Some people insist to act upon only those divine orders, which are mentioned in clear specific words. Therefore, since the word ‘insurance’ is not mentioned anywhere, they do not hesitate to declare it out of Islam.
Not only insurance but there are many others acts not mentioned clearly in Quran but still they are proved from the acts of the Prophet Mohammad (ﷺ). For example:

• Not all the 5 time prayers except one or two are mentioned in the Quran. Neither their timings nor the method of offering. Still, all over the world, the prayers are performed simply because it is proved from the practice of the Prophet Mohammad (ﷺ).

• Third Khalifah Osman bin Affan (رضي الله عنه) started the first Azaan (Call for the prayers) for the Friday prayers. In the period of the Prophet it was not practiced.

• Second Khalifah Omer bin Khattab (رضي الله عنه) started the Taraweeh (Ramadan’s night prayers) with congregation while the Prophet Mohammad ﷺ did not do that. Neither the Taraweeh is mentioned anywhere in the Quran. Still the Taraweeh prayers are performed all over the world.

• Khalifah Omer bin Khattab (رضي الله عنه) ordered to reduce the amount of Mehr to the minimum. (although this order was challenged by an old lady and he had to take back his order)

• Khalifah Omer (رضي الله عنه) had terminated the punishment of cutting hands of the thief during the drought.

• The procedure to select the Khalifah was mentioned neither in the Quran nor in the Ahadith although it was an utmost important need for the existence of Islamic state. All the Khulfah Ar Rashideen had adopted their own ways to choose the Khalifah.

• The compilation of Quran and Ahadith was not done by the Prophet Mohammad(ﷺ). In fact, He was reluctant to allow the compilation of Ahadith. If the Khulfah Ar Rashideen and later Islamic scholars had not done that, the Ummah would have been in the dark age of Jahiliyya (ignorance).

• The beard which is one of the most sensitive issues is mentioned nowhere in the Quran. It is interpreted from some acts and sayings of the Prophet that the beard is obligatory.

Almost 90% of the Fiqh rules are not mentioned by name in the Quran or Hadith but are interpreted out of common sense and Ijtehad from Quran, Ahadith, Seerat and the life of the Companions (رضي الله عنهم) .

The above examples and many more such examples are refuted on the grounds that the above acts were done by the Companions (رضي الله عنهم) of the Prophet Mohammad (ﷺ) whose authority, due to their highest revered status, can not be questioned. It is partially true. The companions (رضي الله عنهم) had lived with the Prophet Mohammad (ﷺ) and observed him closely. Therefore, they were able to interpret the Quran and Ahadith in most appropriate manner. Not everyone can be given authority to do Ijtehad like them, it is an accepted fact. But the point to be noted is that the logic behind the Companions’ (رضي الله عنهم) acts were to comply with the demand of the changing time and its obligations. Their acts clearly show that their decisions were based on the principle of “common-sense should prevail”.
The apparent reason behind the first Azaan of Friday Prayer was that at the time of the Prophet Mohammad (ﷺ) the population of the Muslims had not increased so much as at the time of Khalifah Osman Bin Affaan. The locality was expanding, the people needed to come from far off areas so he introduced the second Azaan ( Allah knows better). Similarly, all the examples quoted above carry a logic of expanding the practicability of Shariah according to the changing times.
In the modern times, the technology like airplanes, cars, computers, etc. are used without any hesitation but the economics and scientific discoveries are not used under the same logic while the Prophet Mohammad (ﷺ) had never used any of the modern time’s inventions.

There are about five hundred verses in the holy Quran which deal with the legalities of Islamic life. There are indeed a number of divine injunctions in the Quran which justify the validity of an insurance contract. A contract of insurance contains the element of mutual cooperation. People should not forget that the Quran is the main guidance to provide an instrumental justification for the application of any subject like banking, insurance, engineering, medicine, high-tech, politics, economics, etc.
One can not find the engineering details in the Quran or Ahadith to construct a building, but an engineer can find guidelines that if he manipulates the proportion of cement and sand, what will be the consequence in the life here and in the life-after.
The Quran is the plain statement and guidance for mankind for their success as the Quran says about itself:

هَـذَا بَيَانٌ لِّلنَّاسِ وَهُدًى وَمَوْعِظَةٌ لِّلْمُتَّقِين
آل عمران:138

“It is a plain statement for mankind, a guidance and instruction to those who fear Allah.”
(Chapter 3, Verse 138)

There are several injunctions in the Quran and Ahadith regarding orphans, widows, poor, sick, etc. It is virtuous to help them with alms and charity but it is more virtuous not to let them beg and help them to stand on their feet. Insurance is the source of this bigger virtue.
The Prophet Mohammad (ﷺ) says:

“The one who looks after and works for a widow and for a poor person is like a warrior fighting for the cause of Allah or like a person who fasts during the day and prays over night.”
(Sahi Bukhari, narrated by Amar bin Saad Bin Abiwiqas)

Quran’s aim is to establish a society with proper distribution of wealth, less number of beggars and more number of donors. With a little common sense and intellect, one can easily interpret Quranic verses which are in support of insurance.

There is a story from the history.
A sage Hazrat Ghayasuddin living somewhere around Bukhara had many pupils. One of the pupils approached to him one day and asked for his permission to travel to another city. The reason was he wanted to earn by himself as he was not happy with the life on the mercy of the donors. Since that place was a small village with no chances of working and earning, he wanted to migrate to a bigger city and become earner. The sage gave him permission.
While travelling, he saw an injured bird falling to the ground. A thought came to his mind that as Allah has promised to feed every bird and animal, so he wanted to see how He will provide the food to this injured bird. Eagerly he sat under the shade of a tree and kept watching. After a little while, another bird descended who had some grains in its beak and started feeding the injured bird. It was a great moment of astonishment and thankfulness for the pupil. He returned back to his old place. When the sage saw him back, he asked the reason. The pupil told the whole story and concluded that it is Allah who provides the food from the sources unknown to human intellect, in the same way as he made a bird source for another injured bird. On hearing the story, the sage smiled and said: “So, you prefer to remain a needy of help from others rather than becoming provider of help to others”. (means, the bird who brings the grain for the injured is better than the injured bird who depends upon the other healthy bird). The pupil understood the message and left again.

The story carries the real message of Islam.
Islam wants to make the people stand on their own feet rather than becoming dependant on others’ alms, Zakat, infaq, charity, etc.

The insurance is based upon the doctrine of ‘Masaleh al Mursal (public interest)‘ for the purpose of eliminating hardship from one’s life.
Allah intends easy life for His creature; He does not want to put them to difficulties.