# Long-,short-term interest rates and inflation

I just read about the "dynamics" of long-, short-term intreset rates and inflation. Does the logic go as follows?

Centrals banks determine interest rates which in turn determine the short term rates. Inflation and expected inflation on the other hand determine how the central bank will most likely change the rates in the future(i.e what the market expects the central banks to do), hence

short term rate + expected inflation= long term rates?

Found the info here,

https://www.investopedia.com/articles/bonds/09/bond-market-interest-rates.asp

Does the logic go as follows?

short term rate + expected inflation= long term rates?

Not exactly. There is some confusion. Inflation distinguishes between real and nominal interest rates, whereas the discrepancy between short- and long-term interest rates is derived from the preclusion of arbitrage opportunities. It is neither actual nor expected inflation, but the forward rate, what explains the discrepancy between short- and long-term interest rates.

By way of example, suppose the short- and long-term rates currently are 3% and 5%, with short- and long-term referring to 1 and 2 years, respectively. The interest rate $r_{f}$ for the 2nd year should be such that $(1+0.03)(1+r_{f})=(1+0.05)^2$. Any other rate $r_{f}$ would allow either the borrower or the lender to make profit with no risk (or somewhat equivalently, to make profit without having to invest his own funds).

It is true that a lender requires higher return for long-term loans than for short-term ones because a longer term entails greater exposure to risk. But the inflation is just one of multiple risks that the lender factors in when determining at what rate (of return) he is willing to lend money.

Edited on 7/22/2018 (at 3:29PM EDT) to supplement as per 2nd-to-last comment

There is no robust or widely-adopted method for calculation of long-term rates based on estimated future inflation (and/or short-term rates). Nor is there a mathematical expression describing how --or even whether-- market forces use the estimated future inflation for determining the rate at which lenders are willing to commit funds for a long-term period. I personally doubt that such method or expression could ever exist.

One can only resort to qualitative reasoning to sketch the generic effect that some information tends to have on long-term rates. In the case of expected/estimated future inflation, it might be implicitly (at most, but perhaps not at all) reflected in other factors and preferences that influence those market forces. Uncertainty --as you rightly point out in one of your comments--, risk aversion, and cost of opportunity are three of many other factors and preferences that influence how much a long-term rate would differ from the short-term rate.

Although it is reasonable to conjecture that these estimates, factors, and preferences are positively correlated with the duration of a long-term period, it would be very hard (or I would say futile as well as impossible) to reliably model by how much they cause a long-term rate to differ from the short-term rate.

• But this what "Inverstopedia" writes right? And you are saying it is oversimplifying? Jul 21 '18 at 12:55
• @user1 It is right in that a lender's perceived risk of future inflation prompts him to require a higher return. But it would be inaccurate --perhaps an oversimplification-- to interpret the article as if short-term rate + expected inflation = long-term rate.Note how Investopedia says that (1) "Inflation is a bond's worst enemy" (rather than its only enemy), and (2) market forces (rather than the central bank) set the long-term rates. I think the article errs in that current high inflation necessarily increases the expected/future inflation, but that goes beyond the scope of your question. Jul 21 '18 at 14:31
• "Market forces (supply and demand) determine equilibrium pricing for long-term bonds, which set long-term interest rates. If the bond market believes that the FOMC has set the fed funds rate too low, expectations of future inflation increase, which means long-term interest rates increase relative to short-term interest rates" . This is the passage that made me think the above. How about if I change "expected interest" to "what the market expects in terms of interest" ... Jul 21 '18 at 16:53
• @user1 See the explanation in the answer: Long-term rate is given, and from there the forward rate is calculated, not vice versa. "Moreover it is uncertainty we get paid for, not just the loss of value due to inflation": Yes, that's correct, plus other considerations such as cost of opportunity, which could be positively correlated with the length or duration of the long-term. Jul 21 '18 at 18:27
• @user1 We're getting there: Market forces determine the long-term rate (see item (2) of my 1st comment), and from there we obtain 1+ $r_{f}$ = 1.070388. In other words, $r_{f}$=7.0388% is the [short-term rate] that should apply in the 2nd year, as otherwise there would be an arbitrage opportunity given the current short- and long-term rates of 3% and 5%, respectively. Jul 21 '18 at 20:39