On this web page we look at the definition of yield till maturity on bonds, formula for YTM, and an example with YTM calculation using Linear interpolation, and a neat Yield to Maturity Calculator.
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How do you define YTM ?
The market required rate of return on a bond (kd) is more commonly referred to as the bond's yield to maturity. Yield to maturity (YTM) is the expected rate of return on a bond if bought at its current market price and held to maturity; it is also known as the bond's internal rate of return (IRR) . Mathematically, it is the discount rate that equates the present value of all expected interest payments and the payment of the principal (face value) at maturity with the bond's current market price.
What is the YTM Equation or YTM Formula
If we now substitute actual values for I, MV, and P0, we can solve for kd, which in this case would be the bond's yield to maturity. However precise calculation for yield to maturity is rather complex and requires bond value tables, or a sophisticated handheld calculator, or a computer. Linear Interpolation: We can use a trial and error procedure to approximate the yield to maturity.
Illustration with YTM Example
To illustrate this, say we have a $1,000 par-value bond with the following characteristics: a current market price of $761, 12 years until maturity, and an 8% coupon rate (with interest paid annually). We want to determine the discount rate that sets the present value of the bond's expected future cash flow stream equal to bond's current market price.
YTM Calculation
Suppose we start with a 10% discount rate and calculate the present value of the bond's expected future cash flows.
Bond Price @ 10%
| Interest | PVIFA @ 10% | Present Value | |
| 80 | 6.814 | $545.12 | |
| Market Value | PVIF @ 10% | Present Value | |
| $1,000 | 0.31863 | $318.63 | |
| Bond Price | $863.75 | ||
A 10% discount rate produces a resulting present value for the bond that is greater than the current market price of $761. Therefore we need to try a higher discount rate to handicap the future cash flows further and drive their present value down to $761. Let's try a 15% discount rate.
Bond Price @ 15%
| Interest | PVIFA @ 15% | Present Value | |
| 80 | 5.421 | $433.68 | |
| Market Value | PVIF @ 15% | Present Value | |
| $1,000 | 0.186907 | $186.91 | |
| Bond Price | $620.59 | ||
A 15% discount rate produces a resulting present value for the bond that is lesser than the current market price of $761. The rate necessary to discount the bond's expected cash flows to $761 must fall between 10% and 15%. To approximate the rate, we interpolate between 10% and 15% as follows Interpolated discount rate
This is an approximation , the use of a computer provides a precise yield to maturity of 11.82 percent. It is important to keep in mind that interpolation gives only an approximation of the exact percentage
User submitted YTM Questions/Problems
Asked:
What is the YTM for a 5 year, 8% bond(interest is paid annually) that is trading in the market for Rs. 924.2? Kindly tell me how can I easily solve for YTM in a step by step manner.
Replied:
Hi Vikas Gupta
Download MS Excel Worksheet to find YTM with numerical method and MS Excel RATE function
Let me assume that the Maturity Value of the Bond is Rs. 1000.
