# Financial Math FM/Determinants of Interest Rates
## Learning objectives

The Candidate will understand key concepts concerning the determinants of interest rates, the components of interest, and how to perform related calculations.

## Learning outcomes

The Candidate will be able to:
* Define and recognize the components of interest rates including: real risk-free rate, inflation rate, default risk premium, liquidity premium, and maturity risk premium.
* Explain how the components of interest rates apply in various contexts, such as commercial loans, mortgages, credit cards, bonds, and government securities.
* Explain the roles of the Federal Reserve and the FOMC in carrying out fiscal policy and monetary policy and the tools used by the Federal Reserve and the FOMC including targeting the Federal Funds rate, setting reserve requirements, and setting the discount rate.
* Explain the theories of why interest rates differ by term, including liquidity preference (opportunity cost), expectations, preferred habitat, and market segmentation.
* Explain how interest rates differ from one country to another (e.g., U.S. vs. Canada).
* Identify the real interest and the nominal interest rate in the context of loans with and without inflation protection and calculate the effect of changes in inflation on loans with inflation protection.

## Components of interest rates
### Components related to inflation

In this subsection, we assume there is no risk (and there may be inflation). Under this assumption, components of interest rates include real (risk-free) rate and inflation rate. We have discussed these in the Financial Math FM/Time Value of Money chapter.

### Components related to risk

In reality, there is risk for loans, particularly, some borrowers may not fully repay the amount lent and the required interest on time. This behaviour is called default. Because of the possibility of default, the interest rate incorporates default risk premium. So, with annual inflation rate $j$, annual default risk premium $\beta$, and annual real risk-free rate $r$, the annual nominal interest rate $i=(1+r)(1+\beta )(1+j)-1\Rightarrow i\approx r+\beta +j$.
If the rates and premium are compounded continuously, the nominal interest rate $i$ is calculated by $e^{i}=e^{r}e^{\beta }e^{j}\Rightarrow i=r+\beta +j$. In this case, the approximation sign is replaced by equal sign.
Apart from default risk premium, there is also maturity risk premium (or term premium), this premium usually exists for long-term securities because of the uncertainty for holding the securities for long time. Similarly, if this premium is also incorporated by the interest rate, then it is added to the expression of the nominal interest rate.

### Liquidity premium

Apart from the risk and inflation, there is also a premium for liquidity, since the security may sometimes have low liquidity, in the sense that it cannot be bought or sold easily, and thus compensation for this low liquidity is demanded. Similarly, this premium is added to the expression of the nominal interest rate if incorporated.

## The (simplified) role of the U.S. central bank

The discussion here is simplified, since the actual operation of the U.S. central bank is quite complicated, and is out of scope of this book.
First, since banks are required to deposit a certain amount of money to the central bank (because of the reserve requirement), even if some party is unable to fulfill its obligation in a transaction, the central bank can step in and allow the transaction. So, the central bank is essential for the operation of payment system.
Second, the central bank can act as lender of last resort, in the sense that banks can borrow money from the central bank in case of emergency (e.g. many customers withdraw their money, and the reserve in the bank cannot cope with the withdrawal amount). Apart from the central bank, banks can also borrow money from their peers if needed, but the peers may not willing and able to do so, and in this case, banks can still borrow money from the central bank, and that is why it is referred as the lender of last resort.
Third, the U.S. central bank consists of a Board of Governors, regional reserve banks and the Federal Open Market Committee (FOMC). In particular, the FOMC sets (monetary) policy with respect to the central bank's capital market activities. Common tools used by such policy include targeting the federal fund rate, setting reserve requirement and setting the discount rate (this is different from the discount rate in the Financial Math FM/Time Value of Money).
Definition. (Federal fund rate) The federal fund rate is the interest rate at which banks with excess reserve funds at their District's Federal Reserve Bank lend these funds to other banks.
Definition. (Discount rate) The discount rate is the rate at which banks can borrow money directly from the Federal Reserve.
For the reserve requirement, it indicates the minimum amount of reserve that each bank need to maintain.

