Linkages between Equilibrium Asset Yields: the Structure of Interest Rates
- See: Mishkin and Serletis, Ch. 6 and Ch. 18 pp. 481-483.
- Concern: the inter-relatedness of financial markets and its consequences for the
structure of interest rates.
- Why are financial markets interrelated?
- borrowers and lenders view alternative assets as substitutes:
- alternative ways of raising (borrowing) funds;
- alternative ways of saving (lending) funds.
- if the yield on an asset is high compared to yields on substitute assets:
- there will be aflow of lending into that asset (demand shifts right)
i.e. as lenders desert alternative assets.
- borrowers seek to borrow in "cheaper" ways (supply shifts left)
- the yield on the asset (and its price) falls back into line with other
yields as a result of these shifts.
- key point: flows of lending and borrowing between markets link yields.
- Last set of notes: Supply and demand model of an asset market
- position of the demand (lender) and supply (borrower) curves for specific assets
depend on yields of other assets.
- this builds in interdependence.
- equilibrium interest rates will be equal if the assets are ‘perfect substitutes’
- if ‘imperfect substitutes’ equilibrium yields will differ by an amount sufficient to
compensate for the differences between the assets.
- Linkages to be examined: all cases of imperfect substitutes
- Risk premia (relation between assets of different riskiness)
- Term structure theory (between assets with different terms)
- International interest rate linkages (between assets from different
countries)
Risk Premia and the Supply-Demand Framework
- Focus on observed yields: not expected yields.
i.e. no adjustment made for default risk.
- Risk is unattractive to lenders.
- clearly the case for “default risk” (lowers the expected yield)
- arguably the case for risk as measured by “variance” of returns
(assumesrisk averse lenders and concerns asset risk that cannot be
diversified away).
- Say that two assets are identical in all respects except risk.
- If the yield on the safe asset (Rs) is the same as that of the risky asset (Rr):
Rs = Rr (at R0 in diagram)
- lenders will choose to hold the safe asset instead of the risky asset.
- as lenders move between assets:
Rs will fall(demand for safe asset shifts right)
Rr will rise(demand for riskier asset shifts left)
- this continues until Rr is sufficiently higher than Rs to compensate
lenders for risk.
- Risk premium: Rr - Rs
- often quoted in “basis points”: 100 basis points is 1%.
- Equilibrium:
Rr = Rs + risk premium
- Lender decisions will ensure that this condition will hold.
RrRs+ risk premium
- lenders move to the risky asset, Rr falls, Rs rises
until Rr-Rsequals the risk premium.
RrRs + risk premium
- lenders move to the safe asset, Rs falls, Rr rises until Rr-Rs
equals the risk premium.
- Across markets lender behavior will ensure a structure of yields linked to
the riskiness of assets.
- If relative riskiness is not changing this premium will be stable and interest rates on
assets of different riskiness will move together.
i.e. Rr = Rs + risk premium
- so if the premium is stable of 1% rise in Rs will go hand-in-hand with a 1% rise
in Rr.
- Private vs. government debt as an example:
- Lower risk a reason why government yields are usually below corporate yields.
- See text Fig. 7.2: yields on Government vs. Corporate bonds (long-term).
- Here is Federal Treasury Bill vs. Corporate Paper (yields in %):
- Bond rating agencies.
- Canada: Standard and Poor's, Dominion Bond Rating Service, Moodys, Fitch
- rate government and corporate debt according to risk.
- low-risk ratings are associated with lower yields (see text Table 6-1).
- diagram below shows US corporate bond yields by ratings (AAA – low risk;
BBB – medium risk; BB higher risk):
- Risk structure of yields changes as perceived risk changes.
- Recessions of early-1980s and early-1990s
- risk premium on corporate debt rose.
- Enron scandal and BBB vs. AAA spread on corporate bonds.
- old ratings regarded as less reliable: risk premia rise.
- Recent financial crisis
:
- August 2007 and sub-prime crisis (Bear-Stearns collapse).
- unclear who holds the “bad” mortgages.
- jump between yields on 3 month T-Bills and 3-month Corporate
paper jumps:
T-Bill Paper Spread (difference)
July 18, 20074.54% 4.64% 0.10%
Aug. 22, 20073.96% 5.05% 1.09%
- Lehman Brothers collapse (Sept. 2008): further rise in rick premia.
