Convertible notes require registration unless these are otherwise exempt from the registration requirements. This is applicable to both the SEC and California, not Delaware.
Assuming the offering qualifies for a federal exemption, you can file an exemption in California using the same basis as the federal exemption. These are accepted by the California Department of Corporations Securities Regulation Division.
If you are an issuer, I strongly encourage you to enlist the assistance of legal counsel in the preparation of your securities and the private placement memorandum.
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You can file a Form D with the SEC. You should also file a 25102(f) with the Department of Corporations in California. There is a filing fee of $25 for an investment of that size. If you have an EIN for your LLC (as you should), you can file online as the issuer. You will need to also register the Department of Corporations as an agent for service of process, and file a consent to electronic filing. There are instructions to fill in this form, or you can have an attorney do it. It is required to be filed within 15 days of issuance of securities. You will also need to have the proper authority for the LLC to enter into the transaction, normally done by a member meeting or minutes acknowledging the manager's authority to do so. Your family member should also sign an investor certificate certifying that he/she is aware of the risks and and that he/she has the ability to bear the economic risk, etc., as you state above. Lastly, if your investor does convert the note, you will need an Operating Agreement if you do not already have one so that it is clear what each of your voting rights are, rights of distribution, etc.
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First, this convertible note is likely a security. As such, every purchase or sale, and every offer to purchase or to sell, must be registered at both the state and federal levels, unless an exemption applies. Generally, small businesses rely on the Section 4(a)(2) private offering exemption to issue securities. There is a safe harbor for the 4(a)(2) exemption that is contained in Rule 506 where, if you comply with the requirements of Rule 506, then your offering qualifies under the 4(a)(2) private offering exemption. Moreover, securities offerings that meet the 506 requirements will preempt state law registration requirements. Meaning, if you properly meet the Rule 506 requirements, you are exempt with complying with state securities laws (other than certain notices and fees that may apply). This is the reason why this exemption is the most popular; it saves a lot of time and money.
You are supposed to file a Form D within 15 days of the first sale in a Rule 506 offering, but it is not a requirement for the offering to be exempt and I would fathom a guess that more than 50% of the businesses that issue securities in a private offering do not do this or are not even aware of the securities laws implications of what they are doing to begin with.
You should check with a CA attorney to make sure your situation complies with Rule 506 and to confirm whether any state notice filings or fees need to be paid. New York, for example, has quirky rules. CA may be similar. I can't say because I am not licensed there. The other thing to be mindful of here is if the investor is not an "accredited investor" you have certain information obligations that must be provided in writing. This is basically a 10A prospectus, usually referred to as a private placement memorandum or PPM. If the investor is accredited, you may not need to provide a PPM but its usually a good idea to do so anyway. With the size of the offering here, however, the costs with doing this might not be justified, especially if the investor is accredited (i.e., net worth > $1,000,000, excluding residence, or annual income over $300,000 per year for the last 2 years).
Your description of the transaction as "getting a convertible note...from" is a little confusing, but I assume you mean your Delaware corp. is issuing its corporate note to a California resident in exchange for $15,000. The convertible note is a "security", and so subject to the registration/qualification requirements under both federal and (since the issuee is in California) California securities laws; but, you're right that "private offering" exemptions are likely available under both if this is an isolated transaction and other conditions are satisfied. Note, though, that California imposes a non-burdensome "Notice of Issuance" filing obligation, even if exempt.