## How does time affect the value of your money? Time value of money.

**Time Value of money**

Let us learn today something related to finance. “Time Value of Money”.

What exactly is Time Value of Money?

You must have heard the quote “Time is Money”. Well! that’s true…

What would you choose if you have the following options:

a). Get Rs 100 today for no reason

b). Get Rs 100 after one year for no reason

It is your life and you follow your own rules so you can choose anything 😛 but a rational person would always go with option a). This is because if he gets Rs 100 today then he can consume it today. For eg: he can invest Rs. 100 today @ 10% and so he can get 110 after a year. But what if he chooses option b). He will get only 100 at the end of the second year and ultimately he ends up with Rs. 10 less.

Or we can say the current or present value of Rs. 100 (that one will get after a year) will be Rs. 100/1.10 (if interest rate is 10%p.a) i.e Rs. 90.9090 today.

We can say that 90.9090 is the “PRESENT VALUE”(PV) of Rs. 100(that one will get after a year) @ 10%p.a or if we invest Rs. 90.9090 today then we will get Rs. 100 at the end of the first year. So, Rs 100 is the “FUTURE VALUE”(FV) of Rs. 90.9090 invested today @10% p.a.

FV = PV*(1+i)^(n)

i = rate of interest per annum

n = number of payment intervals in 1 year

So, till now I guess that everybody believes that going with option a) is a rational decision.

So I repeat “Time is Money”.

Basic conclusion – all things being equal, it is better to have money now rather than later.

Using this basic we can study the concept of annuities.

Now, what is annuity??

In simple terms, An annuity is a series of payments made at equal intervals. Intervals can be monthly, quarterly, semi-annually, annually and so on….

If a series of payment continues forever then it is known as “perpetuity”.

Now let us see the different types of annuities and calculation related to those annuities.

**Type 1. Ordinary Annuity**

It is that annuity where the payments are made at the end of the interval. For eg: Pay x Rs. to bank at the end of every year.

**Present Value of Ordinary Annuity**

Let say an amount of Re. 1 is paid at the end of every year for next 5 years @ 10% p.a.

i.e 1 at the end of 1^{st} year, 1 at the end of 2^{nd} year and so on till 1 at the end of 5^{th} year.

So to get the present value of the annuity we need to solve the following equation:

P.V =

Re. 1 received at the end of the 1^{st} year will be discounted for 1 year @10%, Re. 1 received at the end of 2^{nd} year will be discounted for 2 years @ 10% and so on.

If you observe this clearly, then it is a Geometric Progression with r = a2/a1 = 1/(1+i)

Now, r < 1, so the sum of the above G.P would be a1

So the sum would be (1/(1+i))*

Solving this we will get P.V formula of ordinary annuity = 1*

**Let us do an example to get a better understanding of ordinary annuity.**

Example: Shyam has to pay five annual instalments of Rs. 1000 each to Ram. Payment has to be made at the end of each period. The prevailing interest rate is 10% p.a. So the P.V of these instalments would be

P.V

P.V = 1000 = 3790.787

Interpretation – Shyam can either pay Ram Rs. 3790.787 today or he can pay Rs. 1000 at the end of every year for next five years

OR

The value of Rs 1000 payable at the end of each year for the next five years is Rs. 3790.787 today.

**Type 2. ****Annuity Due**

It is that annuity where the payments are made at the beginning of the interval. For eg: Pay x Rs. to the bank at the beginning of every year.

**Present Value of Annuity Due**

Let say an amount of Re. 1 is received at the beginning of every year for next 5 years @ 10% p.a

So to get the present value of the annuity we need to solve the following equation:

P.V = 1 +

Re. 1 received at the beginning of 1^{st} year will be not be discounted, Re. 1 received at the beg. of 2^{nd} year will be discounted for 1 year @ 10% p.a and so on till Re. 1 received at the beg. of 5^{th} year will be discounted for 4 years @ 10% p.a.

So the formula for the P.V of the annuity due is = 1*

Let us do an example to get a better understanding of ordinary annuity.

Shyam has to pay five annual instalments of Rs. 1000 each to Ram. Payment has to be made at the beginning of each period. The prevailing interest rate is 10% p.a. So the P.V of these instalments would be

P.V

P.V = 1000* = 4169.865 Rs

Interpretation – Shyam can either pay Ram Rs. 4169.865 today or he can pay Rs. 1000 at the beginning of every year for next five years

OR

The value of Rs 1000 payable at the beginning of each year for the next five years is Rs. 4169.865 today.

