Time Value of Money (TVM) Concepts and Questions

Ever wondered why a rupee today is worth more than a rupee tomorrow? It’s not just a quirky riddle; it’s a fundamental concept in finance called the Time Value of Money (TVM).

Why is TVM important for RBI Grade B aspirants?

TVM is a crucial component of the finance syllabus of the RBI Grade B exam. This knowledge will not only help you prepare for the RBI Grade B exam but also enable you to compare different investment options and make informed financial decisions.

Below, we’ll introduce the concepts of Time Value of Money that’ll help you prepare for the RBI Grade B exam.

Introduction to the Time Value of Money

Time Value of Money (TVM) is a crucial financial concept that explores the idea that the value of money today is not the same. When we say money has time value, it simply means that the amount of money you have today is worth more than the same amount of money in the future.

Imagine you have Rs 10,000 with you today. You have two options:

Option 1: You can spend Rs 10,000 right away on something.

Option 2: You can choose to invest Rs 10,000 in a bank account that offers an annual interest rate of 5%.

If you choose option 2:

• After one year, your money will grow by 5% (Rs 500), and you’ll have Rs 10,500.
• After two years, your initial investment of Rs 10,000 will continue to earn the same interest of Rs 500 per year. So, after two years, you’ll have Rs 11,000.

This is what the basic concept of the Time Value of Money signifies.

Reasons for Time Value of Money

Here are the reasons for the time value of money:

• Investment Opportunities: Money today can be used to grow wealth through investments in stocks, bonds, etc.
• Risk and Uncertainty: Current money is typically less risky than future money due to potential changes in the economy.
• Inflation: Prices rise over time, reducing the purchasing power of money.
• Consumption: People usually prefer to spend money now rather than waiting for the future.

Techniques of Time Value of Money 

First, let’s decode the meaning of Present Value and Future Value.

1. Present Value (PV): The current value of money. (Example: Rs 10,000 you have now.)
2. Future Value (FV): The value of money in the future. (Example: Rs 10,000 growing to Rs 10,500 with 5% interest.)

Future Value = Present Value + Interest

Simple and Compound Interest 

1. Simple Interest

Interest is calculated only on the initial amount. You earn or pay a fixed percentage of the principal, regardless of how much you’ve earned or paid in the past. 

The formula to calculate Simple Interest (SI) is: SI = (P * R * T) / 100

Where:

• SI stands for Simple Interest.
• P is the Principal Amount (the initial sum of money).
• R is the Rate of Interest per year (as a percentage)
• T is the Time period in years for which interest is calculated.

2. Compound Interest

Interest is calculated on both the initial amount and accumulated interest.  

The formula to calculate Compound Interest is: A = P * (1 + R)^(t)

Where:

• A is the future amount, including both principal and interest.
• P is the principal amount (initial sum).
• R is the annual interest rate (in decimal form).
• T is the time in years (here T can also be written as n, both things mean the same)

Key Observation – Interest earned via the compound interest method is bigger as compared to the simple interest method, given the same principal amount, interest rate, and time period. 

Compounding Frequency

1. Annual Compounding: Interest is added once a year.
2. Frequent Compounding: Interest is added more often than yearly.

1. Semi-Annually: Twice a year.
2. Quarterly: Four times a year.
3. Monthly: Twelve times a year.
4. Continuously: Constantly.

So, your new rate of interest will be R divided by the compounding frequency and T will be multiplied by the compounding frequency.

Or, alternatively, A = P * (1 + (r/y))^(y*t)

Where y is the number of times that interest is compounded per year.

Question – Calculate the amount to be received after three years from the following information, assuming the principal is Rs. 10,000 and the rate of interest, being 10% is compounded semiannually, quarterly. 

Case 1: Semi-Annual Compounding

• Wherein the existing Rate of Interest is 10% 
• Compounding Frequency is 2
• Therefore, the new rate of interest will be 10/2 = 5% 

New Time Period will be = Time Period * Compounding Frequency

• Wherein the existing Time period is 3
• Compounding Frequency is 2
• Therefore, the new time period will be 3*2 = 6 

Now, you can use the basic formula, A = P * (1 + R)^(T)

• A = 10,000 * (1 + 0.05)^6
• A = 10,000 * (1.05)^6
• A ≈ 13,400

Case 2: Quarterly Compounding

• Wherein the Rate of Interest is 10% 
• Compounding Frequency is 4
• Therefore, the new rate of interest will be 10/4 = 2.5%

New Time Period will be = Time Period * Compounding Frequency

• Wherein Time period is 3
• Compounding Frequency is 4
• Therefore, the new time period will be 3*4 = 12

Now, you can use the basic formula, A = P * (1 + R)^(T)

• A = 10,000 * (1 + 0.0.25)^12
• A = 10,000 * (1.025)^12
• A ≈ 13,448.8

Conclusion: More frequent compounding (quarterly) leads to slightly more interest earned.

