# Single Period Random Cash-Flow [Basic Concepts]

An investment instrument that can be bought and sold is frequently called an asset. Suppose you purchase an asset at time zero, and one year later you sell the asset. The total on your investment is defined to be

$$total\;return\; =\; \frac{amount\; recieved\; \left( X_{1} \right)}{amount\; invested\; \left( X_{0} \right)}$$

Very often, the term return is used for total return.

The rate of return is

$$rate\; of\;return\; =\; \frac{amount\; recieved\; -\; amount\;invested}{amount\;invested\; }$$

These two terms are related by

$$R\; =\; 1\; +\; r$$

Or it can be written as

$$X_{1}\; =\; \left( 1+r \right)X_{0}$$

This shows that a rate of return acts much like an interest rate.

dividend is a distribution of a portion of a company’s earnings, decided by the board of directors, to a class of its shareholders. Dividends can be issued as cash payments, as shares of stock, or other property.

Portfolio is a bundle or a combination of individual assets or securities. Portfolio theory provides a normative approach to investors to make decisions to invest their wealth in assets or securities under risk. It is based on the assumption that investors are risk averse. This implies that the investors hold the diversified portfolios instead of investing their entire wealth in a single or a few assets. The second assumption of the theory is that the returns of assets are normally distributed. This means that the mean and variance analysis is the foundation of the portfolio decisions.

Suppose that this is done by apportioning an amount $$X_{0}$$ among the n assets. We then select amounts $$X_{0i}$$, = 1,2,3……..,n , such that

$$\sum_{i\; =\; 1}^{n}{X_{0i\; }=\; X_{0}}$$

Where $$X_{0i}$$ represents the amount invested in the $$i^{th}$$ asset. If we are allowed to sell short then some of the $$X_{0i}’s$$ can be negative; otherwise we restrict the $$X_{0i}’s$$ can be non-negative.

The amount invested can be expressed as fractions of the total investment. Thus we write

$$X_{0i\; }=\; W_{i}X_{i}$$,  where i = 1,2,3……n

Where Wi is the weight or fraction of asset i in the portfolio. Clearly,

$$\sum_{i\; =\; 1}^{n}{W_{i}\; =\; 1}$$

Some of the Wi’s may be negative if short selling is allowed.

Let $$R_{i}$$ denote the total return of asset i. then the amount of money is generated at the end of the period by the $$i^{th}$$ asset is $$R_{i}X_{oi}\; =\; R_{i}W_{i}X_{o}$$ . The total amount received by the portfolio at the end of the period is therefore $$\sum_{i\; =\; 1}^{n}{R_{i}W_{i}X_{o}}$$. Hence we find that the overall total return of the portfolio is

$$R\; =\; \frac{\sum_{i\; =\; 1}^{n}{R_{i}W_{i}X_{o}}}{X_{0}}\; =\; \sum_{i\; =\; 1}^{n}{W_{i}R_{i}}$$

Equivalently, since $$\sum_{i\; =\; 1}^{n}{W_{i}}=\; 1$$ , we have

$$r\; =\; \sum_{i\; =1}^{n}{w_{i}r_{i}}$$

We can conclude that ;

Portfolio return both the total return and the rate of return of a portfolio of assets are equal to the weighted sum of the corresponding individual asset returns, with the weight of an asset being its relative weight (in purchase cost) in the portfolio , that is,

$$R\; =\; \sum_{i\; =\; 1}^{n}{W_{i}R_{i}\; ,\; r\; =\; \sum_{i\; =\; 1}^{n}{W_{i}r_{i}}}$$

It can also be explained with the help of an example

 Security No of Shares Price Total Cost Weight in Portfolio A 100 40 4000 .25 B 400 20 8000 .50 C 200 20 4000 .25 Portfolio Total Value 16000 1.00

 Security Weight in Portfolio Rate of Return Weighted Rate A 25 17% 4.25% B 50 13% 6.50% C 25 23% 5.75% Portfolio rate of Return 16.50%

Note: I am assuming that you have covered the concepts Random Variables, Expected values, Variances and Covariance’s in the Statistics Paper .

4 Responses to “Single Period Random Cash-Flow [Basic Concepts]”
1. Mayank singh October 31, 2017
2. Mayank singh October 31, 2017
3. zvodretiluret August 1, 2018
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