 लोग लैज लेग़ान येगा at benefit-prince, अखाग, ल [? औज् ?] अचवार नहीं प्देट हैं give me capital येखि ज्त लेज़ानी अच़़ि कादो� already खई गर配- المगाई लेज़ान की ढकल ज़ान ढ़द़तanish अऐकौपा। अर ये बैसिकली एक अजी फनेंज्यल अंस्तुम्ड़न्ट है जिसके साथ कॉपोंज अतश्ट होतें अग्यन अगर अब आब नी दिखा होगा थे नहीं आप आख्सान में नाशल सेविंटेगट सोते हैं उंगे तो दो रेगडिलर अिंखम सέρ्टिकेट हैए कुछ उंके आँशिके फ्चान्च्यर ूपने टी परकाईत थी आखéta. अगर आप अइ सेस्म के लेण अदापी ब हूँँ, च्वरीखर लेडिल या थी लेअ, वो लोग जी। उस्रीकत अते होते है, मकर उसके साज परफरेटेट का एक सक्छन भना अध अध़ेड होते है, जिसके आनध़ परचिया अध लेड होटी होती है. तिनके अपर नमबा लिखाोते हो थे हर एक परची के अपर तो आप हर साल के बाज जब जाते हो अपना वो सर्टिपिकेर लेके सेविंक सेंटर, तो वहां से वो एक चोटीची परची उतार लेते हैं, और नमबर वाली और पिर नमबर तुवाली, और उसके साप से आपको जोभी शरासुद तो नहीं अपके साथ की होती, अपको पेख अपको पेमंट होगी, आप उनको केश देरो, इसके बतले में वो आपको सर्टन पीरेड अप ताएम के उपर, after a certain point in time, after a certain time period, every 6 months or every year, they give you a fixed rate of interest. And then, when it gets mature, if it is a 5-year bond or a 3-year bond, then you can take back the face value of your face, except for those issues. So, this is how the coupon bond or the bearer bond operates. So, when we talk about the bearer bond or the coupon bond, the time period you are getting the payment and the maturity of the bond, then you are getting the payment. So, you get two types of payments. So, when we calculate the yield to maturity, we have to consider these two aspects. So, basically, the overall strategy that is used for the coupon bond or the bearer bond is that you match the amount of money you are getting today to buy that bond, which you are getting the salana payment and the maturity you are getting. So, you are getting two types of payments. After a certain time period, they can be 5-10 depending upon the amount of money you are getting. And then, on maturity, there are two types of payments. When you buy the bearer bond presently, you match the amount of money you have spent on it. So, on that, you get to know the overall yield to maturity. So, this is how we deal with this particular type of investment bonds. Purely, for the example of the coupon bond, if I am getting the local example of Pakistan, there are Pakistan investment bonds which are based on this particular concept. So, now we will discuss how to calculate the yield to maturity. So, if we assume that our face value, FV, is suppose to be Rs. 1000. And the time period for maturity, that is 10 years. And you are getting a coupon payment of Rs. 1 year every year. For this, you are told that the coupon rate is Rs. 10. i.e., you are buying a coupon of Rs. 1000. So, every year, you will get a total bond duration of Rs. 10% of Rs. 1,100. That you will get at the end of the year. The duration of the total bond, maturity, will come after 10 years. So, if you want to get the present value, then what we have to do is, we use the formula of the present value. The important thing here is that, if you look at the right side at the end, then I have written two terms here. This means that you are getting the present value in the first year of the year. You are getting the present value of Rs. 100 in the second year. You are getting the present value of Rs. 100. You have divided this 100 into 1 plus i raised to the power 2. Similarly, you have said that in the 10th year, the payment of the coupon, the 10% of the coupon rate, you will get it in the 10th year. You will get the present value of Rs. 100 divided by 1 plus i raised to the power 10. Along with this, this extra term is also raised to the power 10. What is that? You are getting the money on the maturity. i.e., you had bought the bond of Rs. 1000, 10 years ago. When the maturity period of 10 years is over, then you will get the return of Rs. 1000. But the more you are away from your money, you had that bond on which you are getting the interest of Rs. 10% every year. So, by considering what is going to be the present value of all these different payments, how many payments are there, the payments of the 10th year, the payments of the 10th year, and the payment of the 11th year when you have to get the return of Rs. 1000. So, you are getting the 11 payments. By considering the 11 payments, when you have to calculate the yield to maturity of the coupon bond, you will get the present value of the 11 payments. Similarly, if you look at the formula, on the bottom, there is a formula written in the general format. On the top, the values I have given were examples. So, if we generally develop a formula, then we can see that if we want to find out the present value of a certain coupon bond, on which there are coupon payments for a certain n number of years, and similarly, on the maturity period, you are getting the total face value. So, you have to discount that too. You have to get that present value. On the maturity period, the 10th or whatever year you are going to spend, all the time periods with you, again, it is not necessary that you are getting the payment of the coupon bond. It can be by annually. It can be after every three months. So, it depends upon the terms and conditions on which you have purchased your coupon bond. So, under this formula, you can calculate the present value using this formula. If you know the present value, you know the future value, you know the amount of coupon payments you are going to get, then you can plug these three values into this formula and calculate the interest rate of your yield to payment. The next thing which I am going to discuss with you is that the yield to maturity is linked up with the prices. Okay, whose bond is the price? So, here you are getting a table in which I have enlisted the price of bonds. In that, I have written that the face value is the bond of Rs.1000, the coupon bond. And you have taken that bond by paying Rs.1000. And in that, the certain maturity time period which has been defined, that is going to be there. And that maturity time period is given as, in this example, it is given as 10 years. The yield to maturity, we have calculated the values. This is the first formula which I have shown you. And we have said that the coupon rate, after every year, if you purchase the coupon bond, you get so much interest that is again assumed given as 10%. Okay, we have assumed that the coupon rate is 10%. The maturity is 10 years and the