# Weighting a Portfolio

Let’s take an example to understand different types of weighting schemes that can be used to construct portfolios. Suppose you are very impressed with government’s announcement regarding recapitalization (injecting more money) of public sector banks. You believe that this will lead to stocks of public sector banks performing well in the future and want to invest in the same. You have read smalltalk’s article on benefits of diversification and know that you need to invest in multiple banks, rather than just one banking stock, to avoid company specific risk. Thus, you decide to invest in the following 5 public sector banks

1. State Banks of India (SBI),
2. Punjab National Bank (PNB),
3. Bank of Baroda (BOB),
4. Bank of India (BOI) and

In total, you want to invest a sum of INR 50,000 in these stocks. Now the question is how much should go into each stock? This is determined by weighting scheme. Generally, there are 3 different types of weighting schemes:

1. Equi-Weighted
2. Market-cap Weighted
3. Custom Weighted

Let’s look at the first two in detail

#### Equi-Weighted

In this weighting scheme, we want to give equal weights to all the stocks. We earlier defined weight of a particular instrument as the portion of total portfolio value represented by it. In this case instruments are the stocks and total portfolio value at the time of investing should be 50,000. We want each stock to represent equal portion of the total portfolio value. This can happen only if total value is dividend equally amount all the stocks, so each stock should have a value of INR 10,000. Thus, the initial investment going into each stock would be INR 10,000 (50,000/5) and weight of each stock would be 20% (10,000/50,000). This would be the portfolio composition:

StockInvestmentWeight
SBI10,00020%
PNB10,00020%
BOB10,00020%
BOI10,00020%
ALBK10,00020%
Total50,000100%

Now in order to execute this strategy in market, we have to buy shares of each stock worth INR 10,000. No. of shares of each stock is calculated by dividing the investment in the stock by its current market price. So, we will have to buy the following number of shares:

StockInvestmentWeightCurrent Market PriceShares
SBI10,00020%164.6560.73
PNB10,00020%76.15131.32
BOB10,00020%139.6071.63
BOI10,00020%87.40114.42
ALBK10,00020%44.15226.50
Total50,000100%

As you can see in the above table, when we divide the investment amount by current market price we get fractional shares. It is not possible to buy fractional shares in the market. For example, you cannot buy 1.6 shares of SBI or 2.2 shares of PNB. For SBI, you will have to either buy 1 or 2 shares and similarly, for PNB either 2 or 3 shares. In order to meet this requirement of non-fractional shares we change the above calculated no of shares to nearest integer. Once we have the no of shares of each stock, investment can be recalculates as shares multiplied by current market price. Final weights would be calculated by dividing the new investment for non-fractional shares by total investment, as shown.

StockInvestment (A)Weight (B = A/sum[A])Current Market Price (C)Shares (D = A/C)Rounded Shares (D1)Actual Investment (E = D1 x C)Final Weight (= E/sum[E])
SBI10,00020%164.6560.736110043.6520.06%
PNB10,00020%76.15131.321319975.6519.93%
BOB10,00020%139.6071.637210051.2020.08%
BOI10,00020%87.40114.421149963.619.90%
ALBK10,00020%44.15226.5022710022.0520.02%
Total50,000100%50056.15100%

In the above example, we wanted to achieve an equi-weighted portfolio where each stock has a weight of 20% but as shares can be bought only in whole numbers, we end up with a close approximation.

Read on to understand what happens when we try to build the same portfolio with Market-Cap or Custom weighting scheme.

#### Market-Cap Weighted

Here we want to give stock weights in proportion to their market capitalization. To understand the meaning of market capitalization, read our article What is an Index. To calculate the weight of each stock, we divide stock market cap by the total market cap of all stocks in the portfolio, as explained in the table below. Also, we know weight of an instrument is the portion of total portfolio value represented by that particular instrument. So the investment into each stock will be its weight multiplied by the total portfolio investment of INR 50,000.

StockMarket Cap in INR cr (C)Weight (C/sum[C])Investment in INR
SBI1,25,33068.76%34,378.81
PNB14,9528.20%4,101.43
BOB32,25317.69%8,847.20
BOI7,0923.89%1,945.38
ALBK2,6511.45%727.19
Total1,82,278100.00%50,000

We will follow the same steps when we used the equi-weighted scheme.

The next step is to calculate shares of every stock, based on the investments that we calculated in the above step. Number of shares would be equal to investment in the stock divided by its current market price. We know that the share calculated in the last step might be fractional and thus non-executable on the exchange. For this reason, we change the fractional number of shares to nearest whole number. Once we have no of shares, actual investment amount is equal to no. of shares multiplied by current market price and final weight would be total actual investment divided by individual stock investment.

StockWeight (A)Investment in INR (B=A x 50,000)Current Market Price (C)Shares (D = B/C)Rounded SharesActual Investment in INR (E = D1*C)Final Weight (E/sum[E])
SBI68.76%34,378.81164.65208.8020934411.8568.90%
PNB8.20%4,101.4376.1553.86544112.108.23%
BOB17.69%8,847.20139.663.38638794.8017.61%
BOI3.89%1,945.3887.422.26221922.803.85%
ALBK1.45%727.1944.1516.4716706.401.41%
Total100.00%50,00049947.95100.00%

In the market cap scheme, SBI has a weight of 68.09%, compared to 20.06% that we calculated in the equal weight scheme. Thus, market cap portfolio will be more exposed and sensitive to movement’s in SBI stock price, compared to equal-weight. This is happening because majority of your money is still concentrated in one particular stock and company specific risk is very high in the absence of proper diversification.

The third type of weighting scheme which we mentioned in our last post is Custom. In the eual-weight scheme the investment was divided equally among all the stocks. In the Market-cap scheme, it was divided based on the market capitalization of individual stocks. In Custom scheme, there is no set patter to derive this amount. Here the decision could be based on some exclusive information or investor’s gut feeling. For example, you might believe that PNB is expected to perform better than other stocks, then you can give a higher weight to PNB, compared to others. Once weighting scheme is decided, rest of the steps are similar to market-cap and equal-weight schemes.

Read the next post to learn how to calculate portfolio risk and return based on a custom index.