Continuous Compounding

Continuous compounding is a theoretical concept used in mathematics and finance to model scenarios where interest is calculated and applied continuously. 

Its formula is A = P * e^(rt)

Where:

• A is the Future Value (the total amount including both principal and interest).
• P is the Principal Amount (the initial sum of money).
• e is the mathematical constant approximately equal to 2.71 (You have to assume this) 
• r is the Annual Interest Rate (in decimal form).
• t is the time the money is invested or borrowed for, in years.

Now, let’s move to the concept of compounding and discounting.

Compounding and Discounting 

Assume that you are investing an amount of Rs. 10,000 in a bank at a 10% annual Compound interest rate for 5 years. Now, calculate the amount you will receive after 5 years. 

• Principal Amount (P): Rs. 10,000
• Rate of Interest (R): 10% (Compound interest)  
• Time (T): 5 years

Using the compound Interest formula: A = P * (1 + R)^T

Substitute the values:

• A = 10,000 * (1 + 0.10)^5 
• A = 10,000 * (1.10)^5 
• Amount ≈ Rs. 16,105.10

Conclusion: After 5 years, you’ll have Rs. 16,105.10, including Rs. 10,000 principal and Rs. 6,105.10 interest. This means the Present Value will be 10,000 and the Future Value will be 16,105. 

Therefore the relation between Future Value and Present Value is,

FV = PV * (1 + r)^n

Where,

• FV stands for Future Value, which is the value of money at a future point in time.
• PV represents Present Value, which is the value of money at the current time or today.
• r is the annual interest rate (expressed as a decimal).
• n is the time period in years.

Important Note: This formula is exactly the same as we have discussed in the case of compound interest the only difference over here is, that you are denoting Principal as Present Value and Amount as Future Value. 

Here’s how compounding unfolds:

1. Starting Point (PV): You begin with a certain amount of money today. This is your PV, the foundation of your financial journey.
2. Investment: You decide to put that money to work. You invest your money and therefore you will start earning interest on it. 
3. Arrival at Future Value (FV): Eventually, after a certain period of time, you reach your destination, which is the Future Value (FV). This is the total amount you end up with, which includes both your initial principal and all the interest accumulated over time.

Discounting is like the reverse journey of compounding in the world of finance. While compounding takes your money from Present Value (PV) to Future Value (FV), discounting does the opposite; it takes your money from Future Value (FV) to Present Value (PV) 

Here’s how discounting works:

1. Future Value (FV): You have a specific amount of money that you expect to receive or spend at some point in the future. This is your FV.
2. Time Factor: Money in the future is not as valuable as money today because you could be using it or earning interest on it. So, you “discount” it to find out what it’s worth today. We will discuss more about it later on. 

Calculating PV: Discounting helps you find out how much money you need to have today to end up with the FV you expect in the future.

Important Note: Discount rate in Time Value of Money represents the rate at which future cash flows are adjusted to their present value.

Practical Application of Compounding and Discounting 

Question on Compounding: If you deposit Rs. 25,000 in a bank that offers an 8% annual interest rate on a four-year time deposit, how much would the deposit grow at the end of four years?

To calculate the future value (or the amount), we shall be using our basic formula only, 

which is FV = PV * (1 + r)^t. Now, let’s calculate it:

• FV = 25,000 * (1 + 0.08)^4
• FV ≈ 25,000 * 1.36049
• FV ≈ Rs. 34,012.25

Question on Discounting: You are expecting to receive Rs. 1,000 one year from today. If you want to determine the present value of that amount today, assuming a discount rate of 8%, what is the present value?

The formula remains the same, which is FV = PV * (1 + r)^t. Now, let’s calculate it:

• PV = 1,000 / (1 + 0.08)
• PV = 1,000 / 1.08
• PV ≈ Rs. 925.93

Introduction to Annuities 

An annuity is like a series of regular payments or receipts that happen at a fixed time interval. Think of it as money coming in or going out regularly, like your monthly rent or your salary. 

For example – let’s say you win a lottery, and they promise to pay you Rs. 1,000 every month for the next five years. That’s an annuity. You’ll receive a fixed amount of money regularly for 5 years. 

Types of Annuities

The following are different types of annuities. 

1. Ordinary Annuity: An ordinary annuity is a type of annuity where the payments or receipts occur at the end of each period. 
2. Annuity Due: An annuity due, on the other hand, is a type of annuity where the payments or receipts happen at the beginning of each period. 

