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Your Money: Finding yield to maturity: The internal rate of return of bonds


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Mixed cash flows, however, have multiple sign changes. Thus, in the case of such cash flows, the analysis may throw up multiple real positive IRRs.

By Sunil K Parameswaran

In finance, risk and return are the two criteria used to evaluate securities and investments. Return is measured as a percentage of the initial investment. Absolute returns can be misleading and hence the focus is on percentage returns. For instance, Project A requires an investment of Rs 500,000 and yields a profit of Rs 50,000, while Project B requires an investment of Rs 100,000 and yields a profit of Rs 25,000. Clearly the latter is superior for it yields more bang for the buck. The first project has an RoI, or return on investment of 10%, whereas the second has a RoI of 25%.

Internal rate of return

The internal rate of return or IRR, is a statistic used to evaluate investments and projects. It is that discount rate that makes the present value of subsequent cash flows from an investment, equal to the initial investment. Thus, if the IRR is greater than the cost of funds or capital, then the investment is worthwhile, else it is not.

The IRR is a solution to a polynomial of degree N, if there are N subsequent cash flows after the initial investment. Thus, it will have N roots. Solving the equation is a complex process, but Excel has an IRR function which simplifies matters. Simply enter the cash flows in a column, where the first cash flow, which represents the initial investment should be negative, and the subsequent elements in the column represent the cash inflows from the project.

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Descartes, the famous philosopher, came up with the rule of signs. It says that the maximum number of real positive roots, while computing the IRR, is equal to the number of sign changes. There are two types of cash flows, Pure cash flows, and Mixed cash flows.

Cash flow stream
In the case of the former, the cash flow stream will have a single sign change, from negative to positive. The initial cash flow will invariably be negative, because a project or a security requires an initial investment. Subsequent cash flows may be positive or negative. Mixed cash flows, however, have multiple sign changes. Thus, in the case of such cash flows, the analysis may throw up multiple real positive IRRs.

Multiple real positive IRRs pose a problem in project evaluation for managers. For instance, a company proposes to fund a project with capital having a weighted average cost of 10.50%. The project has mixed cash flows and there are two IRRs, 8% and 18%. Both are mathematically valid. However, the former will say that the project ought to be rejected, while the latter will say that the project should be accepted. This will obviously put the manager in a quandary.

Yield to maturity
The IRR of a bond is referred to as its Yield to Maturity or YTM. Bonds usually entail pure cash flows and hence the problem of multiple real positive YTMs is not faced in practice. The IRR function in Excel can be used in the case of bonds.

However, the rate function can be used for computing the IRR on the coupon date of a bond. If we are in between coupon dates, the yield function should be used. It should be remembered that the YTM is based on the dirty price of the bond and not on its clean price. Thus, prior to invoking the yield function, the accrued interest should be computed and added to the quoted price, which will obviously be a clean price.

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The writer is CEO, Tarheel Consultancy Services

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