(a) City fund had average daily assets of Rs. 1800 million in the past year. The Fund sold Rs. 600 million and purchased Rs. 500 million worth of stuck during the year. i) What was the portfolio…

(a) Portfolio Turnover Rate and Expense Calculation

i) Portfolio Turnover Rate

The portfolio turnover rate measures the percentage of a fund’s assets that are bought and sold during a year. It is calculated as:

$$\text{Turnover Rate}=\frac{\text{Total Purchases or Sales}}{\text{Average Daily Assets}}\times 100$$

Given:

• Average daily assets = Rs. 1800 million
• Fund sold = Rs. 600 million
• Fund purchased = Rs. 500 million

Since turnover is based on the smaller of purchases or sales, we use Rs. 500 million (purchases).

$$\text{Turnover Rate}=\frac{500}{1800}\times 100=27.78\%$$

Answer: The portfolio turnover rate was 27.78%.

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ii) Total Fees and Administrative Expenses

The expense ratio (1.2%) covers both management fees (0.8%) and administrative expenses (remaining 0.4%).

Step 1: Calculate Total Fees Total fees are based on average daily assets (Rs. 1800 million).

$$\text{Total Fees}=\text{Average Assets}\times \text{Expense Ratio}=1800\times \frac{1.2}{100}=\text{Rs. 21.6 million}$$

Step 2: Calculate Management Fees $$\text{Management Fees}=1800\times \frac{0.8}{100}=\text{Rs. 14.4 million}$$

Step 3: Calculate Administrative Expenses $$\text{Administrative Expenses}=\text{Total Fees}-\text{Management Fees}=21.6-14.4=\text{Rs. 7.2 million}$$

Answer:

• Total fees paid = Rs. 21.6 million
• Administrative expenses = Rs. 7.2 million

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(b) Performance Evaluation Using Treynor and Sharpe Measures

Given Data:

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i) Calculate Treynor and Sharpe Measures

1. Treynor Measure (Reward-to-Volatility Ratio)

$$\text{Treynor Ratio}=\frac{R_p-R_f}{{\beta }_p}$$

For Portfolio P: $${\text{Treynor}}_P=\frac{35\%-6\%}{1.2}=\frac{29\%}{1.2}=24.17\%$$

For Market: $${\text{Treynor}}_{\text{Market}}=\frac{28\%-6\%}{1.0}=22\%$$

2. Sharpe Measure (Reward-to-Volatility Ratio)

$$\text{Sharpe Ratio}=\frac{R_p-R_f}{{\sigma }_p}$$

For Portfolio P: $${\text{Sharpe}}_P=\frac{35\%-6\%}{42\%}=\frac{29\%}{42\%}=0.69$$

For Market: $${\text{Sharpe}}_{\text{Market}}=\frac{28\%-6\%}{30\%}=\frac{22\%}{30\%}=0.73$$

Summary of Results:

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ii) Performance Comparison and Conflicting Results

Performance Evaluation:

• Treynor Measure: Portfolio P (24.17%) outperforms the market (22%) because it provides a higher return per unit of systematic risk (beta).
• Sharpe Measure: Portfolio P (0.69) underperforms the market (0.73) because it has a lower return per unit of total risk (standard deviation).

Why Conflicting Results?

1. Different Risk Adjustments:

   • Treynor adjusts for systematic risk (beta) only.
   • Sharpe adjusts for total risk (standard deviation).
   • If a portfolio has high unsystematic risk (diversifiable), Sharpe may penalize it even if it efficiently manages systematic risk.

2. Portfolio Characteristics:

   • Portfolio P has higher volatility (42%) than the market (30%), which hurts its Sharpe ratio despite higher returns.
   • However, its beta (1.2) is higher, meaning it takes on more systematic risk, which Treynor rewards.

Conclusion:

• If an investor believes systematic risk is the only relevant risk, Treynor suggests P outperforms.
• If total risk matters, Sharpe suggests P underperforms.
• The choice depends on whether the fund is diversified (reducing unsystematic risk) or not.