Questions:
What are the formulas for Sharpe ratio, Treynor's ratio and Jensen's alpha?

Nick and Harry are two portfolio managers who have diversified portfolios. Their portfolios performed as follows:   

Nicks' portfolio returned an 18% annualized return in 2023 with a standard deviation of 16% and a beta of 0.8.  
Harry's portfolio had an annualised return of 15% with a standard deviation of 14% and a beta of 1.1. 
The current risk free rate in the market is 3% and the current market return is 10% with a standard deviation of 20%. 

Who performed better according to each of those metrics?

Reference answers:
The formulas of the ratios are as follows: 

Sharpe Ratio = (Rp - Rf) / σp  
Rp = Return of the portfolio 
Rf = Risk free return 
σp = Standard deviation of the portfolio 

Treynor’s Ratio = (Rp − Rf) / βp 
Rp = Return of the portfolio 
Rf = Risk free return 
βp = Portfolio Beta 

Jensen's Alpha = Rp - [Rf + βp ( Rm - Rf)] 
Rp = Return of the portfolio 
Rf = Risk free return 
Rm = market return 
βp = Portfolio Beta

1. Sharpe Ratio
Sharpe Ratio = (Rp - Rf) / σp  
Nick's Portfolio = (Rp - Rf) / σp = (0.18 - 0.03) / 0.16 = 0.9375
Harry's Portfolio = (Rp - Rf) / σp = (0.15 - 0.03) / 0.14 = 0.8571 

Based on Sharpe ratio, Nick's portfolio performed better because it generated higher return relative to the risk of the portfolio.

2. Treynor's Ratio
Treynor’s Ratio = (Rp − Rf) / βp 
Nick's Portfolio = (Rp − Rf) / βp = (0.18 - 0.03) / 0.8 = 0.1875 
Harry's Portfolio = (Rp − Rf) / βp = (0.15 - 0.03) / 1.1 = 0.1091 

Based on the Treynor's ratio, Nick's portfolio had better performance because it generated a higher return per unit of portfolio beta. 

3. Jensen's Alpha
Jensen's Alpha = Rp - [Rf + βp ( Rm - Rf)] 
Nick's Portfolio = Rp - [Rf + βp ( Rm - Rf)] = 0.18 - [0.03 + 0.8 (0.1 - 0.03)] = 6.25% 
Harry's Portfolio = Rp - [Rf + βp ( Rm - Rf)] = 0.15 - [0.03 + 1.1 (0.1 - 0.03)] = 4.3% 

Based on Jensen's Alpha ratio, Nick has a higher excess return over the market return which implies that his portfolio has performed better. 


Nick outperformed Harry in all three ratios.