Yield on the bond is not easy to find as there is no YTM formula available to calculate the yield to maturity. The difficulty in finding the yield on the bond is attributed to the fact that interest rate variable in the bond price variable is not separable from the bond value equation. This requires us to make use of iterative methods to solve the price equation to find the YTM. Here, me Abraham A. will take you on a guided tour to define, explain, and illustrate YTM calculation by using Newton-Raphson method to solve the bond price equation.
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Yield to maturity YTM Formula is used for YTM calculation with Newton Raphson Method illustrated with example calculation.
YTM or yield to maturity is akin to IRR ( internal rate of return ) for bonds. At an rate of return equal to YTM, the present value of interest payments from a bond plus bond's discounted terminal value equals the market price of the bond. Thus YTM provides a bond investor with a rate that she could use in determining whether to invest in the bond. Usually YTM is the same as the coupon rate when the bond is sold at par. Yet if the risks of defualt are high as was the case with recent bond issues by the Irish Goverment then investors would demand a YTM or yield much higher than the interest rate on bond's coupons thus the bond will sell at discount.
We can use TVM equation that is used by MS Excel to find YTM or yield to maturity. This particular equation is a general purpose equation that helps in calculating present value, future value, number of periods, the payment and rate of return. The TVM equation is listed below:
FV + PV (1 + RATE)NPER + PMT [(1 + RATE)NPER -1 ]/RATE = 0
However for our purposes of calculating we will replace some of it's variable names with those associated with bond price calculations
Suppose we have a 4% $1000 bond that pays interest twice a year (semi annual compounding of interest), with 10 years to maturity. Our task is to find market rate of return or the YTM which is yield to maturity. We will get started by first setting up a function of x. The f(x) will be derived from a TVM equation most often used by MS Excel. Then we will find the derivative of f(x) such as f'(x). Once we have the function, we will use the Newton Raphson method to find YTM. This method required starting with a guess rate for YTM such as 10% or 0.10. We then use iterative calculations to check whether the results of the calculation converage to a particular value. If the results do converge, we assume that we have found the rate of return for bond. Yet if the values fail to converage we halt the procedure and restart it using a new seed value. The following step by step illustration highlight Newton Raphson method in finding yield to maturity for our example bond data.
f(x) = 1000 + -900 * (1+x)^20 + 20 [(1+x)^20 - 1]/x
f'(x) = 20 * -900 * (1+x)^19 + 20 * (20 x (1 + x)^19 - (1 + x)^20 + 1) / (x^2)
x = 0.1
f(x) = -3909.25
f'(x) = -97077.7265
x1 = 0.1 - -3909.25/-97077.7265 = 0.0597307219235
Error Bound = 0.0597307219235 - 0.1 = 0.040269 > 0.000001
x1 = 0.0597307219235
f(x1) = -1138.2065
f'(x1) = -46316.0501
x2 = 0.0597307219235 - -1138.2065/-46316.0501 = 0.0351559479556
Error Bound = 0.0351559479556 - 0.0597307219235 = 0.024575 > 0.000001
x2 = 0.0351559479556
f(x2) = -229.7135
f'(x2) = -28881.4785
x3 = 0.0351559479556 - -229.7135/-28881.4785 = 0.0272022873582
Error Bound = 0.0272022873582 - 0.0351559479556 = 0.007954 > 0.000001
x3 = 0.0272022873582
f(x3) = -17.0652
f'(x3) = -24690.6388
x4 = 0.0272022873582 - -17.0652/-24690.6388 = 0.0265111277256
Error Bound = 0.0265111277256 - 0.0272022873582 = 0.000691 > 0.000001
x4 = 0.0265111277256
f(x4) = -0.1165
f'(x4) = -24354.1518
x5 = 0.0265111277256 - -0.1165/-24354.1518 = 0.026506343064
Error Bound = 0.026506343064 - 0.0265111277256 = 5.0E-6 > 0.000001
x5 = 0.026506343064
f(x5) = -0
f'(x5) = -24351.8371
x6 = 0.026506343064 - -0/-24351.8371 = 0.0265063428366
Error Bound = 0.0265063428366 - 0.026506343064 = 0 < 0.000001
YTM = 2.65%
Annual YTM = 5.3%
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