Tutorial assignment week 6 2022
Answer guide
1. Consider the following three hypothetical shapes of the yield curve for government
securities:
a. The yield curve is inverted
b. The yield curve is horizontal
c. The yield curve is U-shaped (downward sloping for the first few years and then
upward sloping thereafter)
How could theory explain the position of each curve? Can you suggest a reason why it is unusual,
historically, to observe a U-shaped yield curve?
Case (a). Recall that the yield curve normally slopes upward, assuming there is a term premium, but
also incorporates expectations of future short-term interest rates. An inverted yield curve could be
explained if the current level of short-term rates exceeds its expected future average by more than
enough to offset the term premium. This might occur if there is some reason for thinking the short-
term policy rate is unusually high; for example if the central bank is responding to temporarily high
inflation or strong business cycle conditions, so the policy rate is currently high but expected to fall in
the future.
Case (b). There are two possible explanations for a horizontal yield curve. One possibility is that there
may be no term premium (hence the expectations theory of the term structure is correct) and the
current short-term rate is equal to its expected future average. This would generate a horizontal yield
curve. An alternative explanation is that there is a term premium, so the yield curve would normally
slope upwards, but the current short-term rate exceeds its expected future average by just enough to
offset the term premium.
Case (c). This would require expectations that the short-term interest rate will fall for a period of time
and then increase over a subsequent future period. It is historically unusual to observe this because
market participants would find it difficult to form well-based expectations a long way into the future.
Typically they will make a judgment about whether the short-term rate is high or low relative to some
normal level, and will expect it to gradually return to normal over a period of time. They don’t have
enough information to form well-based expectations about changes in the direction of movement in
interest rates a long way into the future.
2. Suppose you are an asset manager holding a mix of 10-year bonds and short-term money
market investments. Now you receive a piece of economic news that leads you to expect
future short term interest rates on average to be lower than you previously expected.
Suppose also that 10-year bond yields do not immediately react to this information. In that
situation, how should you adjust your portfolio to optimise your expected returns? Explain
the reason for your answer.
You are starting from a position where you are happy with the mix of investments that you have. If
your revised expectations are correct, the lower than expected future path of short term rates will
eventually become apparent to all market participants, and so 10-year bond yields will fall at some
, future point. This means the price of 10-year bonds will rise at that point. If you are confident in this
expectation, you should increase your holdings of 10-year bonds now so that you can make a capital
gain when the price rises in future.
Another (equivalent) way of thinking about it is that one part of your portfolio is currently being
invested in a succession of short-term money market instruments, the other part is invested in 10-
year bonds. The expected yield on the short-term part of the portfolio has fallen, so if 10-year bond
yields are unchanged, the bonds have become relatively more attractive.
3. How does the expectations theory of the term structure explain key aspects of the
behaviour of yield curves? What is the feature that cannot be accounted for by the
expectations theory? Explain why this is the case, and how an alternative theory can
account for this feature.
The expectations theory explains the following key facts:
• Interest rates on bonds of different maturities move up and down together over time – they
are correlated, but not perfectly. The expectations theory can explain this because short term
rates tend to move slowly over time, so when the current short-term rate changes it influences
expected future short-term rates. Example: suppose the policy rate in Australian were
unexpectedly increased to 1 per cent. Most market participants would expect it to stay at least
at that level and probably to rise further from that level over the next 2-3 years. So yields on
bonds of those maturities would rise to reflect that expectation.
• When short-term rates are low, the yield curve is usually upward sloping, and when short-
term rates are high the yield curve is usually inverted. This can be explained by the fact that if
the short-term rate moves to an extreme position (either high or low) it tends to move back
towards its average over time. So when the short-term rate is low it is expected to rise, giving
an upward sloping yield curve. When it is high it is expected to fall, giving an inverted yield
curve.
• This feature also explains why short term rates tend to have a larger range of variation over
medium to long periods of time than long term rates. Mathematically, if we think of long term
rates as averages of short term rates, we can say that averages are less variable than the thing
they are averaging.
• The expectations theory cannot explain why yield curves slope upwards on average. Why?
Because the short-term rate cannot be permanently below its own average. To explain an
upward sloping yield curve there needs to be some investor preference for shorter maturities.
In other words, given a choice between two bonds with equal expected returns, investors
prefer the one with the shorter time to maturity. This in turn means that longer term bonds
have to have higher expected returns than shorter term bonds in order for investors to hold
them. This is what is incorporated in the preferred habitat theory.
