Modified Dietz method: Difference between revisions

Revision as of 07:55, 5 August 2018

Revision as of 17:38, 14 November 2018

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====Example====

Suppose that at the beginning of the year, a portfolio contains cash, of value $10,000 USD, in an account which bears interest without any charges. At the beginning of the third quarter, $8,000 USD of that cash is invested in some US dollar shares (in company X). The investor applies a buy-and-hold strategy, and there are further transactions for the remainder of the year. At the end of the year, the shares have increased in value by 10% to $8,800 USD, and $100 USD interest is capitalized into the cash account.

What is the return on the portfolio and the cash account over the year, and what are the contributions from the cash account and the shares? Furthermore, what is the return on the cash account?

=====Answer=====

The end value of the portfolio is $2,100 USD in cash, plus shares worth $8,800 USD, which is in total $10,900 USD. There has been a 9 percent increase in value since the beginning of the year. There are no external flows in or out of the portfolio over the year.

:{{nowrap begin}}{{link if exists|weighted flows}} = 0{{nowrap end}}

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:{{nowrap begin}}{{sfrac|{{link if exists|gain or loss}}|{{link if exists|average capital}}}} = {{sfrac|900|10,000}} = 9 %{{nowrap end}}

This 9% portfolio return breaks down between 8 percent contribution from the $800 USD earned on the shares and 1 percent contribution from the $100 USD interest earned on the cash account, but how more generally can we calculate contributions?

The first step is to calculate the average capital in each of the cash account and the shares over the full year period. These should sum to the $10,000 USD average capital of the portfolio as a whole. From the average capital of each of the two components of the portfolio, we can calculate weights. The weight of the cash account is the average capital of the cash account, divided by the average capital (10,000 USD) of the portfolio, and the weight of the shares is the average capital of the shares over the whole year, divided by the average capital of the portfolio.

For convenience, we will assume the time weight of the outflow of $8,000 USD cash to pay for the shares is exactly 1/4. This means that the four quarters of the year are treated as having equal length.

The average capital of the cash account is:

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::{{nowrap begin}}= 8,000 {{link if exists| USD}}{{nowrap end}}

The average capital of the shares over the last quarter requires no calculation, because there are no flows after the beginning of the last quarter. It is the $8,000 USD invested in the shares. However, the average capital in the shares over the whole year is something else. The start value of the shares at the beginning of the year was zero, and there was an inflow of $8,000 USD at the beginning of the last quarter, so:

:{{nowrap begin}}{{link if exists|average capital}}{{nowrap end}}

::{{nowrap begin}}= {{link if exists|start value}} - {{link if exists|time weight}} × {{link if exists|outflow amount}}{{nowrap end}}

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:<math>B = A \times (1+R)+ \sum_{i=1}^n F_i \times (1+R)^ \frac{T - t_i}{T}</math>

For example, suppose the value of a portfolio is $100 USD at the beginning of the first year, and $300 USD at the end of the second year, and there is an inflow of $50 USD at the end of the first year/beginning of the second year. (Suppose further that neither year is a leap year, so the two years are of equal length.)

To calculate the gain or loss over the two-year period,