# Why is the Compound Interest Formula the Central Dogma of Investing?

How the basic compound interest formula is responsible for high end financial calculations and valuation.

Saifuddin

9/15/20221 min read

Compound interest can turn your investments manifold with time. Almost every individual has encountered the compound interest formula at some point in their life. It is interesting to know the importance of the compound interest formula in finance and investment.

“*Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn't … pays it.”* – Albert Einstein

Albert Einstein didn’t exaggerate when he called compound interest the eighth wonder of the world. Compound interest can work wonders for investors. The formula to calculate compound interest is the seed to simplify several complex calculations in investments.

**What is Compound Interest?**

Compound interest is the interest made on the principal amount as well as the accrued interest. It can simply be called ‘interest on interest’ as the rate of interest grows exponentially with time.

**Compound Interest Formula with Example**

The compound interest formula is the most basic thing that is taught at schools and colleges. No matter what background you belong to, you must have come across this formula.

Where P is the Principal Amount or Initial Investment

i is the nominal interest per annum

n is the number of compounding periods

CI is the capital gain due to compound interest after n compounding periods

The principal amount is subtracted again to calculate the amount gained due to compound interest in future. The compounding can occur daily, monthly, yearly, etc and the value of I and n depend accordingly. Let us understand the formula with the help of an example.

**Example**: A man wants to calculate the overall capital gain on an investment of **$10,000 for 4 years compounded yearly at the rate of 6%**.

In this example, the compounding is annual and the interest rate is 6%. P is $10,000 and n is 4.

Compound Interest = $10,000 * [1.0.06]4 – $10,000

= ($10,000 * 1.2624) – $10,000

= $12,624 – $10,000

= $2,624

The capital gain on the $10,000 investment with compound interest will be $2624 after 4 years at the rate of 6% compounded annually.

Now let us understand the same example with quarterly compounding and check the difference.

Due to quarterly compounding, the number of compounding periods will increase, but the rate of interest will be divided into a number of compounding periods per annum.

The formula thus becomes,

Where m is the number of compounding periods per annum. It must be noted that m was kept as 1 when compounding was done annually. In quarterly compounding, the value of m will be 4. Hence,

Compound Interest = $10,000 [1 + 0.06/4]4*4 - $10,000

= $10,000 (1.015)16 - $10,000

= $12, 689 - $10,000

= $2689

**The capital gain on the same investment is higher when compounding was quarterly compared to annually**. Similarly, if the compounding was done daily, m would be 365 and the total gain due to daily compounding on the same investment would be $2712.

For investors, the gain will increase with the increase in compounding periods and vice versa. The table below depicts the gain in this example for different time periods along with a comparison with simple interest.

The table below compares how an investment of $10,000 grows with compound interest and simple interest at an annual compounding rate of 6%.

Number of Years

Compound Interest

Simple Interest

1

600

600

3

1910.16

1800

5

3382.255776

3000

10

7908.476965

6000

15

13965.58193

9000

20

22071.35472

12000

30

47434.91173

18000

40

92857.17937

24000

50

174201.5427

30000

For a realistic idea of compounding, we have kept the interest rate at 6% which is even lower than the inflation rate of many countries presently.

**Compound Interest with Changing Interest Rates**

It is important to understand that the commonly used formula of compound interest can only be used when the interest rate remains the same throughout the investment tenure. Practically, In most financial markets, the interest rates do not remain the same and can change throughout the compounding period. For certain instruments, the rate of interest can also go negative at times.

For such investments, the formula can be brought into the basic form by multiplying each yearly return.

Here, r1, r2, r3….. rn is the rate of yearly interest for the years 1, 2, 3,…… n respectively.

Suppose an investment of $10,000 compounds annually for 5 years at rates of 6%, 8%, 12% -4%, and 10% for the year 1, 2, 3, 4, and 5 respectively.

To calculate the future value of this investment,

At the given rates for each year, the FV of the invested $10,000 will be $13,540 with capital gains of $3,540 due to annual compounding. For convenience in the calculation, we generally take the average of the returns. But practically, average returns do not provide actual gains due to periodic compounding at different rates.

**Power of Compounding**

Successful investors acknowledge the power of compounding and the wonders it can do with time. Compound interest increases the capital at an accelerated rate which grows exponentially with time.