- European government debt: Greek vs German government bonds
- Supply-Demand and changing risk structure:
- initial premium (Rr-Rs) is just sufficient to compensate lenders for risk.
- rise in perceived riskiness causes lenders to shift from the risky to the safe asset
until the new, larger risk premium is just sufficient to compensate them for
the higher level of risk.
- notice that the yields in this case move in different directions.
- Risk structure of interest rates and information:
- observed yield differentials reflect lender assessment of relative riskiness.
- builds in market expectations of things like default probabilities.
- Liquidity premia:
- Can be modeled the same way as risk premia.
- Liquid assets: can be easily converted into money at a "fair" price.
e.g. via an active resale market.
- Liquidity is another characteristic that lenders value:
- two assets identical except one is more liquid;
- less liquid asset will pay a premium.
- Liquidity and risk: could treat the liquidity premium as part of the risk premium.
e.g. illiquid asset has a riskier resale price.
- Crisis and liquidity: some markets “freeze up” – no buyers (lenders).
Term Structure of Interest Rates
Term structure: how do yields vary by term to maturity (other things equal).
Yield curve:
Shows the relationship between the term to maturity and the yield on
otherwise similar assets.
- Focus when measuring term structure is usually on government debt
- Why government debt?
- yields by term are likely to be purely “term effects”
- long and short term assets: similar risk (same issuer)
- good secondary market for government debt
(similar liquidity across assets)
- so: differences in yields by term are likely to reflect only
term structure.
- The term structure arguments are valid for any financial assets that differ by
term to maturity.
- but if the assets differ in characteristics other than term it is harder to
isolate the pure term structure effect.
- Yields curve shapes? (interest rate in % on vertical axis, term on horizontal)
Explanations of term structure
- Theories of term structure can be viewed as applications of the Supply-Demand
model of asset markets.
Expectations Theory
- Key Assumption:many lenders regard short-term and long-term debt as
perfect substitutes (other things equal).
- Consequence?
- over the same period a sequence of short-term assets must
provide the same yield as longer-term assets
e.g., the example treats the bond like a T-Bill (no coupon payment just a single payment,
the argument can be extended to the more general case)
Say the lender wants to save funds for two years:
Option 1:hold one two-year bond
Yield = R2 (NOTE: yield of 5%, R2=.05)
Invest $1 now, get :
(1+R2)(1+R2) = (1+R2)2
in two years.
Option 2:hold two one year bonds sequentially:
Yield on first bond (matures in 1 yr): R1
Expected yield on second bond: E11
(E11 = yield on 1 yr. bond issued one year from now)
So invest one dollar now get:
(1+R1)(1+E11) in two years time.
- Lender will be indifferent between the two options if:
(1+R2)2=(1+R1)(1+E11)
this will have to hold in equilibrium.
e.g., say
(1+R2)2< (1+R1)(1+E11)
- lenders buy one-year bonds, avoid two-year bonds
- so R1 falls and R2 rises until equality holds
- If :(1+R2)2(1+R1)(1+E11)
- lenders buy two-year bonds and avoid one-year bonds
- so R2 falls and R1 rises until equality is established.
(draw the underlying supply-demand diagrams!)
(Demand shifts between markets for 1-yr. and 2-yr. bonds drive the result)
Generally: for an n-year lending period.
Option 1:
- Hold one n-year bond (average annual yield of Rn) for n- years:
In n-yrs. $1 invested now pays(1+Rn)n
Option 2:
- Hold n one-year bonds sequentially with:
- yield of R1 for the first bond and
- yield E1J for the 1 yr. bond issued J years from now.
One dollar invested now pays:
(1+R1)(1+E11)(1+E12)...(1+E1n-1) in n years (note there are n
terms in the expression)
- Equilibrium:
(1+Rn)n= (1+R1)(1+E11)(1+E12)...(1+E1n-1)
- lenders are indifferent between the options
- A popular approximation to the equilibrium condition?
Yield on long-term = average of the yields on short-term assets
Rn = (R1 + E11 + E12 +...+E1n)
n
[ where does this come from? take logarithms of :
(1+Rn)n= (1+R1)(1+E11)(1+E12)...(1+E1n-1)
N∙ln(1+Rn) = ln(1+R1) + ln(1+E11)+ ln(1+E12)+…++ ln(1+E1N)
and note that: log(1+i) i for small i ]
- Similar reasoning links yields and expected yields of any combination of different-term
assets held for the same time period
e.g. Say five years is the period lending occurs over:
- Some options: one 5-year bond;
five one-year bonds;
one 3-yr. bond, then one 2-yr. bond etc.
