 Although the bonds may not be sold at par, but while avoiding the bankruptcy, the bonds basically at maturity goes to the par value. This means that bonds are matured at their par value. Now to account for the current income and the coupon interest payments occurred over the whole life of the bond. We need a certain rate of return that can accommodate these returns. And to measure this rate of return, we have a standard measure and that measure is termed as the yield to maturity or the YTM. By bond yield, we mean a measure of rate of return that account for both current income and the changes in the bond prices over the entire life of the bond till its maturity. And by yield to maturity, we mean the interest rate that makes the present value of a bond payment equal to the price of the bond. In yield to maturity, this is the average rate of return that is earned on a bond if it is bought now and it is held till the maturity. Yeet to maturity is computed by solving the bond price equation for the interest rate given the current market price. If we see the bond valuation formula, there is a discount rate that is denoted as R. So if we have all the values in the equation, we accept the value of R. So while solving the equation with the given values, we can determine the R and that R then is termed as the yield to maturity. We have an example, the yield to maturity of an 8% 30 years bond with power value of $1000 selling at $1276 would be what? So we put these given values in the bond valuation formula and solving for the R which is missing the YTM we get is 0.03 or 6% annually. If we see this, we can see that the YTM is basically the IRR on the investment made in this particular bond. YTM assumes that all bonds coupon payments are reinvested at the discount rate or the yield to maturity rate. By current yield, we means the bond's annual interest payments divided by its current price. For example, we have an 8% 30 years bond which is currently selling at $1276. We have this current price as to determine current yield, we will divide the annual coupon of $80 over this current price. The resulting figure of 6.27% will be termed as the current yield that is an earning that is earning an investor or the bond holder. What is the relationship between bond prices and the YTM? If the YTM is lesser than the coupon interest rate, this means that bond is selling at the premium above the power value, then this bond will be termed as the premium bond. And if YTM is equal to the coupon interest rate of the bond, then that is a power bond or means that bond is selling at its power value or the face value. But if the YTM is greater than the coupon interest rate of the bond, then the bond is selling at the discount. This means that that particular bond is a discount bond. By yield to call, we mean the yield on a bond if it is bought today and it is held till the call made by the company prior to the date of the bond's maturity. What is callable and non-callable bonds and the relation between these two? If interest rate fall, the price of a non-callable bond can rise as that we can see in our diagram in the right below panel where we see that the interest rate falling from right to left, the price of the bond is going upward and that is shown in the left upper quarter of the picture highlighted by below line. The price of a callable bond is flat over a range of low interest rates because of higher risk of repurchases. What is the relationship between callable and non-callable or the straight bonds? For straight bonds, if you see that there are falling interest rates, then the price of such straight bonds can rise and that we can say in the picture here the interest rates are falling from 13 percent towards the 3 percent. The price is going from 200 to 2000 and that is the line that shows the increase in the prices of the straight bond and the price of a callable bond is flat over a range of lower interest rates because of higher risk of repurchase because in these circumstances the issuer thinks it suitable to repurchase back the bond from the bond holder and that flat line we see here in this picture. But if the interest rates are higher, the risk of call is negligible and the values of both the straight and callable bonds converge with each other and that is the conversion area we see because the interest rates are getting higher for both the bonds. So these are converging to a similar level of prices. But at the lower rate of interest, the values of bond begin to diverge and that we see when the interest rates fall from 13 percent towards the 3 percent and that is the diversion area. This diversion is with the difference reflecting the value of the firm's option to reclaim the callable bonds at the call price. So if we conclude this discussion, we can see that the bond market analysts might be more interested in a bond's yield to call rather than the yield to maturity particularly if the bond is likely to be called by the issuer. The realized compound return will be equal to YTM if the reinvestment rate is same as the yield to maturity but there is a problem in this particular situation because the conventional yield to maturity will not equal to the realized compound return because the realized compound return cannot be computed in advance without a forecast of future interest rates or the reinvestment rates. To signify this issue, we have an example where we assume that reinvestment rates are equal to the yield of maturity. We have an example where a two years bond selling at power value paying a 10 percent coupon once a year, the YTM is 10 percent and if the hundred dollars coupon payment is reinvested at an interest rate of 10 percent, the $1,000 investment in the bond will grow after two years to $1,210 and there we work here. For first year the investment value is 1100 and for the second year we have an interest value of 110 so we have a two years bond price at 1210 and if we equate these two values means the value of two years bond equal to the value at 0 which is grown at the reinvestment rate to determine the compound rate of return. We have the unknown value of R and solving this equation for R we get a rate of 10 percent and that 10 percent is equal to the rate of YTM which is again the 10 percent. But if the reinvestment rate is below the yield to maturity, we have another example with little difference. We have a two years bond selling at power value paying a 10 percent coupon once in the year and the YTM we assume is 10 percent if the hundred dollars coupon payment is reinvested at a reinvestment rate of 8 percent, the $1,000 reinvestment in the bond will grow after two years to $1,208 to determine this we have computation here where we have 1100 value at the end of the year 1 then we grow this to 8 percent and the second year value we have $1,208 if we determine the value of R as the reinvestment rate we solve this equation for R and that is here at 9.91 percent which is less than the yield to maturity which is 10 percent. One analysis basically is the forecasting realized compound yield over various holding periods or the investment horizons. The forecast of total return depends on two things. The first is a bond forecast sale price at the end of the bond's life and second is the reinvestment rate at which the coupon are reinvested. The sale price again depends in turn on the YTM at the horizon date.