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**FINC4101 Investment Analysis**

Instructor: Dr. Leng Ling Topic: Term Structure Analysis

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Learning objectives Define term structure of interest rates and yield curve. Explain the purpose of theories of the term structure Describe the expectations theory and discuss its implications. Show how different expectations of future short-term interest rates can lead to different yield curves. Use the expectations theory to infer future short-term interest rates (forward rates). Describe the liquidity preference theory. Define the liquidity premium.

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**Term structure of interest rates**

Relationship between yields to maturity and term to maturity across bonds. Yield curve: The graphical representation of the term structure. Term to maturity is just the time to maturity.

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**Term structure/ yield curve has 4 different ‘looks’**

Term structure/ yield curve can take on one of the following shapes: Rising/ upward sloping (most common) Inverted/ downward sloping/ falling Hump-shaped, rising and then falling Flat The theories of term structure tries to explain why term structure/ yield curve is the way it appears. Upward sloping: long-term bonds have higher YTMs than short-term bonds Inverted: short-term bonds have higher YTMs than long-term bonds Hump-shaped: over a certain range of maturities, longer term bonds have higher YTMs than shorter term bonds. Beyond that range, the reverse is true. Flat: all maturities have the same YTM/ interest rate.

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**Term structure/ yield curve has 4 different ‘looks’**

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**Expectations Theory (1)**

Yields to maturity are determined by expectations of future short-term interest rates. In other words, the shape of the term structure/ yield curve depends on the expected short-term interest rates in the future.

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**Expectations Theory (2)**

Depending on what investors expect the future short-term rate to be: Higher than current short-term rate, or Lower than current short-term rate, or Same as current short-term rate We get different yield curves.

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**Expectations Theory (3)**

market expecting increases in future short-term interest rates leads to upward sloping yield curve. market expecting decreases in future short-term interest rates leads to downward sloping yield curve. market expecting no change in future short-term interest rates leads to flat yield curve. Market: financial market, or investors in the aggregate.

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**How does the expectations theory work?**

Current 1-year interest rate is 8%. Suppose everyone in the market expects that one year from now, the 1-year interest rate will rise to 10%. How does this expectation determine the current 2-year interest rate? All interest rates are quoted on an annual basis. In this situation, investors can choose one of two possible strategies. Here use an example to show how the theory produces the predictions about the yield curve shape.

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**Investment strategies**

For simplicity, assume All bonds are zero-coupon bonds with $1000 face value. You can invest in fractional amount of bonds Rollover: Buy 1-year bond now. When it matures, buy another 1-year bond next year and hold till it matures. Buy-and-hold: Buy a 2-year bond now and hold it till it matures in year 2. The expectations theory says that the two strategies should produce the same expected total returns. The expectations theory says that the two strategies should produce the same total returns. from this, we can derive the two-year interest rate.

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**Deriving the 2-year interest rate (1)**

Suppose you have $1 to invest. With the rollover strategy, at the end of year 1, you get: $1 x (1 + current 1-year interest rate) = 1 x (1.08) = 1.08 Suppose at the end of year 1, the 1-year interest rate is 10% as expected. You invest $1.08 (total proceeds) in a 1-year bond promising 10%. After two years, you get: 1.08 x (1.10) = $1.188

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**Deriving the 2-year interest rate (2)**

Let y2 be the 2-year interest rate. According to the expectations theory, investing $1 in a 2-year bond with an interest rate of y2 must also produce a total return of $ That is, For the theory to hold, investors must be able and willing to exploit mis-alignments in interest rates by means of arbitrage. If 2-year rate is higher than 8.995%, then investors will all want to buy 2-year bonds because the buy-and-hold strategy offers higher total returns than rollover. This increased demand will push up prices of 2-year bond and consequently push down the yield on such bonds until the yield settles down to 8.995%. At this rate, both strategies are equally attractive. Conversely, if 2-year bond interest rate is less than 8.995%, then the rollover strategy will be more attractive. investors will sell 2-year bonds and put their money in the rollover strategy. Demand for 2-year bonds will fall. As demand falls, the price of 2-year bonds also falls while the yield will rise. This continues until the yield on 2-year bond reach 8.995%. At this point, both strategies are equally attractive and investors have no incentive sell 2-year bonds and change prices and thus yields.

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**Compare 1-year and 2-year rates**

1-year interest rate: 8% 2-year interest rate: 8.995% Conclusion: longer maturity bonds have higher interest rates/ yields to maturity. The term structure is upward sloping! * If we observe 1-year rate 8% and 2-year rate 8.995%, then market expects that, one year from now, the 1-year rate will increase from current 8% to 10%.

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Verify for yourself If the market expects future 1-year interest rate to fall to 6%: 2-year interest rate is 6.995%. Yield curve is downward sloping. If market expects future 1-year interest rate to stay at 8%: 2-year interest rate is 8%. Yield curve is flat. Tell student to go back to the example, assume current 1-yr rate is 8%. Tell them to experiment with these two alternative expectations to produce (i) an inverted yield curve and (ii) a flat yield curve. Inverted: ========= (1 + y2)2 = (1.08)(1.06) = y2 = (1.1448)0.5 – 1 = or 6.995% Flat: ===== (1 + y2)2 = (1.08)(1.08) = y2 = (1.1664)0.5 – 1 = 0.08 or 8%

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**Inferring expected future rates (1)**

Expectations theory says that the expected total returns from the buy-and-hold strategy and the rollover strategy are the same. We can use this assumption to infer expected future interest rates, which are called “forward rates”. Forward rate: The inferred short-term interest rate for a future period that makes the expected total return of a long-term bond equal to that of rolling over short-term bonds.

