What's a try (trial?) and error bond?
hi how do i work out trial and error method , the ytm of this 4.5 year bond
for a semi annual coupon bond
price is 104.5 the face value is 100
year---discount factor d(t) --------semi an coupon rate z(t) --------ytm
.6 ----------09999 ----------------------.001700 ----------------- .0017
1 -----------.9884 ----------------------.002400 -------------------.0027
1.5 ---------.9766 ----------------------.004200 ------------------.0042
2 ----------.9696 -------------------- .005400 --------------------. 0053
2.5 ----------.9500 --------------------.006896 -------------------.0069
3 ----------- .9456 ---------------------- .008900--------------------.0090
3.5 ----------.8907 ---------------------- .0104 --------------------.0105
4 ------------ .8806 ------------------- .0113 -----------------------.0110
For me to find YTM with trial and error you need to provide the coupon rate, I used a 5% coupon rate and plugged the values in the online YTM Calculation tool at YTM|Yield till Maturity calculation - ThinkandDone.com that provided the following yield calculations
Bond Price at Lower YTM of 1.41%
Compounding = semi-annual
Par Value = 100
Market Price = 104.50
Coupon Rate = 5%
YTM Low = 1.41%
N = 4.5
Non Zero Bond Price Formula
Coupon Rate x Par Value x PVIFA(ytml%, n) + Par Value x PVIF(ytml%, n)
PVIFA Formula
PVIFA(ytml%, n) = [1 - v] / ytml%
v = 1 / (1 + ytml%)^n
PVIFA(ytml%, n) = [1 - { 1 / (1 + ytml%)^n }] / ytml%
PVIFA Calculation
v = 1 / (1+0.014)^4.5
v = 0.93935381283725
PVIFA(0.014, 4.5) = [1 - 0.93935381283725] / 0.014
PVIFA(0.014, 4.5) = 0.060646187162752 / 0.014
PVIFA(0.014, 4.5) = 4.3318705116251
PVIF Formula
PVIF(ytml%, n) = 1 / (1 + ytml%)^n
PVIF Calculation
PVIF(0.014, 4.5) = 1 / (1+0.014)^4.5
PVIF(0.014, 4.5) = 1 / 1.0645616021716
PVIF(0.014, 4.5) = 0.93935381283725
Non Zero Bond Price Calculation
Price = 0.05 x 100 x 4.3318705116251 + 100 x 0.93935381283725
Price = 21.659352558126 + 93.935381283725
Price @ 0.014 = 115.59
Bond Price at Upper YTM of 6.41%
Compounding = semi-annual
Par Value = 100
Market Price = 104.50
Coupon Rate = 5%
YTM High = 6.41%
N = 4.5
Non Zero Bond Price Formula
Coupon Rate x Par Value x PVIFA(ytmh%, n) + Par Value x PVIF(ytmh%, n)
PVIFA Formula
PVIFA(ytmh%, n) = [1 - v] / ytmh%
v = 1 / (1 + ytmh%)^n
PVIFA(ytmh%, n) = [1 - { 1 / (1 + ytmh%)^n }] / ytmh%
PVIFA Calculation
v = 1 / (1+0.064)^4.5
v = 0.75641942691628
PVIFA(0.064, 4.5) = [1 - 0.75641942691628] / 0.064
PVIFA(0.064, 4.5) = 0.24358057308372 / 0.064
PVIFA(0.064, 4.5) = 3.8059464544331
PVIF Formula
PVIF(ytmh%, n) = 1 / (1 + ytmh%)^n
PVIF Calculation
PVIF(0.064, 4.5) = 1 / (1+0.064)^4.5
PVIF(0.064, 4.5) = 1 / 1.3220178705308
PVIF(0.064, 4.5) = 0.75641942691628
Non Zero Bond Price Calculation
Price = 0.05 x 100 x 3.8059464544331 + 100 x 0.75641942691628
Price = 19.029732272166 + 75.641942691628
Price @ 0.064 = 94.67
YTM Approximation with Linear Interpolation
ytmL = 1.41%
ytmU = 6.41%
pvL = 115.59
pvU = 94.67
ytm = ytmL + [(ytmH-ytmL)(pvL-pvYTM)] / [pvL-pvU]
ytm = 0.014 + [(0.064-0.014)(115.59 - 104.50)] / [115.59-94.67]
ytm = 0.014 + [(0.05)(11.09)] / [20.92]
ytm = 0.014 + 0.5545 / 20.92
ytm = 0.014 + 0.0265
ytm = 0.0405
ytm = 4.05%
@Wilmer
YTM on bonds is akin to internal rate of return at which present value of interest payments and terminal value of bond equals the market price of the bond. We use the Bond Price equation for calculating YTM with Trial and Error method, We select two rates one at which present value of bond is higher than market price of the bond and the second rate at which present value of bond is less than market price of the bond. We then use linear interpolation to arrive at an approximate value of YTM. See the following links
What is YTM, Yield till maturity it's Definition, Formula, an Example Calculation, Linear Interpolation
YTM|Yield till Maturity calculation - ThinkandDone.com
YTM|Yield till Maturity calculator - ThinkandDone.com
We can also use the RATE formula in Excel to find YTM as follows
=RATE(NPER, PMT, PV, FV, type)
here NPER will be number of periods in this case 9 (4.5 x 2)
PMT is the interest payment if coupon rate was 5% then PMT will be 5
PV is the market price -104.50
FV is the terminal value 100
Type is 0 for ordinary annuity (end of period payments)