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The Term Structure of Interest Rates

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I. Yield Curve Yield: The single interest rate that equates the present value of a bond’s payments to the bond’s price. Yield to Maturity: A measure of the average rate of return that will be earned on a bond if held to maturity. Yield Curve:A curve showing the relationship between the yield to maturity and the maturity.

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–Ascending Curve (Normal Curve) Yield Maturity –Descending Curve (Inverse Curve) Yield Maturity

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–Flat Curve Yield Maturity –Humped Curve Yield Maturity

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II. Term Structure Short Rate: The interest rate for a given time interval. Spot Rate: The interest rate appropriate for a given maturity. Spot Rate Curve: The graphical depiction of the relationship between the spot rate and its maturity.

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Theoretical Spot Rate:The theoretical interest rate appropriate for a given maturity. Theoretical Spot Rate Curve: The graphical depiction of the relationship between the theoretical spot rate and its maturity.

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Forward Rate: The interest rate for a future period. Implicit Forward Rate (Implied Forward Rate): The interest rate for a future period, computed on the basis of theoretical spot rates.

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0123...n-1n year Time Line Short Short Short Short Rate 1 Rate 2 Rate 3 Rate n Spot Rate 2 Forward Rate 3 Spot Rate n-1 Forward Rate n

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III. Theoretical Spot Rate On-the-Run Treasury Issues: The most recently auctioned Treasury securities of a given maturity. The Principle Underlying Bootstrapping: The value of the Treasury coupon security should be equal to the value of the package of zero-coupon Treasury securities that duplicates the coupon bond’s cash flow.

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CouponYield to MaturityRateMaturityPrice 0.50.0000.080$96.15 1.00.0000.083 92.19 1.50.0850.089 99.45 2.00.0900.092 99.64 2.50.1100.094 103.49 3.00.0950.097 99.49

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Cash flow for the 1.5-year Treasury: Maturity value = $100.00 Coupon rate = 8.5% MaturityCash Flow 0.5$100 x 0.5 x 8.5% = $4.25 1.0$100 x 0.5 x 8.5% = $4.25 1.5 $100 x 0.5 x 8.5% + $100 = $104.25

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The present value of the cash flow: PV = t [CF t / (1 + z t ) t ], where PV = present value, CF t = cash flow of period t, z t = actual or theoretical spot rate for period t.

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Theoretical 1.5-year spot rate: 0.5-year spot rate = 0.080 / 2 (given) 1.0-year spot rate = 0.083/ 2 (given) 99.45 = [4.25 / (1 + 0.04)] + [4.25 / (1 + 0.0415) 2 ] + [104.25 / (1 + z 3 ) 3 ]. z 3 = 0.04465.

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Cash flow for the 2-year Treasury: Maturity value = $100.00 Coupon rate = 9.0% MaturityCash Flow 0.5$100 x 0.5 x 9.0% = $4.5 1.0$100 x 0.5 x 9.0 % = $4.5 1.5$100 x 0.5 x 9.0 % = $4.5 2.0 $100 x 0.5 x 9.0 % + $100 = $104.5

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Theoretical 2-year spot rate: 0.5-year spot rate = 0.080 / 2 (given) 1.0-year spot rate = 0.083 / 2 (given) 1.5-year spot rate = 0.04465 (derived) 99.64 = [4.5 / (1 + 0.04)] + [4.5 / (1 + 0.415) 2 ] + [4.5 / (1 + 0.04465) 3 ] + [104.25 / (1 + z 4 ) 4 ]. z 4 = 0.046235.

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IV. Implicit Forward Rate (1+ z n ) n (1+ n f t ) t = (1+ z n+t ) n+t, or (1+ n f t ) t = (1+ z n+t ) n+t / (1+ z n ) n, where z n = theoretical spot rate, n f t = the forward rate, n periods from now, at will be carried by loans with maturities of t periods, n, t = 1, 2, 3, ….

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V. Theory of The Term Structure A. The Expectations Theory The expectations theory holds that the shape of the yield curve is determined by the investors’ expectations of future interest movements, and that changes I these expectations change the shape of the yield curve.

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The forward rate equals the market consensus expectation of the future short rate, or i f 1 =E(r i ). The yield to maturity would thus be determined solely by current and expected future one-period rates (r 1, i f 1 ).

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1+ y n = [(1+ r 1 )(1+ 2 f 1 )... (1+ n f 1 )] 1/n. The long-term rate is the geometric mean of the short-term rates. An upward-sloping yield curve would be an indication of higher expected forward rates over time.

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B. The Liquidity Preference Theory Short-term investors dominate the market so that the forward rate exceeds the expected short rate. The excess of f i over E(r i ) is known as the liquidity premium.

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Yield Constant liquidity premium Constant forward rate Yield curve Constant expected short rate Maturity

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Yield Rising liquidity premium Rising forward rate Rising yield curve Rising expected short rate Maturity

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Yield Constant liquidity premium Falling forward rate Humped yield curve Falling expected short rate Maturity

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C. The Market Segmentation Theory (Institutional Theory) The market segmentation theory maintains that market participants have strong preferences for securities of a particular maturity and holds that they buy and sell securities consistent with these maturity preferences.

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Long- and short-maturity bonds are traded in essentially distinct or segmented markets, each of which finds its own equilibrium independently.

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Yield D m S l D l S m S s D s Years to maturity

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D. The Preferred Habitat Theory The preferred habitat theory asserts that investors will not hold debt securities outside of their preferred habitat (maturity preference) without an additional reward in the form of a risk premium. That is, investors prefer specific maturity ranges but can be induced to switch if premiums are sufficient.

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VI. Economic Implications of the Yield Curve A. Yield Curve and Business Cycle Interest rates and the business cycle are procyclical.Increasing interest rates imply that market participants expect a period of economic expansion. Descending yield curve are common near the final phase of a period of economic expansion.

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As the spread between long-term and short- term rates narrows, the market consensus is that the rate of economic expansion will be slowing.

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Interest rate Time Interest rateInterest rate Interest rate MaturityMaturityMaturity

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B. Yield Curve and Financial Intermediaries The more steeply the yield curve slopes upward, the wider the spread earned by depository institutions.

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Interest rate Spread Maturity

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Introduction to Fixed Income – part 2

Introduction to Fixed Income – part 2

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