit rating is a measure of the firm’s default risk in the opinion of the rating agency. The higher the credit rating, the lower the default risk. Cash flow, level of debt, profitability, and variability/riskiness of earnings are all indicators of default riskiness. As conditions change, rating agencies revise credit ratings of debtors. Bonds are called speculative or junk - grade bonds or junk bonds if their Moody's and Standard & Poor's rating is Below Baa (BBB). Bonds are called investment - grade bonds if their Moody's and Standard & Poor's rating is Baa (BBB) and above. Copyright© 2008 John Wiley & Sons, Inc.

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**Copyright© 2008 John Wiley & Sons, Inc.**

Tax Effects on Yields The interest rate most relevant to investors is the rate of return earned after taxes. The after-tax yield, iat, is found by multiplying the pre-tax return by one minus the investor’s marginal tax rate: iat = ibt(1-t) ibt=before tax yield Municipal bonds’ interest income is tax exempt(no tax). Aaa –rated municipal bond has default risk but is tax exempt. Treasury bond has no default risk but is taxed. Corporate bond has default risk and is taxed. Copyright© 2008 John Wiley & Sons, Inc.

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**To Buy a Municipal or a Corporate Bond?**

Investors with high tax brackets(e.g. big firms),holds municipal bonds because of their higher after-tax yield compared to taxable bonds of same risk and maturity. Copyright© 2008 John Wiley & Sons, Inc.

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**Impact of Marketability on Interest Yields**

Marketability – The costs and speed with which investors can resell a security. Cost of trade. Physical transfer cost. Search costs. Information costs. Securities with good marketability have higher demand and so higher prices, hence lower yields. E.g. Treasury bills are the most marketable securities. Copyright© 2008 John Wiley & Sons, Inc.

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**Contract Options and Yields**

Various option provisions may explain yield differences between securities An option is a contract provision which gives the holder or the issuer the right, but not the obligation, to buy, sell, or convert an asset at some specified price within a defined future time period. Copyright© 2008 John Wiley & Sons, Inc.

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**Contract Options and Yields**

A call option permits the issuer (borrower) to call (buy back) the bond before maturity at a pre-specified price (call price). Callable bonds are sold at a higher market yield than comparable non-callable bonds because they are to the benefit of the issuer. Bondholder demand a call interest premium (CIP). CIP = ic – inc ic =yield on callable bonds, inc = yield on non-callable bonds. Issuers call bonds if interest rates decline and thus bondholder in callable securities bear the risk of losing their high-yielding security. Copyright© 2008 John Wiley & Sons, Inc.

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**Contract Options and Yields**

A put option permits the bondholder(lender) to sell the bond back to the issuer at a pre-specified price before maturity. The yield on a putable bond, (ip,) will be lower than the yield on the non-putable bond, (inp)because they are advantage to the bondholder. The difference in interest rates between putable and nonputable contracts is called the put interest discount (PID). PID = ip - inp Bondholders are likely to put their bonds during periods of increasing interest rates. Copyright© 2008 John Wiley & Sons, Inc.

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**Contract Options and Yields**

A conversion option permits the bondholder to convert a bond into another security (usually common stock). Convertible bonds generally have lower yields, (icon) than non-convertibles, (incon )because it is an advantage to the bondholder. The conversion yield discount (CYD) is the difference between the yields on convertibles relative to non-convertibles. CYD = icon – incon Bondholders tend to use the conversion option when the stock market prices are rising and bond prices are declining. Copyright© 2008 John Wiley & Sons, Inc.

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