### Effects of monetary policies

When the (targeted) federal fund rate or discount rate increases (decreases), it is more (less) costly for banks to run a shortfall in their reserves, and thus they are incentivized to keep more (less) money in the reserve. As a result, less (more) loan is provided, decreasing (increasing) the money supply. With the decrease (increase) of money supply (and assuming money demand is the same), the interest rate charged on loans increases (decreases) (knowledge of Macroeconomics can help understanding this reasoning).
When the reserve requirement decreases (increases), it is more (less) costly for banks to run a shortfall in their reserves, and thus they are incentivized to keep more (less) money in the reserve. As a result, less (more) loan is provided, decreasing (increasing) the money supply. With the decrease (increase) of money supply (and assuming money demand is the same), the interest rate charged on loans increases (decreases) (knowledge of Macroeconomics can help understanding this reasoning).

## Theories explaining why interest rate differ by term
### Expectation theory

According to expectation theory, the relative attraction of short and longer term investments will vary according to the expectation of future movements in interest rates.
An expectation of decrease (increase) in interest rate at the future makes short-term investment less (more) attractive, and longer term investments more (less) attractive.
Thus, because of the change in demand, the yields on short-term investments will increase (decrease) and that on long-term investment will decrease (increase).

### Liquidity preference theory

According to liquidity preference theory (or opportunity cost theory), longer dated bonds are more sensitive to interest rate movements than short dated bonds, since investors prefer liquidity, and thus will be more sensitive about the return of long-term bonds before locking their money in long-term bonds. Also, it is assumed that risk averse investors will require compensation (in the form of higher yields) for the greater risk of loss on longer bonds. It may explain some of the excess return offered on long-term bonds over short-term bonds.

### Market segmentation theory

According to market segmentation theory, bonds of different terms are attractive to different investors, who choose assets that are similar in term to their liabilities. That is, if liabilities of an investor are mainly short-term (long-term), then short-term (long-term) bonds will be chosen by the investor.
Thus, the demand for bonds will differ in different terms. Also, the supply of bonds will also vary by term, because of the strategies used by the seller of bonds, e.g. companies and government.
As a result, due to different forces of supply and demand, the effective interest rate will vary according to the term.

### Preferred habitat theory

This theory is similar to the market segmentation theory, but for this theory, each lender and borrower in the market just prefer bonds with a certain term,and is willing to buy or sell bonds with other term, given that there are sufficient incentive to do so. and
Exercise. We have mentioned that the effective interest rate typically increases when the term is longer.

## Difference between interest rates of U.S. and Canada
### Quotation bases

There are different conventional quotation bases for U.S. treasury bills and Canada treasury bills.
Definition. (Quoted rate on U.S. treasury bills) The quoted rate on U.S. treasury bills is ${\frac {360}{\text{days to maturity}}}\times {\frac {\text{dollar amount of interest}}{\text{maturity value of the treasury bill}}}.$
Remark.
* "days to maturity" refers to the remaining number of days until maturity. In particular, for $n$-day treasury bill, days to maturity for calculating the quoted rate at issue is $n$ days.
Definition. (Quoted rate on Canada treasury bills) The quoted rate on Canada treasury bills is ${\frac {365}{\text{days to maturity}}}\times {\frac {\text{dollar amount of interest}}{\text{current price of the treasury bill}}}.$

## Inflation protection

Inflation protection for a loan is to deal with the uncertainty in inflation for the lender of the loan. For inflation protection, the amount repaid by the borrower depend on the actual inflation over the loan term, through a certain index of inflation (e.g. consumer price index, or CPI).
For this kind of loan, rates before the inflation adjustment (say $i$) are specified for the loan. For the adjustment, the rate is multiplied by the inflation rate (say $j$) from the beginning to the time at which repayment is made, with reference to the index. As an approximation, the rates after the inflation adjustment $i'\approx i+j$, and $i'=i+j$ if all rates involved are continuous.
For the loan with inflation protection, some specified rates before the adjustment may be negative, and thus if the inflation rate turns out to be zero, then there is a loss for the lender of loan, which show that there is a cost for inflation protection. However, if the inflation rate is high, then there is a protection against a potential great loss of purchasing power.
* ↑ since investors can take out money from short-term investment more quickly and invest at a higher future interest rate at the future
* ↑ since investors can enjoy the current interest rate (which is expected to be higher than future interest rates) for a long time if they invest in longer term investments
* ↑ If there is not inflation protection for the loan, the rates specified for the loan may be lower than the rates after the adjustment for the loan with inflation protection.