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The purpose of this paper is to find the optimal hedging strategy when an investor has budget constraints on both the initial capital and the future cash flow.

The paper follows the utility minimization of the total cost, using convex utility functions on both initial capital and future cash flows.

Closed‐form solutions of optimal hedging strategies are found in some specific but popular cases. It is also found that this method corresponds to the local risk minimization method in quadratic hedging.

Hedging strategies are calculated for only two popular choices. One may want to calculate hedging strategies for other popular utility functions such as power utility or HARA utility.

When a trader has some budget constraint in both initial capital and future cash flows, this paper gives a simple alternative.

Budget constraints on both initial capital and future cash flow are new to this kind of study. Connection to the local risk minimization strategy is original too.

The purpose of this paper is to describe a generalization of the familiar two‐sample t‐test for equality of means to the case where the sample values are to be given unequal weights. This is a natural situation in financial risk modeling when some samples are considered more reliable than others in predicting a common mean. We also describe an example with real credit data showing that ignoring this modification of the two‐sample test can lead to the wrong statistical conclusion.

We follow the analysis of the classical two‐sample tests in the more general situation of weighted means. We also test our methods against some market data to assess the importance of the findings.

We formulate some explicit test statistics that should be used when the sample values are to be assigned differing known weights. Different cases are presented depending on how much is known about the variances. In the most typical case (the unpooled two‐sample test), we approximate the test statistic with a t‐distribution. Proofs are given where possible.

In the unpooled case, we still only have an approximate t‐distribution. This is related to the classical Behrens‐Fisher problem, which is still not fully solved. We also focus on the case where the sample values are normally distributed. It would be valuable to see how far the discussion can be extended to non‐normal distributions.

Researchers should use the two‐sample test statistics given in this paper instead of the standard ones when testing for equality of weighted means.

Weighted means occur frequently in situations when the credibility or reliability of data vary. However, standard tests for equality of means do not take weights into account. These results will be of value to any researchers studying statistical means of data of varying reliability, such as corporate bond spreads.