To understand bond pricing we will first have to understand time value of money which is basics of financial management. Time value of money is nothing but valuing money based on time at which we receive it.

To understand bond pricing we will first have to understand time value of money which is basics of financial management.

Let us understand the concept of ‘Time value of money’ with an example,

If I ask a question, would you prefer 100 rupee today or 100 rupee after 1 year? I am sure you will say 100 rupee today and simple explanation for this will be that you will invest that 100 rupee in bank and at the end of the year you will get 4% interest and 100 would become 104.

What happened here? Let’s carefully understand this. You preferred to have X amount of money today rather than having the same after 1 year because you can invest the same today and earn some minimum risk free returns and after 1 year that money would be greater than X.

Now let’s change the situation a bit. What if I give you a choice of 100 rupee today or 104 after 1 year? In this situation you will be indifferent because if you choose option 1 you will invest 100 in bank and at the end of the year you will get 104, which is equal to option 2.

Now let us look at the 3rd situation, in this I offer you 100 rupee today or 105 after 1 year what will you choose? In this situation you will go for option 2 because it is giving you more money compared to option 1(105 > 104) after one year.

Let us summarize this

- Money received today is more dearer than same money received after 1 year
- If money received after 1 year is equal to money received today plus interest of 1 year, we are indifferent.
- If money received after 1 year is more compare to money received today plus interest of one year, we will prefer 1st option.

All this is nothing but time value of money. Valuing money based on time at which we receive it. But how do we calculate this value? Well don’t worry I have got that covered for you in next section. But before that please understand these concepts very carefully because time value of money is the foundation of everything in finance and by everything I mean literally everything.

This is very simple. If I tell you that you will receive 10% interest on 1000 rupee invested today, what will be amount after one year?

Simple ((0.01*1000) +1000) =1100.

Let us compress this and put it into a formula

1000 is the money which we have today; we will call it present Value or PV. We want to know the future value (FV) of this PV. We have got the rate of return(R). Time period for this is 1 year.

** 𝑭𝑽=𝑷𝑽∗(𝟏+𝑹) ^{𝑻} **

Why did we add 1 in the formula? Well, because we are getting our initial money also back.

Let’s put our example into this formula,

**1100=1000*(1+0.1) ^{1 } **

Now what will be FV of 1000 at R of 10% for 2 years?

** 1210=1000*(1+0.1) ^{2} **

Now in the above example, we got 210 as interest for 2 years which is 10 rupee extra why?

Because of compounding, 1st year we earned interest of 100 rupee, that 100 was again invested so at the end of 2nd year we received interest on 1100 (1000 original amount plus 100 interest for 1st year).

Now in the above examples we made an assumption that interest is compounded annually. What if it is compounded semi-annually or quarterly or monthly, then how will we calculate that? We have to make two changes here 1st divide R by the no. of times we are compounding interest so in case of semi-annually we will divide it by 2, in quarterly divide by 4 and in monthly we divide it by 12. 2nd change we have to make is we have to multiply the time with no. of time we are compounding the interest. Let’s continue with the same example of 1000 and see the power of compounding.

We can clearly see that when we did compounding on annual basis we got 210 as interest and when we compounded to monthly basis, we got 219.42 as interest. You might think that difference is not that big but think in this way only on 1000 rupee difference is 9.4 rupee what will be the difference when figures are in lakhs and crores and when time period is 10 or 15 years. This is the power of compounding.

A fixed sum of money which you receive or pay at regular interval is called annuity. For example if you receive 100 rupee every month as pocket money, it is called an annuity for both you and your parents because it fulfills two conditions, fixed sum of money that is 100 rupee in this case and regular interval which is monthly in this case. There are two types of annuity:

- Ordinary Annuity
- Due Annuity

Ordinary annuity is nothing but you start your cash outflow from the end of the year. Opposite to ordinary is annuity due annuity, in which we start our cash flow from the beginning of the year.

To understand the concept of calculating future value of annuity, we take our 1000 rupee example further. Now what if you deposit 1000 rupee every year in your bank account which gives 10% return for next 10 years, what will be the sum after 10 years?

One way to approach this problem is to individually calculate all the Future values and adding them.

Same is shown below

We invested our first amount at the end of first year 9 years, 2nd year amount stays for 8 years and last amount would not stay in our account because we are depositing it at the end of the year hence no interest for last 1000. After 10 years you will have 15937.42 in your bank account. This was a small problem with relatively easy figures now if we change the magnitude of the problem it would become difficult to calculate the figures individual. To understand annuity problems we use a bit modified formula which is:

If we put all the values in the above equation we will get the same answers.

If we put all the values in the above equation we will get the same answers.

Now let us focus on the other side of the table. Now if I ask you 10 years from now I will give 50,000 rupees, what will be the present value of that? Or you want to withdraw 1000 every year from your account how much should you put in it? Seeing the difference earlier we were calculating future values and now we want to know the present value. Don’t worry all the concepts are same. Use the below two formulas to calculate the Present value of lump sum and annuity.

Before we proceed further, I would recommend everyone to practice some questions with respect to time value of money because as mentioned earlier it is the foundation for all financial matters. Valuation of bonds will start from calculating Present value only.

Perpetuity is regular stream of cash flow for lifetime. For example, if I tell you that I will give you Rs. 1000 every year for the rest of your life, what is the present value of the stream (Rate of return assumed 10%)?

Our first thought will be that we cannot calculate this value because we will receive money forever. But think it in this way, what amount you should deposit in bank so that you receive 1000 rupee every year. Formula to calculate he PV of perpetuity is:

** 𝑷𝑽𝒐𝒇 𝑷𝒆𝒓𝒑𝒆𝒕𝒖𝒊𝒕𝒚=𝑪 / 𝑹**

Continuing with the above example, now what if the growth rate of 1000 rupee perpetuity is 2% then what will be its present value. We can calculate this value with below formula:

**
𝑷𝑽𝒐𝒇 𝑮𝒓𝒐𝒘𝒊𝒏𝒈 𝑷𝒆𝒓𝒑𝒆𝒕𝒖𝒊𝒕𝒚=𝑪/(𝑹−𝑮)
𝑮=𝑮𝒓𝒐𝒘𝒕𝒉 𝑹𝒂𝒕𝒆
**

Now we go to the interesting part, how to value the bond. Two types of cash flow come from the bond. One is the periodic payment and other is the principal we receive at the end of the maturity period. Now to find the current price of the bond, we have to discount the above two cash flow.

Let’s solve with an example suppose there is a 1000 rupee 10% 10 Year bond which pays coupon annually, what will be its current price at Rate of return 7%?

We will break down this into two parts first we will calculate the P of 1000 rupee which we will receive after 10 years and then we will calculate PV of 100 rupee which we will receive every year for next 10 years and then add both the figures.

Every bond would not be of the same type, some may give semi-annual coupon or even monthly but the way of calculation is the same. I recommend all the readers to solve one question of each type.

As explained earlier YTM tells us how much we have totally earned from a bond including both principal and interest if we buy the bond now and hold it till maturity. Most of the investors rely on this number to compare the bonds. Formula given below gives us the closet value of YTM. There is another trial and error method but I think for initial understanding, this formula method is good.

We calculate YTM based on the assumption that you will hold the bond till the end of the maturity. But what if bond is callable and may be withdrawn from the market before the maturity. In such scenario we calculate YTC. Why do companies call the bond before the maturity? If market interest rate reduces, it would make bond expensive, then it make sense for the company to raise money from a less expensive source and pay back bond holders. Calculating YTC is similar to calculating YTM just assume that instead of call, the bond actually matured on that day and calculate the figure.

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