Let's solve the performance attribution problem step by step. ### **Inputs** | Region | Index Weight (\(w_i\)) | Index Local Return (\(r_i\)) | \(E_0\) | \(E_1\) | Manager Weight (\(w_{i,m}\)) | Manager Local Return (\(r_{i,m}\)) | |-------------|------------------------|------------------------------|---------|---------|-------------------------------|------------------------------------| | Europe | 0.30 | 20% | 0.90 | 0.95 | 0.35 | 18% | | Australasia | 0.10 | 15% | 1.00 | 1.05 | 0.15 | 20% | | Far East | 0.60 | 25% | 1.10 | 1.20 | 0.50 | 20% | **Assuming \(E_0\) and \(E_1\) refer to the exchange rates at the start and end of the period, respectively.** --- ## **Step 1: Calculate Index Returns in USD** The total return in USD for each region is: \[ \text{USD Return} = (1 + \text{Local Return}) \times \left( \frac{E_1}{E_0} \right) - 1 \] #### **Europe** - Local Return: 20% = 0.20 - FX: \(E_1/E_0 = 0.95/0.90 = 1.05556\) \[ \text{USD Return} = (1 + 0.20) \times 1.05556 - 1 = 1.20 \times 1.05556 - 1 = 1.26667 - 1 = 0.26667 = 26.67\% \] #### **Australasia** - Local Return: 15% = 0.15 - FX: \(1.05/1.00 = 1.05\) \[ \text{USD Return} = (1 + 0.15) \times 1.05 - 1 = 1.15 \times 1.05 - 1 = 1.2075 - 1 = 0.2075 = 20.75\% \] #### **Far East** - Local Return: 25% = 0.25 - FX: \(1.20/1.10 = 1.09091\) \[ \text{USD Return} = (1 + 0.25) \times 1.09091 - 1 = 1.25 \times 1.09091 - 1 = 1.36364 - 1 = 0.36364 = 36.36\% \] --- ## **Step 2: Calculate Manager’s Returns in USD** Same formula as above, but with manager’s local return and their weights. #### **Europe** - Manager Local Return: 18% = 0.18 - FX: \(0.95/0.90 = 1.05556\) \[ \text{USD Return} = (1 + 0.18) \times 1.05556 - 1 = 1.18 \times 1.05556 - 1 = 1.24556 - 1 = 0.24556 = 24.56\% \] #### **Australasia** - Local Return: 20% = 0.20 - FX: \(1.05/1.00 = 1.05\) \[ \text{USD Return} = (1 + 0.20) \times 1.05 - 1 = 1.20 \times 1.05 - 1 = 1.26 - 1 = 0.26 = 26\% \] #### **Far East** - Local Return: 20% = 0.20 - FX: \(1.20/1.10 = 1.09091\) \[ \text{USD Return} = (1 + 0.20) \times 1.09091 - 1 = 1.20 \times 1.09091 - 1 = 1.30909 - 1 = 0.30909 = 30.91\% \] --- ## **Step 3: Calculate Total Returns** ### **Index USD Return** \[ \text{Index USD Return} = \sum w_i \times \text{USD Return}_i \] \[ = 0.30 \times 0.26667 + 0.10 \times 0.2075 + 0.60 \times 0.36364 \] \[ = 0.080001 + 0.02075 + 0.218184 = 0.318935 \text{ or } 31.89\% \] ### **Manager USD Return** \[ = 0.35 \times 0.24556 + 0.15 \times 0.26 + 0.50 \times 0.30909 \] \[ = 0.085946 + 0.039 + 0.154545 = 0.279491 \text{ or } 27.95\% \] --- ## **Step 4: Attribution Calculations** ### **Brinson-Fachler Attribution** Let’s define: - \(w_i\): Index weight for region \(i\) - \(w_{i,m}\): Manager weight for region \(i\) - \(R_i\): Index USD return for region \(i\) - \(R_{i,m}\): Manager USD return for region \(i\) - \(R_{index}\): Index USD total return **Country Allocation (Country Selection) Effect:** \[ \sum (w_{i,m} - w_i) (R_i - R_{index}) \] **Stock Selection (Within Country) Effect:** \[ \sum w_{i,m} (R_{i,m} - R_i) \] **Currency Effect:** We can also separate the return into: - Local return - Currency effect (FX movement) - Interaction effect But with this data, let's do the standard Brinson-Fachler attribution. --- ### **1. Country Selection Effect** \[ \text{Country Effect} = \sum (w_{i,m} - w_i) \times (R_i - R_{index}) \] First, calculate \(R_i - R_{index}\) for each region: - Europe: \(0.26667 - 0.318935 = -0.052265\) - Australasia: \(0.2075 - 0.318935 = -0.111435\) - Far East: \(0.36364 - 0.318935 = 0.044705\) Now, \(w_{i,m} - w_i\): - Europe: \(0.35 - 0.30 = 0.05\) - Australasia: \(0.15 - 0.10 = 0.05\) - Far East: \(0.50 - 0.60 = -0.10\) Now, multiply: - Europe: \(0.05 \times -0.052265 = -0.00261325\) - Australasia: \(0.05 \times -0.111435 = -0.00557175\) - Far East: \(-0.10 \times 0.044705 = -0.0044705\) Sum: \[ -0.00261325 + (-0.00557175) + (-0.0044705) = -0.0126555 \] So, **Country Effect = -0.01266 = -1.27\%** --- ### **2. Stock Selection Effect** \[ \text{Stock Selection Effect} = \sum w_{i,m} \times (R_{i,m} - R_i) \] - Europe: \(0.35 \times (0.24556 - 0.26667) = 0.35 \times -0.02111 = -0.0073885\) - Australasia: \(0.15 \times (0.26 - 0.2075) = 0.15 \times 0.0525 = 0.007875\) - Far East: \(0.50 \times (0.30909 - 0.36364) = 0.50 \times -0.05455 = -0.027275\) Sum: \[ -0.0073885 + 0.007875 + (-0.027275) = -0.0267885 \] So, **Stock Selection Effect = -0.02679 = -2.68\%** --- ### **3. Currency Effect** Alternatively, if we want to separate the currency effect, we can use: - Currency effect = (Manager Weights) × (Index FX return - Index Local return) - But as usually presented, the above country and stock effects already use USD returns, so the currency effect is embedded. But, **to answer as per standard Brinson attribution with USD returns, these are the two effects