How exactly are correlated defaults used/analyzed?
我已經閱讀了很多有關相關預設值的內容，但我似乎無法理解它們是如何在投資組合理論設置中實際使用的。假設我有兩個（？）公司，X 和 Y，以及每個公司的歷史預設資訊。了解 X 和 Y 之間的違約相關性如何幫助我建構我的投資組合/損失曲線？如果我有一籃子證券，情況會如何變化？如果需要，我可以添加更多詳細資訊。
相關性對於線性投資組合（例如 CDS 指數）沒有任何作用，但是對於非線性依賴於基礎實體損失的投資組合，例如 CDO 或 $ m $ -th 到預設交換，相關性起作用。在這裡，可能需要某些技術，例如 copula，具體取決於結構的複雜性。
對相關性進行建模的方式可能取決於您擁有的投資組合。
例如，您可以使用混合二項式模型對一小部分貸款組合的預設分佈進行建模。在這種情況下，經濟狀況決定了違約機率，但在每種情況下，個別違約事件具有相同的機率（同質性）。因此，我們通過對一個因素的共同依賴在預設事件中具有相關性。這是通過使混合參數隨機獲得的：p（違約機率）是隨機的。讓 $ p ̃ $ 是我們假設的隨機違約機率分佈在
$$ 0, 1 $$它可以是連續的（即，由密度 f 給出）或離散的。以價值為條件 $ p ̃ $ 違約數量遵循機率參數的二項分佈 $ p ̃ $ . 例如，在離散情況下，有兩種可能的情況，N 個違約中的 k 個違約的機率由下式給出：
$ P(D=k)=f(p_1)\binom Nk p_1^k (1-p_1 )^{N-k}+f(p_2)\binom Nk p_2^k(1-p_1)^{N-k} $
違約次數的變異數為：
[數學處理錯誤] $ Var[D]=Np ̅(1-p ̅ )+N(N-1)Var[p ̃] $
The first term is the variance that would apply if the default probability were fixed (i.e. without correlation). The second term is additional variance. Variation in [Math Processing Error] $ p ̃ $ is an important contributor to the variance in the number of defaults because it determines the correlation between defaults events. Let [Math Processing Error] $ X_i $ denote the default indicator of issuer i (equal to 1 if i defaults and 0 otherwise), we have:
[Math Processing Error] $ ρ(X_i ,X_j )=\frac{Var[p ̃ ]}{p ̅(1-p ̅)} , i≠j $
Note that, since the variance in [Math Processing Error] $ p ̃ $ determines the correlation between default event indicators, we can model the correlation by properly chose the distribution of [Math Processing Error] $ p ̃ $ .
It can be proved that for homogeneous large portfolios of loans the distribution of the loss function is equal to the distribution of [Math Processing Error] $ p ̃ $ . Merton model has an economic meaning and fit whit in this framework so it is often used for this purpose. Assume all firms have the same correlation [Math Processing Error] $ ρ $ and same default probability [Math Processing Error] $ p ̅ $ . The “Fraction of defaults” cumulative distribution for a Large homogeneous loan portfolio may then be written as:
[Math Processing Error] $ P[p(M)≤θ] = N(\frac{1}{\sqrt{ρ}} (N^{-1} (θ) \sqrt{(1-ρ)}-N^{-1} (p ̅ ))) $
This is also known known as one-factor Gaussian copula model and has many applications with CDO.