#
Interest Rate

The rate of interest is the amount charged on any debt by the creditor to the borrower.

## What Is an Interest Rate?

You may consider the interest rate as payment for forgoing today's consumption in favor of future consumption. People who earn money can spend some of it and keep the rest.

The individual will need to save during their working life and invest this money to ensure adequate funds during their retirement years.

The real interest rate is the rise in the consumption opportunity relative to present investment. It can be earned by investing money into certain financial assets and real assets such as stocks, bonds, buildings, etc.

In actuality, the interest rate is thought of as a gradual rise in the worth of money.

The calculation of interest rate =

*Interest rate = Interest payment/ Amount borrowed *

The yearly rate is typically used to express interest rates.

For example, ABC Ltd deposits $60,000 in a bank, and the bank promises to repay $66,000 at the end of the year.

*Interest amount = $66,000 - $60,000 = $6,000*

*Interest rate = $6,000/$60,000 = 10%*

## Key takeaways

- Interest is the compensation received while investing money or the compensation paid while borrowing money.
- Future value is the balance amount at the end of the investment period. Present value is the amount of investment that will provide a known future amount.
- The present value of an amount PV at the end of n years, with interest compounded in m times a year, is given by:
*PV*_{n}= FV / (1 + r/m)^{mn}

- The present value of an amount FV at the end of n years, with interest continuously compounded, is given by:
*PV = FV * exp*^{-rt}

- A risk-free interest rate is a return that comes from an investment whose return is known and certain.
- Government security yields, interbank rates such as LIBOR and MIBOR, and overnight repurchase agreement rates (repo rates) are considered risk-free rates.

**Simple interest **Rate

Above, it was assumed that the deposit length is twelve months, and the interest is calculated on a single interest payment at the end of 365 days. Suppose the amount is invested for two years; what will be the interest amount for year 1 and year 2?

If the amount deposited is dollar P and the rate of interest is r% per year, the interest amount for year 1 will be

*P × t x r/100 *

The depositor has two options while the interest is paid: he can keep the account balance the same or can hold the interest together with the balance at

*[P + (P x r/100)]*

In the first case, where the interest amount is withdrawn from the account, the principal amount at the end of year 1 remains at dollar P, and the interest amount for year 2 will be dollar P x r/100.

In this case, the interest amount for year 1 and year 2 will be the same and equal to dollar P x r/100. This is known as simple interest.

**Example: **John deposits $25,000 in a bank that provides 8% interest for 2 years.

Then,

*Simple Interest = P x t x r/ 100*

*= $25,000 x 2 x 8%/100 = $4,000*

*$2,000 for year 1 and $2,000 for year 2*

## Compound interest Rate

In this situation, the interest amount for 2 years can be greater than the interest amount for the primary 12 months.

The amount of extra interest is the interest on the interest that was received in year 1. This is known as compound interest.

*For second year, Compound interest = P (1 + r/n) ^{nt }- P*

Example: John deposits $25,000 in a bank that provides 8% interest for 2 years compounded quarterly.

And,

*Compound Interest for 2 years = $25,000 x (1 + 0.08/4) ^{4x2 }*

*= $29,291.48 - $25,000*

*= $4,291.48*

## Investments with Interest Rates

When the rate of interest is stated as compound interest, it means that the interest for any given period will also earn interest in future periods. This process is known as compounding.

If the rate of interest is compounding, traders would be interested in knowing in advance the stability of their money owed at a potential future date.

The price of an investment at a future time is referred to as the destiny fee of the funding made on the cutting-edge time.

Future value can be calculated as follows:

If A is the amount of investment, r is the rate of interest, and FV is the future value at the end of the n years, then

*FV _{n} = A x (1 + r)^{n}*

For example, Mohan deposited $50,000 on January 1, 2009, in an investment that promised 12% compound interest. What will be its future value on December 31, 2010?

*Future value on December 31, 2010 = $50,000 x (1 + 0.12) ^{2} = $62,720*

Present value refers to the amount that is to be deposited today in order to receive a known amount at a known future time. Present value is very important in the valuation of financial securities, including derivative securities.

## Note

The value of any financial security is calculated as the present value of all the future cash flows provided by the security. The calculation of present value is known as discounting.

The present value of a future amount at the end of n years is calculated as

*Present value = FV /(1 + r) ^{n}*

The expression (1 + r)n is known as the present value index factor.

For example, Oliver wants to accumulate $125,000 in three years to buy a car. So, the amount invested by him if the interest rate is compounded annually at 9% will be:

*Present value = 125,000 /(1+0.09) ^{3 }= $96,522.94*

## Present Value For Different Compounding Periods

The amount of the payment or investment, the interest rate, and the period of compounding are only a few of the variables that affect the present value of a future payment or investment.

How frequently interest is calculated and applied to the investment is referred to as the compounding period.

Depending on the chosen compounding period, a future sum of money has a different present value. In general, the current value increases with the frequency of compounding. This is because interest is earned at a higher rate when compounding occurs more frequently.

