FIXED-INCOME SECURITIES
Chapter 3
Term Structure of Interest Rates: Empirical Properties and Classical Theories
Outline
• • • • •
Types of TS Shapes of the TS Dynamics of the TS Stylized Facts Theories of the TS
Types of Term Structures • The term structure of interest rates is the series of interest rates ordered by term-to-maturity at a given time • The nature of interest rate determines the nature of the term structure – The term structure of yields to maturity – The term structure of zero coupon rates – The term structure of forward rates
• TS shapes – – – –
Quasi-flat Increasing Decreasing Humped
Quasi-Flat US YIELD CURVE AS ON 11/03/99 7.00%
6.50%
par yield
6.00%
5.50%
5.00%
4.50%
4.00% 0
5
10
15 maturity
Quasi-Flat Quasi-Flat
20
25
30
Increasing JAPAN YIELD CURVE AS ON 04/27/01 2.50%
2.00%
par yield
1.50%
1.00%
0.50%
0.00% 0
5
10
15 maturity
Increasing Increasing
20
25
30
Decreasing UK YIELD CURVE AS ON 10/19/00 6.00%
par yield
5.50%
5.00%
4.50% 0
5
10
15
20
maturity
Decreasing Decreasing(or (or inverted) inverted)
25
30
Humped (1) EURO YIELD CURVE AS ON 04/04/01 5.50%
par yield
5.00%
4.50%
4.00% 0
5
10
15
20
maturity
Humped Humped (decreasing (decreasingthen then increasing) increasing)
25
30
Humped (2) US YIELD CURVE AS ON 02/29/00 7.00%
par yield
6.50%
6.00%
5.50% 0
5
10
15
20
maturity
Humped Humped(increasing (increasing then thendecreasing) decreasing)
25
30
Dynamics of the Term Structure • The term structure of interest rates changes in response to – Wide economic shocks – Market-specific events
• Example – On 10/31/01, Treasury announces that there will not be any further issuance of 30 year bonds – Price of existing 30 year bonds is pushed up (buying pressure) – 30 year rate is pushed down
Example – US YTM TS US term structure of government YTM
7
6 YTM 5 0 4 4 J-01
8 12
S-00 16
maturity
M-00 J-00
20 S-99
24 M-99
28 J-99
time
Stylized Facts (1) : Mean Reversion • Mean reversion: high (low) values tend to be followed by low (high) values • Example: 10 Y swap rate versus Dow Chemical 50 45 40 35 30 Dow Chemical en US $ Taux de swap 10 ans en %
25 20 15 10
01/01/1999
01/01/1998
01/01/1997
01/01/1996
01/01/1995
01/01/1994
01/01/1993
01/01/1992
01/01/1991
0
01/01/1990
5
Stylized Facts (2) : Correlation • Rates with different maturities are – Positively correlated one another – Not perfectly correlated though (more than one factor) – Correlation decreases with difference in maturity
• Example: France (1995-2000) 1M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y
1M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y 1 0.999 1 0.908 0.914 1 0.546 0.539 0.672 1 0.235 0.224 0.31 0.88 1 0.246 0.239 0.384 0.808 0.929 1 0.209 0.202 0.337 0.742 0.881 0.981 1 0.163 0.154 0.255 0.7 0.859 0.936 0.981 1 0.107 0.097 0.182 0.617 0.792 0.867 0.927 0.97 1 0.073 0.063 0.134 0.549 0.735 0.811 0.871 0.917 0.966 1
Stylized Facts (3) • The evolution of the interest rate curve can be split into three standard movements – Shift movements (changes in level), which account for 70 to 80% of observed movements on average – A twist movement (changes in slope), which accounts for 15 to 30% of observed movements on average – A butterfly movement (changes in curvature), which accounts for 1 to 5% of observed movements on average
• That 3 factors account for more than 90% of the changes in the TS is valid – Whatever the time period – Whatever the market
Shift Movements Upwar -Downward Shift Movements 7
6
yield (in %)
5
4
3
2
1
0 0
5
10
15 maturity
20
25
30
Twist Movements Flattening - Steepening Twist Movements 8
7
6
yield (in %)
5
4
3
2
1
0 0
5
10
15 maturity
20
25
30
Butterfly Movements Concave - Convex Butterfly Movements 6
5.5
5
yield (in %)
4.5
4
3.5
3
2.5
2 0
5
10
15 maturity
20
25
30
Theories of the Term Structure • Studying the TS boils down to wondering about the preferences of participants' for curve maturities – Investors – Borrowers
•
Indeed, if they were indifferent in terms of maturity – Interest rate curves would be invariably flat – Notion of TS would be meaningless
• Market participants' preferences can be guided – By their expectations – By the nature of their liability or asset – By the level of the risk premiums they require
Theories of the Term Structure • Term structure theories attempt to account for the relationship between interest rates and their residual maturity • They fall within the following categories – Pure expectations – Pure risk premium • Liquidity premium • Preferred habitat • Market segmentation
• To these main types, we can add – The biased expectations theory, that combines the first two theories
Theories of the Term Structure • Remember: 1+R0,t = [(1+ R0,1)(1+ F1,2)(1+ F2,3)…(1+ Ft-1,t)]1/t
• The pure expectations theory postulates that forward rates exclusively represent future short term rates as expected by the market • The pure risk premium theory postulates that forward rates exclusively represent the risk premium required by the market to hold longer term bonds • The market segmentation theory postulates that – Each of the two main market investor categories is invariably located on a given curve portion (short, long) – As a result, short and long curve segments are perfectly impermeable
Pure Expectations • TS reflects market expectations of future short-term rates – An increasing (resp. flat, resp. decreasing) structure means that the market expects an increase (resp. a stagnation, resp. a decrease) in future shortterm rates
• Example: from a flat curve to an increasing curve – The current TS is flat at 5% – Investors expect a 100bp increase in rates within one year – For simplicity, assume that the short (resp.long) segment of the curve is the one-year (resp. two-year) maturity – Then, under these conditions, the interest rate curve will not remain flat but will be increasing – Why?
