Term Structure Theories and rate risk

Question 1

What is the Term Structure of Interest Rates

Answer 1

The relationship between bond yields and maturity over various maturities, holding all other bond characteristics constant

Question 2

What is a yield curve?

Answer 2

• Yield curve: a graph of the yield-maturity relationship
• Yield curves make it clear that interest rates change at different maturities and over time

Question 3

What is the yield-maturity relationship?

Also known as the yield-to-maturity (YTM) relationship

Answer 3

It is a fundamental concept in finance and fixed-income securities. It describes the relationship between the yield or rate of return an investor can expect to receive from holding a fixed-income investment (such as a bond) and the length of time until the investment matures.

Question 4

What is a spot rate?

Answer 4

Spot rate: interest rate today for a certain time period: 𝑦0,𝑇

They call the yield to maturity on zero-coupon bonds the spot rate, meaning the rate that prevails today for a time period corresponding to the zero’s maturity.
In contrast, the short rate for a given time interval (e.g., one year) refers to the interest rate for that interval available at different points in time. In our example, the short rate today is 5%, and the short rate next year will be 7.01%.
Not surprisingly, the 2-year spot rate is an average of today’s and next year’s short rates.

Question 5

It is often said that an inverted yield curve is a reliable predecessor to a recession. What does it mean for a yield curve to be “inverted”, and why might such a yield curve indicate a recession in the near future according to the Expectations and Liquidity Preference hypotheses? Briefly explain your answers. (4 points)

Answer 5

An inverted yield curve is a yield curve that shows longer term rates that are lower than shorter term rates. According to the Expectations Hypothesis, lower rates in the future indicate declining forward rates, which represent expectations that the economy will decline. If we acknowledge the liquidity preference hypothesis, the liquidity premium might be pushing those future rates upward, and the expected short rate might be even lower and thus might more strongly indicate a future economic downturn, as low interest rates in the distant future may imply that investors are investing in longer-term bonds and thus avoiding in investments that rely on the near future.

Question 6

Do bond investors prefer a bond with high convexity over low convexity (assuming all other factors are equal)? Why or why not? If there is a preference for either, why might it not be advantageous to seek out bonds with the preferred degree of convexity (high or low) in practice? Briefly explain your answer.

Answer 6

Investors will prefer more convex bonds over less convex bonds because more convex bonds increase their price by more when rates fall and their price decreases by less when rates rise. However, it might not be practical to always seek out more convex bonds because it is likely that the market incorporates this benefit into the bond’s price and thus the price will be higher, potentially offsetting the benefit of higher convexity.

Question 7

What is the relation (positive/negative/neither) between yield to maturity and bond price sensitivity to interest rate changes? What about for the relation between coupon rate and price sensitivity? What intuitive explanations can be provided to explain these relations? Briefly explain your answers.

Answer 7

Yield to maturity: A higher YTM indicates a lower duration. When YTM is higher, cash flows from the distant future are discounted more than those in the near future. Therefore, the “average maturity” will be lower and thus the price sensitivity will be lower. Coupon rate: A higher coupon rate indicates a lower duration. When coupon rate is higher, a larger proportion of the cash flows come earlier in the bond’s life. This will lead to price being more heavily influenced by cash flows in the near term that are not as heavily influenced by discount rates and thus the price sensitivity will be lower.

Question 8

When immunizing from interest rate risk, we calculate duration based on the assumption that the yield curve is flat. Why does this assumption lead to a need for frequent rebalancing in a changing interest rate environment? If we calculate duration using yields according to the current term structure of interest rates, will the need for frequent rebalancing over time still be an issue? Briefly explain your answers

Answer 8

The assumption of a flat yield curve is directly violated when interest rates change throughout the life of a bond. Therefore, the use of duration as a single measure to immunize interest rate risk is very weak given the unrealistic nature of a core assumption. If we immunize our portfolio based on the actual yield curve as opposed to a hypothetical flat yield curve, this does not mean that we will not need to rebalance in the future. It is very likely that the yield curve will change over time and thus duration will need to be recalculated and the portfolio rebalanced.

Question 9

The liquidity premium hypothesis also states that issuers of bonds prefer to issue long-term bonds to lock in borrowing costs. How would this preference contribute to a positive liquidity premium?

Answer 9

If issuers prefer to issue long-term bonds, they will be willing to accept higher expected interest
costs on long bonds over short bonds. This willingness combines with investors’ demands for
higher rates on long-term bonds to reinforce the tendency toward a positive liquidity premium.

Question 10

Answer 10

From the lens of the Expectations Hypothesis, it is possible that investors expect an economic downturn in the near future, but do not expect it to last very long. Afterward, the economy is expected to expand for the foreseeable future. When applying the Liquidity Preference Hypothesis, it is possible that the downturn could last longer than suggested by the Expectations Hypothesis as the yield curve might be pushed upward due to the liquidity premium, not recovering conditions. It is also possible (in an extreme case) that instead of an expectation of a steady recovery after year 3, investors could expect the economy to stay in a downturn but with increasing liquidity concerns for longer term bonds that scales with the residual maturity.