- You can write down an equilibrium condition for each pair of options, i.e. they
should all pay the same in equilibrium.
Some Implications of the Pure Expectations theory?
(1) Expectations of future yields determine the shape of the yield curve:
- Flat: expect stable future short-term yields
- Upward slope: future short-term yields will be higher than current yield.
- Downward slope: future short-term yields will be lower than the current yield.
i.e.,
(1+R2)2 = (1+R1)(1+E11)
R1=E11 then R1=R2
R1>E11 then R1>R2
R1<E11 then R1<R2
(Yield curves in the past few years: very low short-term yields -- can't fall further!
So future yields almost certainly higher than now → upward slope)
- More complicated yield curve shapes? Also reflects the pattern of expected
future rates.
- Consider rate on R3 vs R2. Can write an equilibrium condition:
(1+R3)3 =(1+R2)2 (1+E12)
Then:R3=R2 if R2=E12 (flat curve between 2 and 3 yrs)
R3>R2 if R2<E12 (rising curve between 2 and 3 yrs)
R3<R2 if R2>E12 (falling curve between 2 and 3 yrs)
- So between 1 yr and 2 yrs yield curve shape reflects R1 vs. E11 and
between 2 yrs and 3 yrs it reflects R2 vs. E12
- Odd shapes are possible with the right pattern of expected yields:
- Say R1<E11 so R2>R1 and R2>E12 so R3<R2.
The yield curve slopes up then down! (plot this)
e.g. R1=.02, E11=.04 (implies R2=.03)
then if E12=.02 have R3=.0267
- Say R1>E11 so R2<R1 and R2<E12 so R3>R2.
The yield curve slopes down then up! (plot this)
e.g. R1=.02, E11=.01 (implies R2=.015)
then if E12=.03 have R3=.02
(Note for a 3-yr. Period: (1+R3)3 =(1+R1)(1+E11)(1+E12) also holds)
(2) Yields on bonds of different terms to maturity will tend to move together (given
expected future yields)
(1+R2)2 = (1+R1)(1+E11)
(1+R3)3 = (1+R1)(1+E11)(1+E12)
If R1 rises (given expected yields) R2 and R3 must rise.
If R3 falls, (given expected yields) R1 and R2 must fall.
- The effect is stronger if current short-term and expected short-term yields move
together e.g. if rise in R1 is typically associated with a rise in E1.
(3) Given expectations, yields on short-term assets will be more volatile than long-term
yields:
(1+Rn)n = (1+R1)(1+E11)(1+E12)...(1+E1n-1)
- a change in Rn is magnified by its being to the
power of n.
- a large change in R1 is needed to balance it if expected yields are stable:
e.g., say that have an equilibrium:
R1 = .05 , R2 = .05 and R3 =.05
E11 = .05 , E12 = .05
(1+R2)2 = (1+R1)(1+E11) = 1.1025
(1+R3)3 = (1+ R1)(1+E12)(1+E13) = 1.157625
- Now R1 doubles to .10
- With same expected future rates:
R2 = .0747(7.47%)
R3 = .0664 (6.64%)
for equilibrium to hold.
- Doubling of R3 to .10 (expectations constant) would raise R1 to .2073 for
equilibrium.
- An implication for monetary policy?
- Monetary policy typically targets short-term interest rates e.g. R1.
- Effect on Rn will be muted unless expected yields change in the same
direction as R1.
- Central banks can have greater effects on long-term interest rates if they can also
affect expected future interest rates.
e.g. announce the likely path of future rates; ‘forward guidance’
What determines expectations of future short-term yields (E1J) ?
- Generally: anything which affects the position of future borrower (supply) and lender
(demand) curves.
- Possible distinction: real and nominal rates
- Anything which changes the “expected real interest rate” or expected real yield.
e.g., expected future business conditions
demographics (lifecycle theory of saving)
business cycles, changes in risk & liquidity
- post-2009 years: did low long-term yields reflect pessimism about
economic recovery? e.g. Paul Krugman
- Late-2016 to early 2017: do rising long-term yields reflect optimism
about future growth?
- Expected Inflation:
- determines the difference between the real and nominal yield.
- works through the Fisher effect.
- often discussed as if it is a key determinant of future yields.
