Yield curve risk

[firefirehelphelp]

Hi everyone,

Suppose we were given the following set of factor movements:

Year 1 // 5 // 10

Parallel 1 // 1 // 1

Steepness -1 // 0 // 1

Can someone explain the following? " When the steepness makes an upward shift of 100 bps, it would result in a downward shift of 100 bps for the 1-year rate, resulting in a gain of $1, and an upward shift for the 10-year rate, resulting in a loss of $10. The change in value is therefore (1 - 10)."

Why does the downward shift of 100bps for the 1-year rate result in a gain? Given that the sensitivity is -1, shouldn’t a downward shift result in a loss?

Thanks.

[S2000magician]

The sensitivity isn’t negative one.

The interpretation of the steepness factor is that a +100bp steepness change means that the 1-year rate decreases by 100bp (= −1 × 100bp), the 5-year rate is unchanged (= 0 × 100bp), and the 10-year rate increases by 100bp (= 1 × 100bp). Thus, 1-year bonds should increase in value, 5-year bonds should remain as is, and 10-year bonds should decrease in value.

[MikeHams]

@firefirehelphelp

Hi everyone,

Suppose we were given the following set of factor movements:

Year 1 // 5 // 10

Parallel 1 // 1 // 1

Steepness -1 // 0 // 1

Can someone explain the following? " When the steepness makes an upward shift of 100 bps, it would result in a downward shift of 100 bps for the 1-year rate, resulting in a gain of $1, and an upward shift for the 10-year rate, resulting in a loss of $10. The change in value is therefore (1 - 10)."

Why does the downward shift of 100bps for the 1-year rate result in a gain? Given that the sensitivity is -1, shouldn’t a downward shift result in a loss?

Thanks.

I literally just finished working through this and I think I understand it. The reason why there is a gain and loss is because we are multiplying the factor movements to the year in which they occur. Steepness has a positive factor movement in each period so it is equally affected at each along the time (1 year X 1 + 5 year X 1 + 10 year X 1) = 16 which is then used to find the level change. For the steepness we have a change of -1 X 1 year - 1 X 10 year = 9 which is then used to find the steepness change and so on.

The reason for the price change is because when IR decrease our bond price increases which is why when the steepness factor is 1 in year 10 we consider this as 1 X 10 = 10, but since it’s a positive factor we know that IR have increased here which will make our bonds price decrease resulting in a loss.

[firefirehelphelp]

Thanks for the replies. Can someone please elaborate on this “The reason for the price change is because when IR decrease our bond price increases which is why when the steepness factor is 1 in year 10 we consider this as 1 X 10 = 10,”

I don’t understand why we need to multiply the year by the steepness factor even after reading the curriculum.

[S2000magician]

@MikeHams

@firefirehelphelp

Hi everyone,

Suppose we were given the following set of factor movements:

Year 1 // 5 // 10

Parallel 1 // 1 // 1

Steepness -1 // 0 // 1

Can someone explain the following? “When the steepness makes an upward shift of 100 bps, it would result in a downward shift of 100 bps for the 1-year rate, resulting in a gain of $1, and an upward shift for the 10-year rate, resulting in a loss of $10. The change in value is therefore (1 - 10).”

Why does the downward shift of 100bps for the 1-year rate result in a gain? Given that the sensitivity is -1, shouldn’t a downward shift result in a loss?

Thanks.

I literally just finished working through this and I think I understand it. The reason why there is a gain and loss is because we are multiplying the factor movements to the year in which they occur. Steepness Parallel has a positive factor movement in each period so it is equally affected at each along the time (1 year X 1 + 5 year X 1 + 10 year X 1) = 16 which is then used to find the level change. For the steepness we have a change of -1 X 1 year - 1 X 10 year = 9 which is then used to find the steepness change and so on.

The reason for the price change is because when IR decrease our bond price increases which is why when the steepness factor is 1 in year 10 we consider this as 1 X 10 = 10, but since it’s a positive factor we know that IR have increased here which will make our bonds price decrease resulting in a loss.