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**Inferring expected future rates (2)**

In general, we obtain the forward rate by equating the return on an n-period zero-coupon bond with that of an (n – 1)-period zero-coupon bond rolled over into a one-year bond in year n: (1 + yn)n = (1 + yn-1)n-1(1 + fn) YTM of n-period zero coupon bond YTM of (n-1)-period zero coupon bond Forward rate for period n.

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**Inferring expected future rates (3)**

Thus, the forward rate formula is Re-arranging the formula gives us the forward rate formula.

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**Applying the forward rate formula**

Two-year maturity bonds offer yield-to-maturity of 6%, and three-year bonds have yields of 7%. What is the forward rate for the third year? Verify that the forward rate is 9.03% f3 = (( )3/ ( )2 ) – 1 = ( /1.1236) – 1 = – 1 = = or 9.03%

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Problems (1) The current yield curve for default-free zero-coupon bonds is as follows: Assume that the zero-coupon bonds have a face value of $1000. Maturity (Years) YTM 1 10% 2 11 3 12 BKM EOC chap 9, Q37

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**Problems (2) What are the implied one-year forward rates?**

Assume that the pure expectations hypothesis is correct. If market expectations are accurate, one year from now what will be the yields to maturity on one- and two-year zero coupon bonds? If you purchase a two-year zero-coupon bond now, what is the expected total rate of return over the first year? What if you purchase a three-year zero-coupon bond? Use the forward rate formula. 1-year forward rate from end of year 1 to end of year 2, f2 = (1.11)2/(1.1) – 1 = or 12.01% 1-year forward rate from end of year 2 to end of year 3, f3 = (1.12)3/(1.11)2 – 1 = or 14.03% 2) Forward rates = expected interest rates. Therefore, YTM on 1-year zero = f2 = 12.01% To find YTM on 2-year zero, make use of the idea that rolling over 1-year bonds must produce same total expected return as buying and holding a 2-year bond. Let y2 be YTM on 2-year zero. Total return from buy-and-hold = (1 + y2)2 Total return from rollover = (1.1201)(1.1403) = Therefore, y2 = ( )0.5 – 1 = or 13.02% 3) Assume $1000 par value. 2-year zero coupon bond. Current price = 1000/(1.11)2 = At the end of the year, the bond becomes a 1-year zero. So use the expected YTM on 1-year bond next year (equal to 1-year forward rate) of 12.01% Year end price = 1000/(1.1201) = Expected total rate of return = (892.78/811.62) – 1 = = 0.10 or 10% 3-year zero coupon bond. Current price = 1000/(1.12)3 = At the end of the year, the bond becomes a 2-year zero. So use the expected YTM on 2-year bond next year of 13.02% Year end price = 1000/(1.1302)2 = Expected total rate of return = ( / ) – 1 = = 0.10 or 10%. So this is consistent with the expectations theory.

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**Liquidity Preference Theory**

Investors prefer to hold short-term bonds because short-term bonds have more ‘liquidity’ than long-term bonds. Short-term bonds offer greater price certainty and trade in more active markets. Investors like these attributes. Because long-term bonds are less liquid by comparison, investors demand a liquidity premium for holding long-term bonds. Liquidity premium: extra expected return demanded by investors as compensation for the lower liquidity of long-term bonds.

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**Liquidity Preference Theory**

The theory says that investors demand a liquidity premium on long-term bonds. Thus, long-term bonds have higher interest rates than short-term bonds. This gives rise to an upward sloping yield curve. Even if future short-term interest rates are expected to remain unchanged, yield curve will be upward sloping because of the liquidity premium. In contrast, under the expectations theory, if future short-term rates are expected to stay the same, yield curve will be flat.

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Liquidity premium The spread between the forward rate of interest and the expected short-term rate: Liquidity premium = forward rate – expected short-term rate Re-arranging the formula, we get Forward rate = expected short-term rate + liquidity premium So, forward rate > expected short-term rate So, if the liquidity preference theory is true, forward rate > expected short-term rate. In contrast, if the expectations theory is correct, forward rate = expected short-term rate.

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More problems (1) The yield to maturity on one-year zero coupon bonds is 8%. The yield to maturity on two-year zero-coupon bonds is 9%. a) What is the forward rate for the second year? a) The forward rate (f2) is the rate that makes the return from rolling over one-year bonds the same as the return from investing in the two-year maturity bond and holding to maturity: 1.08 (1 + f2) = (1.09)2 f2 = = 10.01%

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More problems (2) b) If you believe the expectations theory, what is your best guess as to the expected short-term interest rate next year? c) If you believe the liquidity preference theory, is your best guess as to next year’s short-term interest rate higher or lower than in (b)? b. According to the expectations hypothesis, the forward rate equals the expected value of the short-term interest rate next year, so the best guess would be 10.01%. c. According to the liquidity preference hypothesis, the forward rate exceeds the expected short-term interest rate next year, so the best guess would be less than 10.01%.

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**More problems (3) Which of the following statements is true?**

The expectations theory indicates a flat yield curve if anticipated future short-term rates exceed current short-term rates. The basic conclusion of the expectations theory is that the long-term rate is equal to the anticipated short-term rate. The liquidity preference theory indicates that, all other things being equal, longer maturities will have higher yields. The liquidity preference theory states that a rising yield curve implies that the market anticipates increases in interest rates. Answer is (3).

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**Summary Term structure of interest rates.**

Theories of the term structure try to explain the shape of the term structure/ yield curve. Pure Expectations theory and its implications. Calculate forward rates using observed interest rates of different maturities. Liquidity Preference theory and its implications. Relationship between forward rate, expected short-term rate and liquidity premium.

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Practice 8 Chapter 10: (a)(b), 41.

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