Calculating the Present Values and Future Values of Annuities 

Question – You are depositing Rs. 25,000 “every year” in a bank that offers an 8% annual interest rate on a four-year time deposit, 

Now, if I ask you, is this a case of annuity? 

Yes, certainly it is!

Because now you are depositing Rs. 25000 “every year”, and remember annuity is just a series of payments made at “equal intervals”. 

Future Value of Ordinary Annuity 

Question – Suppose you invest 1000 at the end of each year for the next 5 years with an interest rate of 5%. What would be the total value of money at the end of 5 years?

We can use the following formula – FV(Ordinary Annuity)=C*[{(1+i)^n – 1}/i]

Where, 

• C = Cash flow per period = Rs. 5000
• i = interest rate = 5%
• n = number of payments = 5

Just plug in the values now, and you will get 

• FV = 1000 [(1+.05)5–1] / .05
• FV = 1000 [1.27628 – 1] / .05
• FV = 1000 [.27628] / .05
• FV = 276.28/.05
• FV = 5525.6

Now, let’s move to the calculation of the future value of the annuity due.

Future Value of Annuity Due 

Here, we can directly use the shortcut method to calculate the future value of Annuity Due.

FV of Annuity Due= (1+r) x P*[{(1+r)^n – 1}/r]

Where, 

• P=Periodic Payment
• r= rate per period
• n= number of periods

Present Value of Ordinary Annuity 

Question – What is the present value of the project which pays cash flows of 1000 at the end of each year for the next 5 years assuming the interest rate to be 5%?

You can use the following formula directly:

 PV(Ordinary Annuity)=C*[{1 – (1+i)^-n}/i]

Where, 

• C = Cash flow per period = Rs. 1000
• i = interest rate = 5%
• n = number of payments = 5

Apply the values in the formula, you will get  

• PV = 1000 * [(1 – (1+.05) -5] / .05
• PV = 4329.47

Present Value Annuity due 

For ordinary annuity, payments are made at the end of each period. However, for the Annuity Due, payments are made at the beginning of each period.

We can directly use the shortcut method to calculate the present value of Annuity Due.

 PV(Annuity due)=C*[{1 – (1+i)^-n}/i]*(1+i)

Present Value and Future Value Factors 

To define, Present Value and Future Value Factors, are often referred to tables which are precomputed values used in finance to simplify the calculation.

For example, look at the following table

This table helps you in the calculation.

Question – If you deposit Rs. 25,000 in a bank that offers an 8% annual interest rate, how much would the deposit grow at the end of two years?

Solve this question by yourself first. 

Explanation – If we solve this question, by using the relationship between FV and PV, we will get, 

FV = PV * (1 + r)^t

On further solving, Now, let’s calculate:

• FV = 25,000 * (1 + 0.08)^2
• FV = 25,000 * (1.08)^2
• FV ≈ 25,000 * 1.1664 (in the table above this exact value is given)
• FV ≈ Rs. 29,160

Other Related Concepts 

In this section, we will understand two basic concepts – 

Rule of 72, 114, and 144

• Rule of 72: By dividing 72 by the annual rate of return, investors can get a rough estimate of how many years
• Rule of 114: To estimate how long it takes to triple your money, divide 114 by your expected interest rate (or rate of return)
• Rule of 144: To estimate how long it takes to Quadruple your money, divide 144 by your expected interest rate (or rate of return)

Effective Interest Rate (EIR)

The Effective Interest Rate (EIR), also known as the Annual Equivalent Rate (AER) or Annual Percentage Yield (APY), is a measure that reflects the true or effective annual interest rate earned or paid on an investment, loan, or financial product, taking into account the effects of compounding.

In simpler terms, the effective interest rate represents the total interest or return you’ll receive or pay over the course of a year, factoring in how often interest is added to the principal amount. It provides a more accurate picture of the actual financial impact of a transaction because it considers the frequency of compounding.

The formula to calculate EIR is given below 

EIR=(1+r/m)^m -1

Where, 

• r=Nominal rate of interest (Yearly)
• m=Frequency of compounding per year

Click on the link, “RBI Grade B Time Value of Money PYQs”, to practice the actual bond-related questions asked in the exam.

Click on the link, “RBI Grade B Time Value of Money Concepts” to download the above-mentioned bond concepts in a PDF format.

Conclusion

We hope this article has helped you grasp the fundamental concepts of the Time Value of Money. Now, to solidify your TVM understanding and prepare for the RBI Grade B exam, it’s essential to practice as many questions as possible on this topic.

Once you’ve mastered TVM, delve into other crucial finance topics, such as bonds, derivatives, primary and secondary markets, alternative sources of finance, and more. We’ve provided comprehensive resources, including concepts, PYQs, and practice questions, for each of these finance topics to help you excel in your RBI Grade B preparation.