Pure Expectations (Cont’) • Consider a long-term investor (2-year horizon) – His objective is maximizing his return on the period – Either invests in a long 2-year security or invests in a short 1-year security, then reinvests in one year the proceeds in another 1-year security
• Before interest rates adjust at the 6% level – First option returns an annual return of 5% over two years – Second option returns 5% the first year and, according to his expectations, 6% the second year, i.e., 5.5% on average per year over two years
• Investor will thus buy short bonds (one year) rather than long bonds (two years) – Similar behavior for the short-term investor (return on 2-y bond after 1 year is 4.05%=(5+105/1.06-100)/100 < 5% (return on 1 year bond) – As a result • The price of the one-year bond will increase (its yield will decrease) • The price of the two-year bond will decrease (its yield will increase) – The curve will steepen
Pure Expectations (Cont’) • In summary, market participants behave collectively to let the relative appeal of one maturity compared to the others disappear • In other words, they neutralize initial preferences for some curve maturities • The pure expectations theory has an important limitation – Investors behave in accordance with their expectations for the unique purpose of maximizing their investment return – They are risk neutral - They do not take into account the fact that their expectations may be wrong – The pure risk premium theory includes this contingency
Pure Risk Premium • Indeed, if forward rates were perfect predictors of future rates, future bond prices would be known with certainty • Unfortunately, it is not the case – Future interest rates are unknown (re-investment risk) – Future bond prices are unknown (market risk)
• Example: an investor with a 3 years horizon – May invest in a 3-year zero coupon bond and holding it until maturity – May invest in a 5-year zero coupon bond and selling it in 3 years – May invest in a 10-year zero coupon bond and selling it in 3 years
• What would you prefer? – Return of the first investment is known ex-ante with certainty – Not the case for the 2nd and 3rd – We don’t know the price of these instruments in three years
Pure Risk Premium (con’t) • However, we know something about their risk (volatility) • A bond price risk measured by price volatility – Tends to increase with maturity (P’(r)>0) – In a decreasing proportion (P’’(r)<0)
• Assume interest rates increase to 6% – The long bond price will fall to 5/1.06+ 105/1.062 = $ 98.17 – The short one will fall to 105/1.06 = $99.06. – Decrease in 2-year bond price nearly twice as big as decrease in 1-year bond price
• Pure risk premium theory: TS reflects risk premium required by the market for holding long bonds • The two versions of this theory differ about the shape of the risk premium
Pure Risk Premium - Liquidity • Risk premium increases with maturity in a decreasing proportion • Formally 1+Ro,t = [(1+ R0,1) (1+ E(R1,2)+ L2) (1+ E(R2,3)+ L3)… (1+ E(Rt-1,t)+ Lt)]1/t
• Lk is the liquidity (actually risk) premium required by the market to invest in a bond maturing in k years 0 = L1 < L2 < L3 < ... < Lt L2 -L1 > L3 -L2 > L4 -L3 > ... > Lt -Lt-1
• Hence, an investor will be interested in holding all the longer bonds as their return contains a high risk premium, offsetting their higher volatility • Implies that a “normal” TS is increasing
Preferred Habitat • Postulates that risk premium is not uniformly increasing • Indeed, investors have a preferred investment horizon dictated by the nature of their liabilities (shorter is not always better) • Nevertheless, depending on bond supply and demand on specific segments – Some lenders and borrowers are ready to move away from their preferred habitat – Provided that they receive a risk premium that offsets their price or reinvestment risk aversion
• Thus, all curves shapes can be accounted for
Market Segmentation • Extreme version of pure risk premium theory – Investors never move away from preferred habitat (infinite risk premia) – Commercial banks invest on a short/medium term basis – Life-insurance companies and pension funds invest on a long term basis
• Shape of the curve determined – By supply and demand on short and long-term bond markets – Insurance comp., pension funds are structural buyers of long-term bonds – Commercial banks' behavior is more volatile: banks prefer to lend money directly to corporations and individuals than invest in bond securities
• Their demand for short-term bonds is influenced by business conditions – During growth periods, sell bonds to meet corporations' and individuals' demand for loans => relative increase in short-term yields – During slow-down periods, corporations and individuals pay back their loans, thus increasing bank funds; then banks invest in short-term bonds => relative decrease in short-term yields compared to long-term yields
Biased Expectations Theory and Stochastic Approach • All these theories are not mutually exclusive • Biased expectations theory is an integrated approach – Combines pure expectations theory and risk premium theory – Postulates that TS reflects market expectations of future interest rates as well as permanent liquidity premia that vary over time
• Thus, all curve shapes can be accounted for • Stochastic Approach – – – –
Uncertainty about future interest rates is not implicit in current TS Difficult to correctly anticipate future interest rates driven by surprise effects TS modeled as a predictible term plus a stochastic process This theory represents an alternative to traditional theories generally used for pricing and hedging contingent claims
Synthesis â&#x20AC;&#x201C; Explanation of TS Shapes