 If coupon rate on a bond is equal to the current market interest rate and the bond is selling at the par value, then in this situation, the coupon payments are offering to the bond holder a fair rate of return on her investment in the particular bond. But if the coupon rate is lesser than the market interest rate, this means that the mere coupon payments are made to the bond holder. In this situation, these coupon payments are entirely insufficient to compensate the bond holder at the competitive rates. Now in this situation, it is desired by such bond holder to have certain price appreciation. And for that purpose, bonds are offered at discount to these bond holders. In this way, an opportunity of built-in capital gain is provided to the bond holder through selling them the bond at the discount rate. Now how to compute built-in capital gain? We have an example. Here we have outstanding 7% bond with $1000 par value. Bonds remaining life is 3 years and the market interest rate is 8%. Now if we solve this bond valuation equation using the given data, we have at P0 or the current market price of bond at $974.23. And this is the current price of the bond. And if we want to determine the value of the bond after the one year or the before two years of the bond, then we have the value of bond at $982.17. So we have two points price at current market level, which is the price not or the P0 price after year one, which is P1. To determine the capital gain, we need to deduct the current price or the P0 from P1 and that capital gain we have at $7.94. Now to determine a total rate of return on this bond, we need to add this capital gain of $9.74 to the annual coupon payment of $70. The sum of these two figures we will divide over the current price which is P0 and the resulting rate of return is the 8%. And that 8% is exactly equal to the market interest rate. Discount bonds provide capital gain through augmenting below market coupon rate to provide a fair total rate of return to the bondholders. But so far as the premium bonds are concerned, these bonds provide interest income greater than what is available in the market. So near maturity, the prices of premium bonds fall because of remaining few coupon payments which are unpaid. And second, the resulting capital loss offset the large coupon payments because the price of the bond is falling near the maturity. So the capital gains which are available in the initial life of the bonds are going on depletion and they are converting into the capital loss. But that capital loss is set off against the larger coupon payments made to the bondholders across the life of the bond. Bondholders receive only a competitive rate of return in this particular case of premium bonds. So how bonds price move over the timeline of the bonds life? We have price paths in this regard. These price paths of two bonds we have in the picture at the below of the screen. Each bond is selling at a YTM of 8%. For bond 1, we have coupon rate greater than 8% and that 8% coupon rate is depicted here with a thick below line. This bond is suffering from capital losses. For second bond which is denoted as B2, we have coupon rate less than 8% and that is enjoying capital gain because of price is steadily approaching to the power value. We see that bond price approaches power value as the maturity date approaches. So these are the path both the bonds are approaching to their power value. Each bond offers investors the same total rate of return. What is the relationship between yield to maturity at holding period return? We see that fluctuations in yields will change our bonds rate of return that we have already seen. An increase in the bonds yield will reduce its price which reduces the holding period return and the vice versa. For yield to maturity, this is the average return if the bond is held till the maturity of the bond. But holding period rate of return is the rate of return over the particular investment time period. YTM depends on coupon rate, current price and its power value at maturity. But holding period returns depend on the bond's price at the end of the holding period. And that bond price is in fact an unknown future value. For yield to maturity, all of the values like current prices, coupon rate and power value at maturity are readily available in the market. But for holding period rate of return such values can only be forecasted. Now come to zero coupon bonds. Zero coupon bond carry no interest at all means these are offering no interest payments to their holders. These zero coupon bonds provide all of their return in the form of price appreciation. These zero bonds provide only one cash flow to their bond holders at the maturity of the bond. The example is the US Treasury bills. That is the example in the universe of zero coupon bonds. Treasury steps where these are long term zero coupon bonds are commonly quoted from the coupon bearing notes and the bond. Each coupon bond may be stripped off its semi-annual coupons. Then each coupon payment would be treated as a standalone zero coupon bond. The final repayment of principal on a zero coupon bond would be treated as another standalone zero coupon. The Treasury program under which coupon stripping is performed is called as STRIPS. And these zero coupon securities are also called as Treasury STRIPS. There is a question that what should happen to prices of zero coupon bonds as the time passes away. We see that zero coupons must be sold for power value on their maturity. These bonds may be sold at a discount before maturity due to the effect of time value of money. And as the time passes on, the prices should approach power value on these bonds. And that we see here in the example when time moves from current to the maturity, the price of the bond goes on their power value. To understand this in further, we have an example where we have a zero coupon bond with maturity life of 30 years. Market interest rate is 10% and at current level, the price of the bond is 57.31 dollars. And after the one year means for the remaining 29 years, the price of the bond is 63.04. So we see that the power value which we discount here that is discounted for less year because its price increased by one year discount factor. And that increase in the price level is exponential that we see in the graph. This is not linearly until its maturity.