There is no direct way to compute YTM, instead we resort to using linear interpolation to approximate the actual YTM
It starts off with an initial guess of the YTM at which we check to see if the computed price of the bond is either above or below the Market Price of the bond. If the calculated price at this rate is higher than the market price, we select a second rate at which the calculated price comes below the market price and vice verca
P = INTEREST x PVIFA(i%, n) + Maturity Value x PVIF(i%,n)
PVIFA(i%, n) = [1 - { 1 / (1+i)^n }] / i
PVIF(i%, n) = 1 / (1+i)^n
INTEREST = Coupon Rate x Maturity Value
Let's start off with a rate of 5%
| INTEREST | PVIFA(5%, 5 Years) | Present Value |
| 80 | 4.329476671 | Rs. 346.36 |
| Maturity Value | PVIF(5%, 5 Years) | Present Value |
| 1000 | 0.783526166 | Rs. 783.53 |
| Bonds Calculated Price | Rs. 1,129.88 |
At 5% , the calculated bond price is Rs. 1,129.88 which is higher than bond's current market price of Rs. 924.20 Thus we select a higher rate of 15%
| INTEREST | PVIFA(15%, 5 Years) | Present Value |
| 80 | 3.352155098 | Rs. 268.17 |
| Maturity Value | PVIF(15%, 5 Years) | Present Value |
| 1000 | 0.497176735 | Rs. 497.18 |
| Bonds Calculated Price | Rs. 765.35 |
At 15% , the calculated bond price is Rs. 765.35 which is lower than bond's current market price of Rs. 924.20
Now we can use linear interpolation to approximate the YTM
P0 = iL + [(iH-iL)(pvL - pvYTM)] /(pvL - pvH)
P0 = 0.05 + [ (0.15 - 0.05) (1129.88 - 924.20) ] / (1129.88 - 765.35)
P0 = 0.05 + [ (0.10) (205.68) ] / (364.53)
P0 = 0.05 + [ 20.568 ] / (364.53)
P0 = 0.05 + 0.05642334
P0 = 0.10642334
The YTM comes out to approximately 10.64%
=RATE(5,80,-924.2,1000)
Asked:
Thatcher Corporation's bonds will mature in 10 years. The bonds have a face value of $1000 and an 8 percent coupon rate, paid semi-annualy. The price of the bonds is $1,100. The bonds are callable in 5 years at a call price of $1,050. What is the yield to maturity? What is the yield to call
Replied:
Hi Rizalyn Arda
YTC Calculation
Let's start off with a rate of 5%
| INTEREST | PVIFA(5%, 20) | Present Value |
| 40 | 12.46221034254 | $498.48 |
| Maturity Value | PVIF(5%, 20) | Present Value |
| 1000 | 0.376889482873 | $376.88 |
| Bonds Calculated Price | $875.36 |
At 5% , the calculated bond price is $875.36 which is lower than bond's current market price of $1,100 Thus we select a lower rate of 2.5%
| INTEREST | PVIFA(2.5%, 20) | Present Value |
| 40 | 15.589162285647 | $623.56 |
| Maturity Value | PVIF(2.5%, 20) | Present Value |
| 1000 | 0.6102709428588 | $610.27 |
| Bonds Calculated Price | $1,233.83 |
At 2.5% , the calculated bond price is $1,233.83 which is higher than bond's current market price of $1100
Now we can use linear interpolation to approximate the YTM
P0 = iL + [(iH-iL)(pvL - pvYTM)] /(pvL - pvH)
P0 = 0.025 + [ (0.05 - 0.025) ($1,233.83 - 1,100) ] / ($1,233.83 - $875.36)
P0 = 0.025 + [ (0.025) (133.83) ] / (358.47)
P0 = 0.025 + [ 3.34575 ] / (358.47)
P0 = 0.025 + 0.009333417022345 P0 = 0.0343
The YTM comes out to approximately 3.43%
=RATE(20,40,-1100,1000)
YTM=3.31%
YTC Calculation
Let's start off with a rate of 5%
| INTEREST | PVIFA(5%, 10) | Present Value |
| 40 | 7.7217349291848 | $308.87 |
| Maturity Value | PVIF(5%, 10) | Present Value |
| 1000 | 0.6139132535407 | $613.91 |
| Bonds Calculated Price | $922.78 |
At 5% , the calculated bond price is $922.78 which is lower than bond's current market price of $1,050 Thus we select a lower rate of 2.5%
| INTEREST | PVIFA(2.5%, 10) | Present Value |
| 40 | 8.7520639309709 | $350.08 |
| Maturity Value | PVIF(2.5%, 10) | Present Value |
| 1000 | 0.7811984017257 | $781.19 |
| Bonds Calculated Price | $1,131.27 |
At 2.5% , the calculated bond price is $1,131.27 which is higher than bond's current market price of $1050
Now we can use linear interpolation to approximate the YTC
P0 = iL + [(iH-iL)(pvL - pvYTM)] /(pvL - pvH)
P0 = 0.025 + [ (0.05 - 0.025) ($1131.27 - 1,050) ] / ($1131.27 - $922.78)
P0 = 0.025 + [ (0.025) (81.27) ] / (208.49)
P0 = 0.025 + [2.03175] / (208.49)
P0 = 0.025 + 0.009745071706077 P0 = 0.03474
The YTC comes out to approximately 3.47%
=RATE(10,40,-1050,1000)
YTC=3.40%
Asked:
Calculate YTM on
A 14% 8 year bond with a current price of 976
A 12% 5 year bond with a current price of $1054.