bond you have bought is Rs.1000. Okay, so now I have made a column here, price of bond. Before discussing this, I will tell you how the price of the bond is determined. So, the face value is not the price of bond. Okay, the face value means that you have bought the bond. On top of that, it is written as Rs.1000 bond. But the price of the bond is determined by the interaction of market supply and demand. Right? So, that will tell you the price of the bond. So, we are going to take into account different prices. Prices are set from the demand and supply forces. And we have said that the bond price is Rs. 1200, Rs. 1100, Rs. 1000, that is, it is equal to the face value. And it can be Rs. 900. It is less than the face value. It can be Rs. 900 and Rs. 800. When it is Rs. 800 or Rs. 900 or Rs. 1200, then the yield to maturity is also calculated. By using the old formula, they have been estimated as if the price of bond is Rs. 1200, then we have estimated the yield to maturity in percentage as Rs. 7.13. After that, we have said that if the price of bond is Rs. 1100, then it was estimated as Rs. 8.48. If the price of bond is equal to its face value. Right? This is important. So, if our price of bond is equal to the face value, then you can see that the yield to maturity becomes equal to this 10% that is written. This was your coupon rate. Right? The coupon rate was 10%. So, if from the interaction of market forces, the price of bond becomes equal to the face value, then the yield to maturity becomes equal to the coupon rate. Right? But if the price of bond is lower than the face value, then the yield to maturity is higher than the coupon rate. Right? But if the price of bond is higher than the face value, then the yield to maturity is lower than the coupon rate. Further, the price of bond has declined from Rs. 1200 to Rs. 1100. So, the yield to maturity is smaller than the coupon rate. It is smaller than your coupon rate. And if the face value becomes equal to the price of bond, then the coupon rate and the yield to maturity, the two becomes equal, become equal to each other. But if your price of bond is lower than the face value, then your yield to maturity increases from your coupon rate. Right? Fine. Why is this? To discuss this, what did we do? We said that basically, when the price of bond is lower than the face value in the market, then people start selling it. Right? And when they start selling it, then what will happen? We calculate the overall yield to maturity value. Then you will find that its yield to maturity seems to be larger as compared to the coupon rate. But if the price of a bond becomes higher, then people retain. They prefer to retain those bonds. And as a consequence, what happens? The higher the retention, the lower the yield to maturity value. So, this is the trade-off that we saw that when the yield value is equal to the coupon rate, then we tell this particular situation that bond is at par. And when we see the price and yield here, that they are negatively related, prices are higher, then yield to maturity falls and vice versa. And the third important thing that we have noted here, that is that if the yield is greater than the coupon rate, then your yield, if the yield is greater than the coupon rate, then your bond's prices are lower than the par value. So, this is an interesting phenomenon. Basically, our take away of this full discussion is that we can see with the help of this example that our yield to maturity and our price of bond are both inversely related. If one goes up, then the other goes down, and vice versa. With this, there is another financial instrument from the coupon bond, which is important to mention, and that is called a perpetuity. But there is no maturity in perpetuity. So, if you have bought a coupon bond, then you will get every fixed amount of fixed period of time, after that, according to the coupon rate, you will get a sood, but you will never be able to sell it back, that issuance authority, from which you have bought it. So, such a financial instrument, the longer you keep it with you, according to the coupon rate, you will get an interest on it. That is called a perpetuity. The bad side is, it cannot be, it is not redeemable. But perpetuity, if you go to the market, which is your financial market, in this, you go and buy and sell it. Its face value will not be there. Its market price, you can sell it on it, you can buy it, it may cause you a loss, it may benefit you, the amount of money that you took, perpetuity, you can get less than that, you can sell it on more money, it will depend on the conditions of the market. But till the time, any perpetuity, investor, till then, it will be the payment of the coupon rate. So, that is another interesting financial instrument that is available in the market. So, when we talk about perpetuity, and to discuss its yield to maturity, let's see how we can calculate it. So, for that, I have put up a formula, in which, the price of perpetuity, is the ratio of yearly payment and yield to maturity of the perpetuity, you can explain it, if you know the price of perpetuity, you know the yearly payment, so, you can yield to maturity of the perpetuity, you can calculate the yield to maturity, similarly, if you know the price of the perpetuity then, you can calculate the yearly payment. To further explain this, I have taken an example, and in this example, you can see that, suppose, we have a zoomed situation, in which, we have a bond, which is perpetuity, and its price is Rs. 2000, and the continuous constant payment is Rs. 100 annually. So, if we want to know in this situation, that the yield to maturity means, how much interest we are paying, for that, you simply have to do, the annual payment, which is C, you have to divide the price from perpetuity, and you will get the value for the yield to maturity. In this case, in this example, I was getting the annual payment, the price, which I had paid, to buy perpetuity, that was Rs. 2000, and if I want to know or calculate the yield to maturity, then it would be Rs. 100 divided by Rs. 2000, which gives me a 5% yield to maturity. So, this is how we have got the formulas to assess the yield to maturity or the yearly payment or present value. You can calculate the yield to maturity, but remember that for different types of financial instruments, we are using different types of formulas and different types of concepts. Depending upon the nature of that financial product.