- Expectations theory suggests that the observed term structure can
be used to identify current lender expectations (yields).
e.g. R1 > R2 implies that E11 > R1 : short-term (one-year are expected to rise).
i.e., the term structure can be used as a way of forecasting future rates.
- Sometimes these are used to estimate expected inflation
- assumes that expected real yield is constant.
Liquidity Preference, Risk and Term Structure:an Extension of Expectations Theory
- Long and short-term bonds are not perfect substitutes.
- There may be preference for short-term assets.
- Longer term bonds may be regarded as riskier. Why?
- More exposed to inflation risk (greater uncertainty regarding future
inflation when time horizons are long).
- For those with uncertain (or short) holding times:
- “Thin” long term markets add to uncertainty of resale prices.
- Changes in interest rates affect resale price (interest rate risk).
- These problems are likely more sever the longer term to maturity.
- A premium on long-term bonds may be required if lenders think this way.
- So when:
(1+Rn)n = (1+R1)(1+E11)(1+E12)...(1+E1n-1)
- the typical lender may prefer the sequence of one period bonds:
- so: Rn will rise, R1 will fall until lenders are
indifferent between the two options:
until a liquidity premium is paid.
- Equilibrium:
(1+Rn-LPn)n = (1+R1)(1+E11)(1+E12)...(1+E1n-1)
LPn = liquidity premium on the n-period bond (text calls this ln).
- This premium will likely be higher the longer the term to maturity (text Fig 6-4)
- On its own, liquidity preference would lead to an upward
sloping yield curve.
e.g., if yields were expected to be stable in the future
- Expectations hypothesis: predicts a flat yield curve
- Liquidity preference: predicts an upward slope.
- The most common shape of yield curve is upward sloping.
- an advantage of the liquidity premium view.
Preferred Habitat theory
- Some lenders prefer long and others prefer short term bonds (given same
yield)
- Prefer short term: liquidity preference problem
- Prefer long term: concern about uncertainty of future short-term yields (if intend to lend long)
- Consequence?(new prediction)
- the relative supply of long and short term assets will affect the shape of
the yield curve.
- relative to distribution of lender preferences
e.g., say half of lenders prefer short and half prefer long lending:
- If 1/4 of bonds are short term and 3/4 of bonds are long
term:
- some lenders with preference for short term bonds
will hold long term bonds.
- premium will be paid on long term bonds to attract lenders
- yield curve tends to slope upward.
- If 3/4 of bonds are short term and 1/4 of bonds are long term:
some lenders with preference for long term bonds will hold short
term bonds.
- premium will be paid on short term bonds to attract
lenders
- yield curve tends to slope downward.
Evidence on Term Structure Theories?
- Pure expectations:
- yields on bonds of different terms do seem to move
together as the theory predicts
- greater volatility of short term rates seems
consistent with the theory
- expectations and the yield curve shape:
- expectations are difficult to measure
- hard to test this prediction of the theory
- Liquidity preference
- advantage: it gives a reason for bias in favor of upward sloping yield
curves (see handout)
- Preferred Habitat
- suggests that a bias toward upward sloping yield curves might be due to
greater supply of long-term debt than lenders with preferences for
long-term debt.
- unique prediction: relative supplies of assets of different
terms will affect yield curve shape
- empirical studies divided on this prediction.
- Yield curve slope as a leading indicator?
- Some measure of its slope is sometimes used in forecasting.
- "inverted" (downward sloping) yield curves are associated with the start of a
recession.
(Text mentions a fourth theory: ‘Segmented Markets’ in which there are no linkage
between assets of different terms to maturity, i.e. lenders do not regard them as substitutes. Implausible given the co-movement in rates)
Exchange Rates and International Yield Differentials
- If similar domestic and foreign financial assets are viewed as good substitutes then
substitution should link yields on domestic and foreign assets.
- Small open economy model of the Canadian financial markets: an extreme case
- Canada too small to affect world interest rates (big changes in Canada barely
budge world supply and demand curves).
- Large flows of lending and borrowing between Canadian and world financial
markets should keep Canadian yields tied to world yields.
e.g. Say Canadian yield higher than world rate on similar assets.
- Foreign lenders switch to Canadian asset;
- Canadian borrowers borrow abroad.
- Rise in lending (asset demand), fall in borrowing (asset demand)
in Canada lowers Canadian interest rate to world level.