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#### What Is Meant By The Term Yield To Maturity? Describe The Formula For Calculating It

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Yield to maturity or YTM is also known as the redemption yield. As the name suggests, if any investment is held till the maturity of the same, return ate that will be generated by it is the YTM. In other words, YTM is total return expected from any bond that is held till its maturity. It is considered as the long term yield from the bond but is expressed as the annual rate. YTM is the internal rate of return (IRR) of the investment in the bond if the bond is held by the investor till maturity, with all the payments made as per schedule and reinvested at same date. YTM is similar to the current yield that divides the annual cash inflows from the bond by the market price of the bond for determining how much money one will be able to make through buying and holding the same for 1 year. However, unlike the current yield, YTM considers the present value of the bond’s payments for future coupon. To be more specific, it factors in time value of the money whereas simple current yield computation does not. Likewise it is often considered as more thorough means for computing return from the bond. YTM of the discounted bond that does not pay the coupon is the good start for understanding some of the issues those is more complex with the coupon bonds.  YTM is useful in estimating whether purchase of a bond is good investment or not. An investor can determine the required yield that is the return on the bond that will make the bond worthwhile. Once the investor determines the bond’s YTM that is considered for buying, investor is in the position to compare YTM with required yield for determining if the bond will be good buy or not. As the YTM is expressed as the annual rate irrespective of the maturity term of the bond, it can be used for comparing the bonds those have different coupons and maturities since values of the different is expressed by YTM in same annual terms.  However, the major limitation associated with computation of YTM is that it generally does not consider the taxes payable by the investor on the bond. Apart from that YTM makes assumption regarding future that cannot be known in advance in actual.

Formula for computing the YTM for a discount bond is –

Where, n = number of years

Face value = maturity value of the bond or par value

Current price = price of the bond today

As the YTM s the rate of interest that will be earned by the investor through reinvesting each coupon payment from bond at the constant rate of interest until the maturity date of the bond, present value of all future cash flows is equal to the market price of the bond, its maturity value and coupon payments, but discount rate cannot be computed directly. However, trial and error method is there to find YTM with the present value formula as follows –

Bond Price =

Each one of future cash flows of bond is known and as the current price of the bond is known, trial and error process can be applied to variable of YTM in the equation until present value of payment stream is equal to the price of the bind.

Solving the equation through hand requires the understanding of relationship among the bond's price and the yield, in addition to different types of the bond pricings. Bonds can be priced at a discount, at par or at a premium. While any bond is priced at par, the rate of interest of the bond is equal to the coupon rate. If the bond priced above par, it is called as premium bond and it has the coupon rate higher as compared to the realized rate of interest rate. If the bond is priced below par, it is called as discount bond, has a coupon rate lower as compared to the realized rate of interest. If an investor computes YTM for the bond priced below par, he or she would solve the equation through plugging in various annual interest rates that were higher than the coupon rate until finding a bond price close to the price of the bond in question.

Computations of YTM assumes that all the coupon payments are reinvested at same rate as the current yield of the bond and considers the current market price of the bond, its coupon interest rate, par value, and terms to the maturity. YTM is just the snapshot of return from the bond as the coupon payments cannot be reinvested always at same rate of interest. As the rate of interest goes up YTM also goes up and as the rate of interest goes down the YTM will fall. The complex procedure for determining the YTM means that often it is difficult to compute the precise value of YTM. Rather one may approximate the YTM through using the table for bond yield, financial calculator or any other software. Although YTM represents the annualized return rate on the bond, payments on coupon are generally semiannually. Hence, YTM is computed on six-month basis as well.

Example – computation of YTM through trial and error method

Say an investor currently holds a bond whose par value is \$100. The bond is currently priced at a discount of \$95.92, matures in 30 months and pays a semi-annual coupon of 5%. Therefore, the current yield of the bond is (5% coupon x \$100 par value) / \$95.92 market price = 5.21%.

To compute the YTM here, cash flows shall be determined 1st. Every six months (semi-annually), the bondholder would receive a coupon payment of (5% x \$100)/2 = \$2.50. In total, he or she would receive five payments of \$2.50, in addition to the face value of the bond due at maturity, which is \$100. Next, we incorporate this data into the formula, which would look like this:

Now we shall solve for the rate of interest "YTM," which is where things get tough. Yet, we do not have to start simply guessing random numbers if we stop for a moment to consider the relationship between bond price and yield. As was mentioned above, when a bond is priced at a discount from par, its interest rate will be greater than the coupon rate. In this example, the par value of the bond is \$100, but it is priced below the par value at \$95.92, meaning that the bond is priced at a discount. As such, the annual interest rate we are seeking must necessarily be greater than the coupon rate of 5%.

With this information, we can calculate and test several bond prices by plugging various annual interest rates that are higher than 5% into the formula above. Using a few different interest rates above 5%, one would come up with the following bond prices:

 Annual % Rate Semi-annual % rate Bond price 10.0% 5.0% \$89 9.0% 4.5% \$91 8.0% 4.0% \$93 7.0% 3.5% \$95 6.0% 3.0% \$97

Increasing the interest rate up by one and two percentage points to 6% and 7% yields, bond prices becomes \$98 and \$95, respectively. Because the bond price in our example is \$95.92, the list indicates that the interest rate we are solving for is between 6% and 7%. Having determined the range of rates within which our interest rate lies, we can take a closer look and make another table showing the prices that YTM calculations produce with a series of interest rates increasing in increments of 0.1% instead of 1.0%. Using interest rates with smaller increments, our calculated bond prices are as follows:

 Annual % Rate Semi-annual % rate Bond price 7.0% 3.50% \$95.48 6.9% 3.45% \$95.70 6.8% 3.40% \$95.92 6.7% 3.35% \$96.15 6.6% 3.30% \$96.37

Here, it can be seen that the present value of our bond is equal to \$95.92 when the YTM is at 6.8%. Fortunately, 6.8% corresponds precisely to our bond price, so no further calculations are required. At this point, if we found that using the YTM of 6.8% in our calculations did not yield the exact bond price, we would have to continue our trials and test interest rates increasing in 0.01% increments.

It shall be clear regarding the fact that why most of the investors prefer using special programs for narrowing down possible YTMs rather than computing through trial and error as the computation required for determining YTM can be quite lengthy as well as time consuming.

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