There is always mathematical and physical foundation for truth and so do money and interest rate. The concepts of money and interest rate are simple and are provided as below in hope that people will have some true knowledge to fight against jargon of politicians.

Money

Money is not wealth unless it is the real limited resource or the piece of tokens pegged with real limited resource. It is mainly a mechanism to save transaction effort compared with barter where a miller can only have his hair cut when a barber is hungry for a bread. In case money is real and therefore wealth, it results from the nature of trust and concept of proof-of-work and the law of conservation of energy; the best way to comprehend real money is to think it as energy battery. People's money number can increase or decrease proportionally without impact on anyone's wealth. Prosperity can only be measured in real term, not by money. Measure economic growth by money is terribly wrong; we could announce that the money will shift one digit right so that we have a 10 times GDP. The fear about deflation is also terribly wrong; in case that the price of daily life times 0.01 and the wage times 0.1, people are in fact getting richer and behind the scene there must be some improved efficiency/manufacture technology in which on average each person holds 10 times of real resource than before.

So why do we need to increase society's money volume when the money is not the real wealth?

Imagine the two fictional and funny stories:

1. Till now, we all people use gold as money. The circulating gold volume is fixed in our planet for thousands of years. With improved manufacture technology, we can have 1000 times of clothes compared with thousand years ago. But because the gold volume is fixed, the clothes price must be 1/1000 compared with thousand years ago, otherwise we won't have enough gold for transaction. But then we come to a trouble that it is difficult to put a tiny gram of gold in the pocket, the tiny gram of gold is so easy to get lost in the pocket. So the government announce, the government will issue papers which one unit represents 1/100000 gram of gold so that people can put the paper in their pocket. At the same time, all people wealth is measured in unit number of the paper instead of gram. Literally, a person had 1 (gram of gold) before and now have 100000 (unit of paper).
2. Till now, all people's money number is kept in a central database and everyone uses debit card to do transaction just like thousand years before. All goods are sold in supermarket where there are electronic price tags for all goods. However, there is only 2 decimal digits on the device. With improved manufacture technology, we can have 1000 times of clothes compared with thousand years ago, the clothes price must be 1/1000 compared with thousand years ago. Some clothes were priced 0.12 before and now shall be priced at 0.00012. But because the electronic tag can not support the required 5 decimal digits, the transaction can not be done. So the government announce, everyone's money number will be times 1000 so that the electronic tag can show 0.12 instead of 0.00012.

It is all about physical constraint and nothing to do with economics principle that we need to increase money. The first story is about weight and size, the second story is about price tag device and our eyesight. If we all people understand the nature of money and we have a money technology that can bypass these constraints, we can keep money volume fixed. Until then, assuming the civilization is advancing, we may still need to increase the society's money volume and the distributing of new money shall follow the proportion law as mentioned. People will get real rich simply because doing transaction with others or introducing new real resource into the society with advanced technology (for the above clothing example, it is the sericulture technology that we can have 1000 times of silk), not because the distribution of the new money.

Interest rate

Consider the case that money is back by the proof-of-work. Its saving interest rate shall be zero to concur the law of conservation of energy even its volume is increasing by introducing into the economy. For example, digging the gold in the gold standard money system or producing the energy barrel in the battery money system.

Consider the case that money is not back by the proof-of-work.

Suppose the money volume is fixed. Because all the transaction cash flow of the whole society sums to zero, there won't be any additional money for deposit interest, in other words, the saving interest rate is zero. To lend some money, it is possible the borrower does not repay properly. Suppose one borrower will default among 101 borrowers, then the interest rate that a lender would charge is at least 1% so that the lender will not lose money in the long run.

Suppose the money volume is increasing at the speed of 5% annually, then the saving interest rate is 5% simply because the 5% new money distributes by proportional law and it is mathematically impossible to offer 5.01% saving interest rate universally. When the borrower will never default, the lending interest rate is called risk-free rate, it is at least 5% to be fair.

