Alice would like to purchase a £100 concert ticket, but the tickets are nearly sold out and she does not have the cash. Bob, on the other hand, does have the cash. Alice is quite trustworthy, and payday is tomorrow, but she's afraid there will be no tickets left by then and so has asked Bob to loan her the money. Bob could simply give her the money and receive it a day later, end of transaction. But he too has wants and needs. Bob could purchase the video game he is wanted for some time and will play it all night. By loaning the money to Alice he loses that night of fun. By lending his money, Bob incurs an opportunity cost (the cost of a lost opportunity). To recoup his losses, Alice agrees to make the deal sweeter and to give Bob an extra £1 tomorrow in addition to the £100 she will owe him. Bob doesn't find that to be enough, and so after some back and forth, they settle on £5, the cost to Alice of borrowing £100 for one day. Alice’s word is her bond and, indeed, the word bond is the name we give for this contract between Alice and Bob.
What we have just seen is the development of an interest rate. Here, the £5 is the cost of borrowing and, relative to the principle of £100, gives an interest rate of 5% over one day. Two people, because of their differing financial situations and current desires, agreed to exchange cash for a price. They both voluntarily engage in the market because the transaction makes each better off: Alice find the concert ticket worth more than the £105 she is paying and Bob finds the £5 more valuable than one night of video games. If they didn’t find these deals beneficial, they wouldn't engage in the transaction.
However, there are only two people. Given that Alice is a friendly person, she could have asked her social network for that £100. Carol, who doesn't like video games in his feeling ill at the moment, has little desire to spend her money right now. Carol would have lent the money to Alice for £3 instead. If Alice had asked more people than Bob alone, she could have gotten a better price . She would have entered more fully into the interest rate market. Bob and Carol would now be competing to sell their money to Alice, hence there is a marketplace.
In fact, being so sociable, Alice is attending the concert not alone, but with Dan and Erin. Separately, Alice, Dan, and Erin Who are all momentarily short on cash, each message their friends asking for £100 each. As we noted before, Alice is considered by many to be very trustworthy. Dan also works in the same store as Alice and so will also be paid tomorrow. He is considered an honest person and so is expected to pay back the principle and the interest without a fuss. However, Erin is currently unemployed. She claims that her grandmother is sending her money early for her upcoming birthday and should be in tomorrow. Her friends are neither sure of the story, the accuracy of the delivery, nor even the amount Erin will receive. Erin could pay late, causing a further opportunity cost on the lender, or pay only partially (or not at all) which of course incurs not only an opportunity cost to the lender but a financial one. To be fair, Erin isn't as untrustworthy as we've made her out to be, but there are concerns. So, what do Alice, Dan, and Erin pay in the marketplace?
Alice is quick to jump. She is the first to message everyone and sees that Carol, who is not spending money anytime soon, will give her the money for 3%. That now is the current interest rate for a riskless loan. It is not that there is no risk involved, but Alice is trusted enough that the lender Carol need not think that she will default, i.e. not pay. Alice is expected to be paid tomorrow and, short of that, can easily sell one of her fashionable jackets quickly to get the money. In fact, she could give the jacket to Carol now, as collateral, and in lieu of payment Carol has something equally, or nearly, as valuable.
The market interest rate of 3% is what we can observe but is not what Dan will pay; it is only the latest value of a riskless loan dependent on the financial state of the market players. Carol has given up her £100 and has nothing left to lend. The next cheapest person is Bob who is asking for 5%. Dan begrudgingly agrees to pay the higher price, the concert ticket is still more valuable to him than the £105 he will pay. Tomorrow, Dan will have to pay back the original £100, the principle, plus the extra £5, £2 more than Alice.
Finally, Erin asks for £100. There are few willing to lend her the money and they are asking for substantially more. If they could have lent to Alice or Dan for at least 5%, surely they need extra to be compensated for the risk that Erin may not pay at all. Erin's lack of complete trustworthiness will ultimately cost her an extra 10%, she will have to pay back a total of £115 tomorrow. It may seem backwards to ask for more from a riskier person, but the lenders are thinking about the long term. If they repeatedly lent money to someone as risky as Erin, they will find themselves on the wrong end of the deal soon enough. To ensure the expected value of the deal is positive, i.e. sensible, they need to ask for more so that when the default happens they are cushioned from the loss by the previous and future loans.
We have just seen two things here.
First, the dynamics of the interest rate. The interest rate on a loan of one day is not constant from each day to the next, indeed from instant to instant. Because the low cost loan was taken by Alice, the supply of available loans dried up just a little and the price increased. It would be hard to say a priori how much the increase would be, though. However, it is possible, and indeed more likely, that Alice and Erin are just a drop in the bucket of all those currently asking for money. They could have a very negligible effect on the current market interest rates for riskless loans. In fact, we could spin an alternate story where Alice, having agreed to pay 3%, misses out on a better deal. Someone gets their pay-check just after Alice strikes the deal and is willing to lend to Erin for 2%. In this we see that the dynamics of the interest rate is governed by a multitude of individual financial circumstances that, for all intents and purposes, is random. In fact, it is a random process as it will change with time. This in turn means the bond price, the price of the contract between borrower and lender, is itself random. We'll return to this in a later post.
The second element of the interest rate market is that the price for money had several components. One component is the time value of money. Without concern for losing the money, people still want to be paid for their loss of opportunity of immediate enjoyment, their opportunity cost . Another component is the risk of not being paid in full or at all. Some people are untrustworthy enough to actually demand a higher price above those very trustworthy people. Lending to the untrustworthy time and again would inevitably incur a loss of money due to failure to pay . That loss should be recouped by charging a higher price.
Viewed another way, an amount of money one day away that will be delivered by a party should be discounted by the value of a bond issued by that same party. The idea of the riskless party enables us to strip away concerns of the particulars of the individual in question and simply ask, what is the time-value of money? We could similarly discount money from a less trustworthy individual, such as Erin, but then we are asking what is the time value of money, given it will be paid by Erin?
As noted above, the bond price is a random variable. Indeed, like the interest rate, it is a random process. The initial value of the bond is only dependent on the interest rate at that time. However, after a half a day there will be a new interest rate. In fact, there will be an interest rate for half day loans themselves. After half a day, the new half day loan and the original full day loan will end at the same time, so purchasing the half day loan when it is struck or the full day loan half way through it maturation should give you the same growth rate. Hence the value of the day loan at half a day after it was struck will have to be the value of the new half day loan. This is an important point which will help to define the price of derivative contracts, a subject for a later post.
In the end, the concert goes well, everyone is paid back, the video game was lots of fun, and Carol has finally beaten her cold.