Replied:
Hi Said Hassan
A 14% 8 year bond with a current price of 976
YTM (annual compounding ) = 14.53%
YTM (semi annual compounding ) = 7.26%
A 12% 5 year bond with a current price of $1054
YTM (annual compounding ) = 10.56%
YTM (semi annual compounding ) = 5.29%
See the attached MS Excel Worksheet for usage of MS Excel RATE function to find YTM
YTM Calculation
Let's start off with a rate of 10%
| INTEREST | PVIFA(10%, 8) | Present Value |
| 140 | 5.3349261979027 | $746.88 |
| Maturity Value | PVIF(10%, 8) | Present Value |
| 1000 | 0.4665073802097 | $466.50 |
| Bonds Calculated Price | $1,213.38 |
At 10% , the calculated bond price is $1,213.38 which is greater than bond's current market price of $976 Thus we select a higher rate of 7.5%
| INTEREST | PVIFA(20%, 8) | Present Value |
| 140 | 3.8371598031931 | $537.20 |
| Maturity Value | PVIF(20%, 8) | Present Value |
| 1000 | 0.2325680393613 | $232.56 |
| Bonds Calculated Price | $769.76 |
At 20% , the calculated bond price is $769.76 which is less than bond's current market price of $976
P0 = iL + [(iH-iL)(pvL - pvYTM)] /(pvL - pvH)
P0 = 0.1 + [ (0.2 - 0.1) (1,213.38 - 976) ] / (1,213.38 - 769.76)
P0 = 0.1 + [ (0.1) (237.38) ] / (443.62)
P0 = 0.1 + [ 22.738 ] / (443.62)
P0 = 0.1 + 0.051256
P0 = 0.151256
The YTM comes out to approximately 15.13%
You can use MS Excel to arrive at an exact value for YTM of 14.53% by using this formula
=RATE(8,140,-976,1000)
YTM=14.53
Asked:
I need to calculate the IRR. With the following information:
Investment made corporate bond
Face value: 1,000,000 payable on maturity
fixed annual coupon rate: 5%
Corporate bond not trade in an active market.
Corporate bond has been trading at a discount due to chnages in the market rates since it was issued 5 years ago. Purchased bond for 750,000 with 40,000 transaction costs.
Replied:
Hi Gursh
IRR for Bond's is called YTM or Yield till Maturity
Couple of things, most US bonds have a semiannual coupon rate and 10 years to maturity
So I am assuming only 5 years are left for Bonds to mature
I used the RATE function MS Excel to compute the YTM on this Bond
For Annual Coupon rate the YTM is 11.92%
And for semiannual Coupon rate the YTM is 5.88%
We don't include the transaction costs in computing the Bond's YTM that is an expense and is not included in Market Price
For calculations, see the attached MS Excel WorkSheet, keep in mind that we can use manual calculations to approximate the YTM with linear interpolation but it is not a straight forward method but more like Trial and Error and is time consuming. If you wish for sample calculations with Trial and Error method let me know and I will create a manual or numerical solution
See the attached MS Excel Worksheet for usage of MS Excel RATE function to find YTM
YTM Calculation
Let's start off with a rate of 5%
| INTEREST | PVIFA(5%, 5) | Present Value |
| 50,000 | 4.3294766706308 | $216473 |
| Maturity Value | PVIF(5%, 5) | Present Value |
| 1,000,000 | 0.7835261664685 | $783526 |
| Bonds Calculated Price | $1,000,000 |
At 5% , the calculated bond price is $1,000,000 which is greater than bond's current market price of $750,000 Thus we select a higher rate of 12%
| INTEREST | PVIFA(12%, 5) | Present Value |
| 50,000 | 3.604776202345 | $180238 |
| Maturity Value | PVIF(12%, 5) | Present Value |
| 1,000,000 | 0.567426855719 | $567426 |
| Bonds Calculated Price | $747664 |
At 12% , the calculated bond price is $747,664 which is less than bond's current market price of $750,000
P0 = iL + [(iH-iL)(pvL - pvYTM)] /(pvL - pvH)
P0 = 0.05 + [ (0.12 - 0.05) ($1,000,000 - 750,000) ] / ($1,000,000 -
$747664)
P0 = 0.05 + [ (0.07) $250,000) ] / ($252,336)
P0 = 0.05 + [ 17500 ] / ($252,336)
P0 = 0.05 + 0.069352
P0 = 0.119352
The YTM comes out to approximately 11.94%
You can use MS Excel to arrive at an exact value for YTM of 11.92% by using this formula
=RATE(5,50000,-750000,1000000)
YTM=11.92%
Asked:
Ackerman Co. has 9 percent coupon bonds on the market with 9 years left to maturity. the bonds make annual payments. if the bond currently sells for $934. what is the ytm?