In the example explained above, it can be seen that there is no major difference between compound and simple interest when the number of years is less than 10. However, after 10 years, you can see a significant difference between the accrued interest over time between the compound and simple interest.

The exponential growth of invested capital over time due to compound interest is commonly called the power of compounding.

**Importance of Compound Interest Formula in Investing**

The compound interest formula is not only used to calculate the future value of investments according to the interest rates and investment tenure. This formula is the seed of many widely used formulas in finance and investments.

**Compound Interest formula in Calculating Net Present Value and Internal Rate of Return**

Net Present Value is the present value of an investment considering all the inflows and outflows of cash during the investment tenure. NPV is used in the capital budgeting of businesses and investments to identify whether it will be profitable or not.

To calculate the NPV, the present values of all the future cash flows are calculated and netted. If the NPV is positive, the investment can be considered profitable as the present value of cash inflows are more than outflows.

The interest rate and time are important variables in the calculation of NPV. The interest rate at which the NPV becomes zero is called as Internal Rate of Return (IRR).

Now, if we keep NPV = 0, We can also calculate for i with this equation. The value of interest rate (i) when NPV = 0 is the Internal rate of Return (IRR). If the interest rate is below IRR, the investment won’t be beneficial as the NPV of the investment will become negative.

For example, Let’s calculate NPV on an investment of $10,000 that pays 4 cash inflows in years 1, 2, 3, 4 of $5000, $6000, $4000, and $3000 at an interest rate of 6%.

Now we will discount each cash flow to its present value according to interest rate and time.

NPV = -10,000 + 5000/(1+0.06) + 6000/(1+0.06)2 + 4000/(1+0.06)3 + 3000/(1+0.06)4

As $10,000 is the cash outflow at the present date, we have kept it negative without discounting. All other cash inflows are in future, hence need to be discounted.

NPV = -10,000 + 4717 + 5340 + 3358 + 2376

NPV = -10,000 + 15,791

NPV = $5,791

The formula to calculate the NPV can be derived from the compound interest formula. In the compound interest formula, the Future value is calculated using the present value. In NPV, the present value is calculated using multiple expected future values of inflows.

Here, CI is the Interest gained due to compounding while P is the principal or present value of the investment. If we add the interest gained with the principal amount, this becomes the future value of the investment. Hence, we can rewrite the above equation as:

As multiple cash flows are involved in calculating NPV, each future value of cash flow needs to be discounted to the present value. All the present values are then netted off to calculate the NPV.

The equation can also be written as

Hence, it can be concluded that the compound interest formula is used multiple times to discount each cash flow while calculating the NPV and IRR.

**Compound Interest Formula for DCF calculation**

The sum of all the cash flows discounted at interest rate i is also called as Discounted Cash Flow (DCF). NPV of the investment can be rewritten as:

Hence, the compound interest formula is used in the calculation of Discounted Cash Flows (DCF).

**Compound Interest Formula for Bond Valuation**

The compound interest formula also forms the base for bond valuation. We explained how the compound interest formula can be used to calculate the present or future values using the other.

For a bond that does not involve coupon payments like the Zero Coupon Bond, the Present value of the bond can be simply calculated using the above equation. The future value in bonds is also called as maturity amount or face value of the bond. In bond valuation, n will be considered as maturity tenure.

For bonds that pay out coupons at regular intervals, each coupon payment cash flow needs to be discounted to the present value using the DCF formula. The sum of face value and all the coupon payments in the future are discounted to the present value to calculate the present value of the bond with coupons.

The formula for compound interest is also used in the Dividend Discount Model (DDM) for stock valuation, Option Pricing Models, CAGR, and various other formulas in finance. The compound interest formula is used whenever there is a change in the invested amount due to interest rate and time.

**Compound Interest in Investments**

Compound interest plays a lead role in any long-term investment. If the chosen investment instrument does not include compounding, investors might be missing out on a lot over the long term. Compounding is the major reason why it is always recommended to start investing early.

At a younger age, investors will have many years to stay invested and reap the benefits of compounding. An investor in his 20s can enjoy more than 30 years of compounding while an investor in his 50s won’t be able to benefit the same.

**Compound Interest on Stocks**

The share price of each stock increases or decreases daily during the active market session. The compounding in stocks can work either way but only if the investor has opted for Dividend Reinvestment Plan.