First, I fixed your third sentence.

Second, your calculations are incorrect. You don’t multiply the number of years by the sensitivity each year.

Reread what I wrote above. Those numbers tell you how much to change the YTM at each maturity for a given change in each factor:

• Given a 100bp parallel change, you change the 1-year YTM by 100bp (= 1 × 100bp), you change the 5-year YTM by 100bp (= 1 × 100bp), and you change the 10-year YTM by 100bp (= 1 × 100bp).
• Given a 100bp steepness change, you change the 1-year YTM by −100bp (= −1 × 100bp), you change the 5-year YTM by 0bp (= 0 × 100bp), and you change the 10-year YTM by 100bp (= 1 × 100bp).

You then use the new YTM at each maturity to determine the effect on your portfolio.

So:

• if you have a portfolio of 1-year bonds and steepness increases by 50bp, then the 1-year YTM decreases by 50bp (= −1 × 50bp), and the value of your portfolio increases
• if you have a portfolio of 5-year bonds and steepness increases by 50bp, then the 5-year YTM doesn’t change (0 × 50bp = 0bp ), and the value of your portfolio stays the same
• if you have a portfolio of 10-year bonds and steepness increases by 50bp, then the 10-year YTM increases by 50bp (= 1 × 50bp), and the value of your portfolio decreases

[self_employed]

Crystal clear!

If we have a mixed porfolio, we weighted average it across maturities right?

[MikeHams]

@S2000magician

@MikeHams

@firefirehelphelp

Hi everyone,

Suppose we were given the following set of factor movements:

Year 1 // 5 // 10

Parallel 1 // 1 // 1

Steepness -1 // 0 // 1

Can someone explain the following? “When the steepness makes an upward shift of 100 bps, it would result in a downward shift of 100 bps for the 1-year rate, resulting in a gain of $1, and an upward shift for the 10-year rate, resulting in a loss of $10. The change in value is therefore (1 - 10).”

Why does the downward shift of 100bps for the 1-year rate result in a gain? Given that the sensitivity is -1, shouldn’t a downward shift result in a loss?

Thanks.

I literally just finished working through this and I think I understand it. The reason why there is a gain and loss is because we are multiplying the factor movements to the year in which they occur. Steepness Parallel has a positive factor movement in each period so it is equally affected at each along the time (1 year X 1 + 5 year X 1 + 10 year X 1) = 16 which is then used to find the level change. For the steepness we have a change of -1 X 1 year - 1 X 10 year = 9 which is then used to find the steepness change and so on.

The reason for the price change is because when IR decrease our bond price increases which is why when the steepness factor is 1 in year 10 we consider this as 1 X 10 = 10, but since it’s a positive factor we know that IR have increased here which will make our bonds price decrease resulting in a loss.

First, I fixed your third sentence.

Second, your calculations are incorrect. You don’t multiply the number of years by the sensitivity each year.

Reread what I wrote above. Those numbers tell you how much to change the YTM at each maturity for a given change in each factor:

• Given a 100bp parallel change, you change the 1-year YTM by 100bp (= 1 × 100bp), you change the 5-year YTM by 100bp (= 1 × 100bp), and you change the 10-year YTM by 100bp (= 1 × 100bp).
• Given a 100bp steepness change, you change the 1-year YTM by −100bp (= −1 × 100bp), you change the 5-year YTM by 0bp (= 0 × 100bp), and you change the 10-year YTM by 100bp (= 1 × 100bp).

You then use the new YTM at each maturity to determine the effect on your portfolio.

So:

• if you have a portfolio of 1-year bonds and steepness increases by 50bp, then the 1-year YTM decreases by 50bp (= −1 × 50bp), and the value of your portfolio increases
• if you have a portfolio of 5-year bonds and steepness increases by 50bp, then the 5-year YTM doesn’t change (0 × 50bp = 0bp ), and the value of your portfolio stays the same
• if you have a portfolio of 10-year bonds and steepness increases by 50bp, then the 10-year YTM increases by 50bp (= 1 × 50bp), and the value of your portfolio decreases

I understand what you’re saying. You really do make it clear. I used the wrong word, but do we multiply the years by the factor changes and not the sensitivities? They do 16/(300 x .01) to find the I’m assuming overall factor change for that specific factor which is then multiplied by the sensitivities they give us later? That’s the only rationale I have for how they computer the 16, the 9 for the factor sensitivity which then gets multiplied by their each respective small change in factor.

[S2000magician]

@self\_employed

Crystal clear!

If we have a mixed porfolio, we weighted average it across maturities right?

Only if you want to do it correctly.

[S2000magician]

@MikeHams

@S2000magician

@MikeHams

@firefirehelphelp

Hi everyone,

Suppose we were given the following set of factor movements:

Year 1 // 5 // 10

Parallel 1 // 1 // 1

Steepness -1 // 0 // 1

Can someone explain the following? “When the steepness makes an upward shift of 100 bps, it would result in a downward shift of 100 bps for the 1-year rate, resulting in a gain of $1, and an upward shift for the 10-year rate, resulting in a loss of $10. The change in value is therefore (1 - 10).”

Why does the downward shift of 100bps for the 1-year rate result in a gain? Given that the sensitivity is -1, shouldn’t a downward shift result in a loss?

Thanks.

I literally just finished working through this and I think I understand it. The reason why there is a gain and loss is because we are multiplying the factor movements to the year in which they occur. Steepness Parallel has a positive factor movement in each period so it is equally affected at each along the time (1 year X 1 + 5 year X 1 + 10 year X 1) = 16 which is then used to find the level change. For the steepness we have a change of -1 X 1 year - 1 X 10 year = 9 which is then used to find the steepness change and so on.

The reason for the price change is because when IR decrease our bond price increases which is why when the steepness factor is 1 in year 10 we consider this as 1 X 10 = 10, but since it’s a positive factor we know that IR have increased here which will make our bonds price decrease resulting in a loss.

First, I fixed your third sentence.

Second, your calculations are incorrect. You don’t multiply the number of years by the sensitivity each year.

Reread what I wrote above. Those numbers tell you how much to change the YTM at each maturity for a given change in each factor:

• Given a 100bp parallel change, you change the 1-year YTM by 100bp (= 1 × 100bp), you change the 5-year YTM by 100bp (= 1 × 100bp), and you change the 10-year YTM by 100bp (= 1 × 100bp).
• Given a 100bp steepness change, you change the 1-year YTM by −100bp (= −1 × 100bp), you change the 5-year YTM by 0bp (= 0 × 100bp), and you change the 10-year YTM by 100bp (= 1 × 100bp).

You then use the new YTM at each maturity to determine the effect on your portfolio.

So:

• if you have a portfolio of 1-year bonds and steepness increases by 50bp, then the 1-year YTM decreases by 50bp (= −1 × 50bp), and the value of your portfolio increases
• if you have a portfolio of 5-year bonds and steepness increases by 50bp, then the 5-year YTM doesn’t change (0 × 50bp = 0bp ), and the value of your portfolio stays the same
• if you have a portfolio of 10-year bonds and steepness increases by 50bp, then the 10-year YTM increases by 50bp (= 1 × 50bp), and the value of your portfolio decreases

I understand what you’re saying. You really do make it clear. I used the wrong word, but do we multiply the years by the factor changes and not the sensitivities? They do 16/(300 x .01) to find the I’m assuming overall factor change for that specific factor which is then multiplied by the sensitivities they give us later?

They’re assuming that 1 year, 5 years, and 10 years are the respective durations, and that the market value of the portfolio is equally weighted amongst those three maturities.

It sure would be nice if they made that aspect clear. This stuff isn’t difficult at all, but it has to be explained clearly and accurately.