Though these things are basic, yet forms the foundation for every calculation in the world of finance. For eg: When you go to a bank to take a loan, the amount of EMI is estimated using these formulas.

**Case 1. (When payments are semi-annual)**

Payment of Rs. 1000 has to be made at the end of every six months

Term/Duration of the Loan = 10 years

Interest rate = 8% per annum payable half yearly

Now if one has to make the payment half yearly i.e after every six months then the interest rates need to be adjusted accordingly.

If the payment interval is 6 months then the interest rate applied should be of 6 months.

So, in our case, the interest rate for 6 months would be 8/2% i.e 4%

Now we know that the total duration of the loan is 10 years. Our one interval is 6 months i.e ½ year, so there would be a total of 20 (10*2) intervals.

So, the value of loan, in this case, would be computed using the formula = .

= 1000*

**Case 2. (When payments are quarterly)**

Payment of Rs. 1000 has to be made at the end of every three months

Term/Duration of the Loan = 10 years

Interest rate = 8% per annum payable quarterly

Now if one has to make the payment quarterly i.e after every three months then the interest rates need to be adjusted accordingly.

If the payment interval is 3 months then the interest rate applied should be of 3 months.

So, in our case, the interest rate for 3 months would be 8/4% i.e 2%

***(We have divided the interest rate by 4 because there are 4 quarters in a year)**

Now we know that the total duration of the loan is 10 years. Our one interval is 3 months i.e ¼ year, so there would be a total of 40 (10*4) intervals.

So, the value of loan, in this case, would be computed using the formula =

= 1000*

**Case 3. (When payments are monthly)**

Payment of Rs. 1000 has to be made at the end of every month

Term/Duration of the Loan = 10 years

Interest rate = 8% per annum payable monthly

Now if one has to make the payment monthly i.e after every month then the interest rates need to be adjusted accordingly.

If the payment interval is of 1 month then the interest rate applied should be of 1 month.

So, in our case, the interest rate for 1 month would be 8/12% i.e .66667%

***(We have divided the interest rate by 12 because there are 12 months in a year)**

Now we know that the total duration of the loan is 10 years. Our one interval is of 1 month i.e 1/12 year, so there would be a total of 120 (10*12) intervals.

So, the value of loan, in this case, would be computed using the formula =

= 1000*

So, we can see that the interest rate is the number of intervals in a year and the duration is multiplied by the number of intervals in a year. Numbers derived after these adjustments help in the computation of loan amount.

Let us do an example to get a better understanding of the above concepts.

Example: Let say that Ashish has borrowed Rs 80 lakhs from the bank. Bank would be willing to grant him the loan only of he is ready to take it for 20 years @ 8% per annum payable monthly. Ashish wanted to know the monthly instalment amount and so he decided to reach out to a consultant for this. The consultant gave him the required result using the following method:

Loan amount = 80,00,000

Duration = 20 years

Interest rate = 8% per annum payable half yearly i.e 8/12% per month.

So let say the amount that Ashish needs to pay to the bank is Rs x per month for 20 years.

Consider this to be the case of ordinary annuity where Rs. x needs to be paid at the end of every month.

Total payment intervals would be 20*12 = 240

So the following equation will the value of the monthly payment i.e x

80,00,000 = x*

So x = 66916 (approx….)

I have also attached an excel file showing the complete computation of the above-discussed example.

See the complete loan schedule in the attached excel file. Click here to download the file.

You can make changes in the yellow boxes to get your results.

Now let’s have a brief description of the schedule:

**For Year 1**

In any year of during the term of the loan, Ashish will have to pay a total amount of Rs. 802982.5. It can be seen in cell C17 of the excel sheet. Some part of this is the **interest paid** during the year and the remaining part is the **principal loan amount repaid** during the year.

To calculate the principal repaid during the year, we need to calculate the present value of the annuity payment at the end of year 1.

i.e 66915.02* ————– (1)*

Now Principal Repaid in the 1st year is = 8000000 – (1)*………… (cell c16)

And Interest Paid in 1st year is = Total amount paid in 1st year – Principal repaid in 1st year ………..(cell c15).

**For Year 2**

The total amount paid during the year would be the same i.e Rs. 802982.5 (cell D17)

Principal Repaid in 2nd year = Principal Amount left after the 1st year (cell c19) – Principal Amount left after 2nd year (cell D19).

Interest paid during the year = Total amount paid in 2nd year – Principal repaid in 2nd year

And that is how the full schedule in made.

You can also change the value in the yellow cells and get the desired results.

Try it with different figures, solve it manually and compare your answers !!

GOOD LUCK!

-by Himanshu Goyal