In short, the saving interest rate is the same as the speed of money volume increasing rate and the lending interest rate is somewhat higher based on risk consideration.

So the risk-free rate is also the saving interest rate? No. Even lending gold to a risk-free borrower still requires some fee which effectively implies a small positive risk-free lending rate although the saving interest rate is zero.

In economics literature, the risk-free rate is explained somewhat tautologically by two words: "time preference" and "utility function" which can be dropped off because both concepts are in fact a natural corollary of Darwin theory. Suppose we live by water only and survival means not running out of water, denote the following:

• is the water a person possesses
• is time
• ) is the survival probability with specific water volume and at a specific time
• is the random variable of daily change of the water volume, something like daily rainfall minus daily water usage
• is the probability density of the daily change of the water volume
• is the volume this person lending to his neighbor today
• is the volume that his neighbor repays days later
• this neighbor never defaults

Then we have the following equations about survive. By lending,

Or by keeping

Note that the value of lending formula is monotonically increasing with B and smaller value than that of keeping formula when B is equal to A. Therefore must be greater than to hold this equality which means lending is not hurting compared with keeping. The "time preference" is in fact and positive; in an alternative method of safety inventory, it means the cost to establish the inventory. Typically the more certain of the distribution of the water daily change volume, the smaller the time preference and there is no time preference only when the two stochastic events are the same:

If here survival means to survive for a long time or forever, then the time is irrelevant and the $U$ function is the "utility function"; if someone has a "utility function" differs from this survival probability function a lot, then he/she will go extinct and objectively we won't see this gene and its resulting preference about water; credit: C.Chen 1994.

In addition, when moving energy accross time, there will be some loss so that the amount of disposable energy is a little bit lower. The borrower will pay a little bit more to compensate this loss. Therefore a positive small rate is witnessed by the borrower.

In summary:

• The risk-free lending rate is a little bit higher than the saving interest rate due to lender's survival concern and loss compensation accross time; the difference in between is the time preference.
• Saving interest rate is the same as the printing money speed where "printing money" means "creating money from air" and the proof-of-work amount is diluted usually.
• In case the money is a real money, the saving can not have any interest because it results from constant proof-of-work; although increasing gold or battery volume, their saving interest rate is still zero unless the saving interest rate serves the purpose of distributing new money as the consequence of economy expansion and money with constant purchasing power.
• When Albert saves his money in a saving account, he is saving his wealth, the real wealth remains the same while the money number changes at the printing speed of money.
• Saving interest rate has nothing to do with time preference.
• As long as it is lending, its rate is higher than the risk-free lending rate, the riskier it is, the higher it is.
• Saving means storage of one's own wealth, not a lending to others even the borrower never defaults in this lending.

Sometimes, we may name the lending "investment" or "buying bond" politely although it is logically a redundancy or tautology. Many financial instruments are not money but treated as money and therefore people tragically underestimate the risks of these financial instruments and a terrible credit risk wipe out is sure to happen if some translation is not in place to make things clear:

• When Albert invests in government bonds, he is lending government and facing the risk that the government might default.
• When Albert parks his money in a money market account whose deposit interest is back by a pool of corporate bonds, he is lending these corporate and facing the risk that his money might be gone.
• A check is like a zero-coupon bond, not a money. When Albert accepts a check from Bob, Albert is buying a bond issued by Bob and is in fact lending Bob which is why Albert does not receive any cash from Bob at the time of transaction. Albert faces the cascade credit risk of Bob; Albert might realize the check only after Bob realizes a check from Candy.
• the "cashback" of a credit card is not money. When Albert accepts Bob's credit card and this card is issued by bank Candy, Albert is lending Candy by two transactions, one with the number of "cashback", the other with the number of the payment. At the same time Candy is lending Bob the number of the payment.

If we were using sound money, everything would be simpler and no average Joe would be the victim. The world would be much simpler if the debt is distinguished from the money and the money follows the proof-of-work principle.