The daily gain or loss on the stock in percentage is actually the change from the price of the previous closing. Hence, compounding occurs daily in stocks. But if dividends are not reinvested, investors won’t be able to enjoy the benefits of compounding in stocks.

Stock investments have historically proven to be the most beneficial investment avenue over the long term. However, it involves risk and can also decrease the value of the investment if the chosen company does not perform well.

**Compound Interest in SIP**

Systematic Investment Plan or SIP is a method of investment, commonly used in mutual funds or unit trusts. In SIP, a fixed amount is invested periodically at predefined intervals into the chosen scheme. Each periodic investment is used to buy units of mutual funds at the prevailing rates per unit. The prices of each unit can change each day in a way similar to stocks.

In the initial days, the profits on SIPs due to daily compounding will be normal but in the long term, the monthly gain can become much more than the periodic investment amount.

The power of compounding in SIP can be better understood with the help of an example.

Suppose, **$500 is invested each month in a mutual fund that provides an average gain of 8%**. The table below describes the value of the investment after given years.

Number of Years

Total Invested value ($)

Value of Investment ($)

Gains ($)

1

6,000

6266

266

3

18,000

20,403

2403

5

30,000

36,983

6983

7

42,000

56,403

14,403

10

60,000

92,083

32,083

15

90,000

1,74,173

84,173

20

1,20,000

2,96,474

1,76,474

25

1,50,000

4,78,683

3,28,683

30

1,80,000

7,50,148

5,70,148

It can be observed from the table that SIP investors were able to gain more than the invested amount only after 5 years of investment. After that, the monthly gains were much more than the investment amount. The value of the invested amount gets almost doubled in 15 years at the rate of 8%. The power of compounding is a major advantage in periodic investment.

SIP investments have lesser risk compared to stocks but the risk factor depends on the fund chosen by the investor. Each scheme has different risk factors and accordingly, the expected returns also vary.

**Compound Interest in Fixed Income Securities**

Investors can enjoy the benefits of compound interest at lesser risk with fixed-income securities. Such instruments do not get majorly affected by volatility in the market and provide fixed returns over time.

Fixed income instruments with compound interest include bonds, certificates of deposits, money market accounts, high-yield savings accounts, etc.

Bonds and certificates of deposits can be seen as a loan given to the entity that is selling bonds. These entities can be banks, financial institutions, businesses, or governments. The entity that sells the bonds agrees to pay the interest with or without the principal amount after a predefined time period.

Not every fixed-income security involves compound interest. Investors must ensure that the chosen security involves compounding or regular periodic payout that can be reinvested to enjoy the magic of compounding.

Bonds with one-year maturity with simple interest can also be reinvested each year to gain the benefits of annual compounding.

Fixed-income securities include two major risk elements namely inflation risk and third-party risk. As the interest is to be paid by the major players with very low risk, they provide lesser interest compared to most other instruments. The return rate can sometimes also go below inflation which will reduce the buying power of the invested amount.

The rate of interest is higher on the securities offered by businesses and financial institutions compared to government-issued securities. This is because the government is least likely to go bankrupt and default while the businesses have a higher probability of defaulting. The risk of the security issuer getting defaulted or failing to repay is called a third-party risk.

**Compound Interest in Real Estate**

Real estate investments are low-risk investments with varying returns. Compounding plays an important role in increasing the value of owned asset manifolds with time. Each year, the rate of real estate increases at different interest rates.

For a better understanding of the power of compounding in real estate, try to find out the prices of real estate in your locality and compare it with the same 15-20 years ago. Those who own real estate assets for more than 10 years must be well aware of the capital gains.

However, real estate has liquidity issues as there is no frequent buying and selling. Most real estate assets have very high prices and it is not feasible to invest small amounts in real estate.

REITs are the solution to the liquidity issue in real estate. REITs work similarly to mutual funds but the investment instrument is real estate. The compounding works well in REITs as well.

**Conclusion**

Compound interest can bring impossible-looking financial goals well within reach. But inflation also holds the power of compounding against you. Hence, if you are not earning compound interest above inflation, you are eventually paying it.

No matter what investment instrument you choose for compound interests, the common thing among all is time. Compounding grows stronger with time but in short tenure, compounding holds no power. Make sure to remain invested for the long term in the instruments that deliver compound interests and watch yourself triumph financially in life.