The method for estimating the future value for various compounding periods can also be used to compute the present value for those periods.

The present value of a sum to be received after n years is calculated as

*PV _{n }= FV /(1+r/m)^{mn}*

Where m is the number of compounding periods.

If it is continuous compounding, the present value is calculated as

*PV = FV x e ^{-rt}*

where t is the number of years and e is exponential.

For example, calculate the present value of $200,000 to be received in 3 years if the interest is compounded semiannually at the rate of 10%.

*PV = 200,000/(1+ 0.1/2) ^{2x3} = $149,243.08*

If continuous compounding,

*PV = 200,000 x e ^{-(0.1)x(3)} = $148,163.64*

## Future Value For Different Compounding Periods

The process of adding interest from an investment back into the principal sum is known as compounding interest, which enables the investment to grow even faster over time.

The future value of a sum to be received after n years is calculated as

*FV _{n} = PV x (1+r/m)^{mn }*

Where m is the number of compounding periods.

If it is continuous compounding, the future value is calculated as

*FV = PV x e ^{rt}*

Where t is the number of years and e is exponential.

The frequency of this compounding can significantly affect the investment's potential return.

However, if the interest is compounded quarterly, then after one year, your investment would be worth $10,509.45 ($10,000 x (1 + 0.05/4)^4), which is slightly more due to the compounding effect happening four times a year instead of just once.

As we move towards shorter compounding periods, the future value of the investment will continue to increase.

If the interest is compounded monthly, then after one year, your investment would be worth $10,511.62 ($10,000 x (1 + 0.05/12)^12), which is even more than the quarterly compounding scenario.

If the interest is compounded continuously, then the future value will be,

*FV = 10,000 x e ^{0.05} = $10,512.71*

## Risks Associated With Interest Rate

While the investor is aware of the amount he'll get at the realization of the investment duration, the funding is stated to be risk-free.

Understanding the risks associated with fixed-income security is important before understanding what a risk-free interest rate is. Here, we take into account the dangers of bond investing.

When an investor buys a bond, they will eventually get the bond's face value as well as periodic payments.

The market price in effect at the time would be paid to the investor if they choose to sell the bond before it matures.

But there are some uncertainties concerning the number of fees to be obtained through the bondholder.

The coupon payment that one would get is uncertain if the bond has a variable coupon rate.

Future interest rate ambiguity also has an impact on the bond's value. Bond values will vary, falling as interest prices rise and growing as they upward push, relying on the bond's interest fee.

As a result, an investor who wants to sell the bond earlier than it matures may even deal with the uncertainty of no longer knowing the bond's sale price.

**2. Default risk**

This measures the uncertainty of whether the borrower will be able to make periodic coupon payments as well as the uncertainty of the principal amount at maturity.

Within the case of government bonds, there can be no threat that the government will fail to make its normal payments. However, there's a chance that the business enterprise will not fulfill its duties when it buys organization bonds.

**3. Call risk**

This is a reference to the ambiguity over (i) whether the callable bond's issuer will call the bond before maturity and (ii) the moment at which they would make the call if they decide to do so.

**4. Liquidity risk**

Selling a bond at its fair value will be simple if there is enough demand for bonds.

But, it will be exceedingly challenging to locate a buyer if there is very little demand for a bond. If one needs to sell in such a scenario, one must accept the rate being paid for the bond, which is usually less than its fair cost.

Bonds that are frequently exchanged are said to be illiquid, whilst those that are heavily traded are thought to be extremely liquid.

## Risk-Free Interest Rates

Given that bond, investment involves these risks, a risk-free investment is one that has no liquidity risk, default risk, call risk, and interest rate risk.

Usually, the following are considered risk-free:

**1. Government security rates**

The central government issues government securities in an over-the-counter market that is very active with participation by banks and other financial institutions.

It has no liquidity risk or default risk, but there may be interest rate risk on long-term government securities.

It's widely assumed that short-time period authorities securities are danger-free, and the yield is considered risk-free interest prices. It is used in the valuation of options, futures, and forward rates.

## Note

Non-callable bonds have no call risk.

**2. Interbank rates**

Interbank rates are the rates at which one bank can borrow or lend to other banks. These rates are determined every day on the basis of demand for interbank borrowing and lending.

It has been used as a reference for floating rate loans, interest rate swaps, forward rate agreements, and currency swaps.

The most important international interbank is London Interbank Offer Rate (LIBOR).

**3. Repurchase agreement rates (repo rate)**

A repurchase settlement is a settlement among parties where one party agrees to sell government securities at a specific time and buy them again at a later time at a special rate.

Buying price after selling is higher than the selling price. In practice, there is no exchange of securities at the time of either selling or buying.

The repurchase agreement can be for any period, depending on the needs of the party that wants to enter into the agreement.

## Note

The most common period for repurchase is an overnight repurchase agreement, where the loan is secured only for overnight use.

Since government securities are used as collateral, there will be no default risk, call risk, liquidity risk, or interest rate risk.

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