Question 11

You’ve been exposed to many “rates” in the last few pages. Explain the differences among spot rates, short rates, and forward rates.

15.4 page 488

Answer 11

The n-period spot rate is the yield to maturity on a zero-coupon bond with a maturity of n periods.
The short rate for period n is the one-period interest rate that will prevail in that period.
Finally, the forward rate for period n is the short rate for that period that would satisfy a “break-even condition” equating the total returns on two n-period investment strategies. The first strategy is an investment in an n-period zero-coupon bond; the second is an investment in an (n − 1)-period zero-coupon bond “rolled over” into an investment in a one-period zero. Spot rates and forward rates are observable today, but because interest rates evolve with uncertainty, future short rates are not. In the special case in which there is no uncertainty in future interest rates, the forward rate calculated from the yield curve would equal the short rate that will prevail in that period.

Question 12

If the expectations hypothesis is valid, what can we conclude about the premiums necessary to induce investors to hold bonds of different maturities from their investment horizons?

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Answer 12

The risk premium will be zero.

Question 13

Answer 13

You’d want to benefit from the decrease in yield of the lower maturity bonds, you would buy a lower maturity bond and you can benefit from the increase of longer maturity bonds so you can sell a long maturity bond. Liquidity preference theory: under the LPT investors have short horizons and thus demand a liquidity premium for longer term bonds due to their lower liquidity at the end of the curvenger horizon, driving up long term bond yields.

Question 14

What is the expectations hypothesis (EH)

Answer 14

Forward rates represent the market expectations of future short rates

So (1+y0,3)^3 = (1+Y0,1)(1_y1,2)(1+y2,3)

Main assumption, agents are risk neutral. they do not requrie a risk premium for taking up any risk

Question 15

What is the Liquidity preference hypothesis (LPH)

Answer 15

forward rates represent a blurrred picture of market expectations of future short rates

Same as EH but with a Liquidity premium

Agents are risk averse and require a risk premium

Question 16

When immunizing from interest rate risk, we calculate duration based on the assumption that the yield curve is flat. Why does this assumption lead to a need for frequent rebalancing in a changing interest rate environment? If we calculate duration using yields according to the current term structure of interest rates, will the need for frequent rebalancing over time still be an issue? Briefly explain your answers. (4 points)

Answer 16

The assumption of a flat yield curve is directly violated when interest rates change throughout the life of a bond. Therefore, the use of duration as a single measure to immunize interest rate risk is very weak given the unrealistic nature of a core assumption. If we immunize our portfolio based on the actual yield curve as opposed to a hypothetical flat yield curve, this does not mean that we will not need to rebalance in the future. It is very likely that the yield curve will change over time and thus duration will need to be recalculated and the portfolio rebalanced.

Question 17

Assets: 120 000 D = 9
Liabilities 115 000 D= 2.3

The bank currently holds a pool of mortgages with a present value of €120,000 and a duration of 9 years. Using your answer from the previous subquestion, what interest rate movement (increase or decrease) might pose a problem for the bank? Briefly explain your answer. (3 points)

Answer 17

An increase in interest rates would cause a problem for the bank. Because the duration of the liabilities is less than the duration of the assets, an increase in rates will lower the present value of the assets more than it would for the liabilities.

In simple terms, if the bank has to wait longer to pay its debts than it takes to get paid from its mortgages, and if interest rates rise, the bank would owe less in today’s terms than it would have without the interest rate change. But the amount it would get from its mortgages would not drop as much. This could be good for the bank because it essentially owes less in present value terms, while the value of what it owns (the mortgages) doesn’t drop as sharply.

So, if the duration of the liabilities was higher than that of the mortgages, an increase in interest rates could potentially be favorable to the bank’s financial position, assuming it holds to maturity and there are no other adverse impacts from the rate change.

Question 18

When are you exposed to price risk?

Answer 18

This occurs when there is a risk that the price of a bond will decrease due to an increase in interest rates.

An investor is exposed to price risk when holding a bond whose duration is longer than the investment horizon. If interest rates rise, the price of the bond will fall, and if the investor needs to sell the bond before maturity (which is sooner than the bond’s Macaulay duration), they will realize a capital loss on the sale of the bond.

Question 19

When are you exposed to reinvestment risk?

Answer 19

This risk comes into play when there is a risk that the future cash flows from an investment (like coupon payments) will have to be reinvested at a lower interest rate than the current yield.
An investor is exposed to reinvestment risk when holding a bond whose duration is shorter than the investment horizon, or when interest rates are falling. This is because the investor will receive cash flows (such as coupons or return of principal) that they will have to reinvest at the now-lower prevailing interest rates, which results in a lower income than initially planned.