4. Suppose the expected yields on 1-year government bonds over the next 10 years are given
by the sequence [1.2, 1.4, 1.8, 2.4, 2.7, 3.3, 3.8, 4.6, 5.1, 5.3] per cent. Suppose also that the
liquidity premium for a bond with maturity of n years is given by:
Answer guide
1. Consider the following three hypothetical shapes of the yield curve for government
securities:
a. The yield curve is inverted
b. The yield curve is horizontal
c. The yield curve is U-shaped (downward sloping for the first few years and then
upward sloping thereafter)
How could theory explain the position of each curve? Can you suggest a reason why it is unusual,
historically, to observe a U-shaped yield curve?
Case (a). Recall that the yield curve normally slopes upward, assuming there is a term premium, but
also incorporates expectations of future short-term interest rates. An inverted yield curve could be
explained if the current level of short-term rates exceeds its expected future average by more than
enough to offset the term premium. This might occur if there is some reason for thinking the short-
term policy rate is unusually high; for example if the central bank is responding to temporarily high
inflation or strong business cycle conditions, so the policy rate is currently high but expected to fall in
the future.
Case (b). There are two possible explanations for a horizontal yield curve. One possibility is that there
may be no term premium (hence the expectations theory of the term structure is correct) and the
current short-term rate is equal to its expected future average. This would generate a horizontal yield
curve. An alternative explanation is that there is a term premium, so the yield curve would normally
slope upwards, but the current short-term rate exceeds its expected future average by just enough to
offset the term premium.
Case (c). This would require expectations that the short-term interest rate will fall for a period of time
and then increase over a subsequent future period. It is historically unusual to observe this because
market participants would find it difficult to form well-based expectations a long way into the future.
Typically they will make a judgment about whether the short-term rate is high or low relative to some
normal level, and will expect it to gradually return to normal over a period of time. They don’t have
enough information to form well-based expectations about changes in the direction of movement in
interest rates a long way into the future.
2. Suppose you are an asset manager holding a mix of 10-year bonds and short-term money
market investments. Now you receive a piece of economic news that leads you to expect
future short term interest rates on average to be lower than you previously expected.
Suppose also that 10-year bond yields do not immediately react to this information. In that
situation, how should you adjust your portfolio to optimise your expected returns? Explain
the reason for your answer.
You are starting from a position where you are happy with the mix of investments that you have. If
your revised expectations are correct, the lower than expected future path of short term rates will
eventually become apparent to all market participants, and so 10-year bond yields will fall at some
, future point. This means the price of 10-year bonds will rise at that point. If you are confident in this
expectation, you should increase your holdings of 10-year bonds now so that you can make a capital
gain when the price rises in future.
Another (equivalent) way of thinking about it is that one part of your portfolio is currently being
invested in a succession of short-term money market instruments, the other part is invested in 10-
year bonds. The expected yield on the short-term part of the portfolio has fallen, so if 10-year bond
yields are unchanged, the bonds have become relatively more attractive.
3. How does the expectations theory of the term structure explain key aspects of the
behaviour of yield curves? What is the feature that cannot be accounted for by the
expectations theory? Explain why this is the case, and how an alternative theory can
account for this feature.
The expectations theory explains the following key facts:
• Interest rates on bonds of different maturities move up and down together over time – they
are correlated, but not perfectly. The expectations theory can explain this because short term
rates tend to move slowly over time, so when the current short-term rate changes it influences
expected future short-term rates. Example: suppose the policy rate in Australian were
unexpectedly increased to 1 per cent. Most market participants would expect it to stay at least
at that level and probably to rise further from that level over the next 2-3 years. So yields on
bonds of those maturities would rise to reflect that expectation.
• When short-term rates are low, the yield curve is usually upward sloping, and when short-
term rates are high the yield curve is usually inverted. This can be explained by the fact that if
the short-term rate moves to an extreme position (either high or low) it tends to move back
towards its average over time. So when the short-term rate is low it is expected to rise, giving
an upward sloping yield curve. When it is high it is expected to fall, giving an inverted yield
curve.
• This feature also explains why short term rates tend to have a larger range of variation over
medium to long periods of time than long term rates. Mathematically, if we think of long term
rates as averages of short term rates, we can say that averages are less variable than the thing
they are averaging.
• The expectations theory cannot explain why yield curves slope upwards on average. Why?
Because the short-term rate cannot be permanently below its own average. To explain an
upward sloping yield curve there needs to be some investor preference for shorter maturities.
In other words, given a choice between two bonds with equal expected returns, investors
prefer the one with the shorter time to maturity. This in turn means that longer term bonds
have to have higher expected returns than shorter term bonds in order for investors to hold
them. This is what is incorporated in the preferred habitat theory.
4. Suppose the expected yields on 1-year government bonds over the next 10 years are given
by the sequence [1.2, 1.4, 1.8, 2.4, 2.7, 3.3, 3.8, 4.6, 5.1, 5.3] per cent. Suppose also that the
liquidity premium for a bond with maturity of n years is given by:
