- Thread starter Nicole Seaman
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this is from your Study Notes page 21 on Allen.

If K=100 how can there data points which are 165 periods ago?

I thought we have 250 data points and just look back at the 100 recent and these 100 recent data points we sort descending and do our computations on them.

So the oldest can be max. 100 periods ago?

Alphabet (Ticker: GOOG)

Exxon Mobil Corp (Ticker: XOM)

lambda, λ 0.98

K 100

Periods Hybrid (EXP)

Return Ago Weight Cumul

-5.46%

-5.06% 220 0.02% 0.10%

-3.87% 121 0.18% 0.28%

-3.58% 234 0.02% 0.29%

-3.51% 218 0.03% 0.32%

-2.94% 24 1.26% 1.58%

-2.88% 232 0.02% 1.60%

-2.66% 126 0.16% 1.76%

-2.64% 219 0.02% 1.79%

-2.41% 22 1.32% 3.11

Thank you a lot, Regards

Hi @QuantFFM **Good catch**, my **mistake **(apologies)! I just examined the XLS (it's in the learning sheet at https://www.bionicturtle.com/topic/learning-spreadsheet-allen-chapters-2-3/ ). **The K = 100 is not used at all; it is just mistakenly displayed on the chart. **Per the text ("we retrieved daily returns for the one year ending") the historical window is 250 days. And the 250 days is used to determine the hybrid weights. For example, the worst return of -5.46% is weighted according to (1-0.98)*0.98^(165-1)/(1-0.98^25) which is a product of the typical (1-λ)*λ^(n-1) = (1-0.98)*0.98^(165-1) and 1/(1-0.98^250) which is a slight upward multiplier that "trims" the total weight of 250 (truncated from an infinite series) days from 99.36% to a full 100.0% based on the 250 day window.

Re: "I thought we have 250 data points and just look back at the 100 recent and these 100 recent data points we sort descending and do our computations on them. "

No, my "uninvited" K = 100 is maybe confusing you. There is only a historical window (in this example, the recent T = 250 daily returns). They are sorted and weighted. And you can see from the weight example above, the only inputs to the weight are the actual day (and, if we are trimming up to 100.0% which is optional, then the 250 enters). I hope that explains, thank you again!

cc @Nicole Seaman I'll need to fix this

**Note 04/02/19: This has been fixed in the PDF**

Re: "I thought we have 250 data points and just look back at the 100 recent and these 100 recent data points we sort descending and do our computations on them. "

No, my "uninvited" K = 100 is maybe confusing you. There is only a historical window (in this example, the recent T = 250 daily returns). They are sorted and weighted. And you can see from the weight example above, the only inputs to the weight are the actual day (and, if we are trimming up to 100.0% which is optional, then the 250 enters). I hope that explains, thank you again!

cc @Nicole Seaman I'll need to fix this

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Hi @theapplecrispguy Sorry that's just a mistake in the text (cc @Nicole Seaman ) We've somehow inserted all those extra words uggh , it **should be much simpler**:

So it should read:

**Note: This has been updated in the PDF v7**

- The Macaulay duration of the bond is the sum of the time-weighted present values ($477.7621) divided by the price (the $101.16 just being the sum of the present values)
**times one plus half the yield, which is $100.2 (=101.16 × (1 + (1 + 2.092%/2). It can also be calculated directly as DV01 divided by the price**and is found to be 4.7702. - The Modified duration of the bond is found by dividing the Mac duration by one plus half the yield (4.772 / (1 + 2.092%/2) and is found to be 4.7208.
**It can also be calculated directly as DV01 divided by price and multiplied by 10,000: 0.0473/101.16*10,000 = 4.7208 (inputs rounded).** - The (Mac) convexity of the bond is approximately the sum of time squared weighted present values ($2347. 19) divided by price
**times one plus half the yield**($100.2) and is found to be 23.4354

- "The Macaulay duration of the bond is the sum of the time-weighted present values ($477.7621) divided by the price (the $101.16 just being the sum of the present values) and is found to be 4.7702.
- The Modified duration of the bond is found by dividing the Mac duration by one plus half the yield (4.772 / (1 + 2.092%/2) and is found to be 4.7208. It can also be calculated directly as DV01 divided by price and multiplied by 10,000: 0.0473/101.16*10,000 = 4.7208 (inputs rounded).
- The (Mac) convexity of the bond is approximately the sum of time squared weighted present values ($2347. 19) divided by price ($100.2) and is found to be 23.4354"

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Yes @theapplecrispguy certainly ... thank you again!

On page 9 of the Study Notes for P1.T4 Schroeck's Chapter 14 on Capital Structure in Banks, under section "Describe how economic capital is derived",

"Step 3. Estimation of Unexpected Loss Contribution (ULC) to the lending portfolio as a function of

When I look at the formula for ULCi, I can see the weight of the loan, and the correlation to the portfolio, but I only see it as a function of

Am not sure if it's a missed understanding on my part?

On a related note, do you publish videos or learning spreadsheet for Chapter 14 for P1.T4 or is it intentionally not produced? Always appreciate the rich learning content, so I thought I double check.

Thanks!

Evelyn

On page 9 of the Study Notes for P1.T4 Schroeck's Chapter 14 on Capital Structure in Banks, under section "Describe how economic capital is derived",

"Step 3. Estimation of Unexpected Loss Contribution (ULC) to the lending portfolio as a function of

When I look at the formula for ULCi, I can see the weight of the loan, and the correlation to the portfolio, but I only see it as a function of

Am not sure if it's a missed understanding on my part?

On a related note, do you publish videos or learning spreadsheet for Chapter 14 for P1.T4 or is it intentionally not produced? Always appreciate the rich learning content, so I thought I double check.

Thanks!

Evelyn

Evelyn

Hi @evelyn.peng Yes I agree that the inclusion of "expected loss" is bad, when describing the steps to retrieve the unexpected loss contribution (ULC). It looks like we picked this up from Schroeck page 179, notice his first bullet (EL is implicted indirectly, but that's very confusing!):

I am tagging to revise this for clarity, thank you!

Re: XLS for the section, it looks like we don't have a (polished) version published yet, only a staged version ... so it will be published at some point. Thanks!

"Therefore, considering a loan at the portfolio level, the contribution of a single *UL(i) *to the overall portfolio risk is a function of:

- The loan’s expected loss (
*EL*), because default probability (*PD*), loss rate (*LR*), and exposure amount (*EA*) all enter the*UL*-equation - The loan’s exposure amount (i.e., the weight of the loan in the portfolio)
- The correlation of the exposure to the rest of the portfolio"

Re: XLS for the section, it looks like we don't have a (polished) version published yet, only a staged version ... so it will be published at some point. Thanks!

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- ($100.00 * 0.75%/2 × 0.99925) + ($100.00 * 0.75%/2 × 0.99648) + ($100.00 + $100* 0.75%/2 × 0.99648) = $100.255, or
- [(0.75%/2 × 0.99925) + (0.75%/2 × 0.99648) + ((1.0 + 0.75%/2) × 0.99648)] * $100.00 = 1.00255 * $100.00 = $100.255

- On 5/30/10, for the bond that matures in 11/31/11, there is an initial forward 1.5 year forward rate curve give by {0.193%, 0.600%, 1.080%}. Using these forward rates, it is priced at $100.190. Please note at this time, the
*six month forward rate*, F(0.5, 0.5) = 0.60% and the*one-year forward rate*, F(1.0, 0.5) = 1.080%. - Then we go forward in time six months, to 11/30/11 when the bond has only one year to mature. We can make different assumptions but the scenario assumption of Realized Forwards implies:
- The previous six-month forward rate of 0.60%
**becomes the six-month spot rate**(note: this can also be called the six-month forward rate starting immediately) - The
*previous*one-year forward rate of 1.080%**becomes the current six-month forward rate**; hence the "realization" of the forward rates. The one-year bond (on 11/30/11) is priced with these two rates; I meant to say the 0.60% becomes the six-month spot rate, and the previous one-year forward becomes the six-month forward rate. They do not all become spot rates; rather, the pricing mechanics is exactly the same as in the initial step, we've just moved forward six months so there are only two cash flows to discount. I hope that's helpful, thanks!

- The previous six-month forward rate of 0.60%

Given Jensen's E(Rp) - rf = α + β(p,m)* E(Rm - rf),

And assuming ρ(p,m) = 1.0 in a well-diversified portfolio s.t. β(p,m) = σ(p)/σ(m):

E(Rp) - rf = α + [σ(p)/σ(m)]*E(Rm - rf). Now

[E(Rp) - rf]/σ(p) = α/σ(p) + [σ(p)/σ(m)]*E(Rm - rf)/σ(p) = α/σ(p) + [σ(p)/σ(m)]*E(Rm - rf)/σ(p) =

[E(Rp) - rf]/σ(p) = α/σ(p) + E(Rm - rf)/σ(m). Thanks,

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