Replied:
Hi Ashley
The YTM is 10.15%
YTM Calculation
Let's start off with a rate of 5%
| INTEREST | PVIFA(5%, 9) | Present Value |
| 90 | 7.1078216756441 | $639.70 |
| Maturity Value | PVIF(5%, 9) | Present Value |
| 1000 | 0.6446089162177 | $644.60 |
| Bonds Calculated Price | $1,284.3 |
At 5% , the calculated bond price is $1,284.3 which is greater than bond's current market price of $934 Thus we select a higher rate of 7.5%
| INTEREST | PVIFA(15%, 9) | Present Value |
| 90 | 4.7715839197324 | $429.44 |
| Maturity Value | PVIF(15%, 9) | Present Value |
| 1000 | 0.2842624120401 | $284.26 |
| Bonds Calculated Price | $713.70 |
At 15% , the calculated bond price is $713.70 which is less than bond's current market price of $934
P0 = iL + [(iH-iL)(pvL - pvYTM)] /(pvL - pvH)
P0 = 0.05 + [ (0.15 - 0.05) (1,284.30 - 934) ] / (1,284.30 - 713.7)
P0 = 0.05 + [ (0.1) (350.3) ] / (570.6)
P0 = 0.05 + [ 35.03 ] / (570.6)
P0 = 0.05 + 0.0614
P0 = 0.1114
The YTM comes out to approximately 11.14%
You can use MS Excel to arrive at an exact value for YTM of 10.15% by using this formula
=RATE(9,90,-934,1000)
YTM=10.15
Asked:
A 10 year, 12% semi annual coupon bond, with a par value of RM 1,000, maybe called in 4 years at a call price of RM 1,060. The bond sells for RM 1,100. (Assume that the bond has just been issued).
(a) What is the bond\'s yield to maturity ?
(b) What\'s the bonds current yield ?
(c) What\'s the bond\'s capital gain or loss yield ?
(d) What is the bond\'s yield to call ?
(e) What are the four main types of bonds ? Explain the differences.
Replied:
Hi kawaii88
YTM Calculation
Let's start off with a rate of 5%
| INTEREST | PVIFA(5%, 20) | Present Value |
| 60 | 12.46221034254 | $747.73 |
| Maturity Value | PVIF(5%, 20) | Present Value |
| 1000 | 0.376889482873 | $376.89 |
| Bonds Calculated Price | $1,124.62 |
At 5% , the calculated bond price is $1,124.62 which is greater than bond's current market price of $1,100 Thus we select a higher rate of 7.5%
| INTEREST | PVIFA(7.5%, 20) | Present Value |
| 60 | 10.194491359192 | $611.67 |
| Maturity Value | PVIF(7.5%, 20) | Present Value |
| 1000 | 0.2354131480606 | $235.41 |
| Bonds Calculated Price | $847.08 |
At 15% , the calculated bond price is $847.08 which is less than bond's current market price of $1,100
P0 = iL + [(iH-iL)(pvL - pvYTM)] /(pvL - pvH)
P0 = 0.05 + [ (0.075 - 0.05) ($1,124.62 - 1,100) ] / ($1,124.62 - 847.08)
P0 = 0.05 + [ (0.025) (24.62) ] / (277.54)
P0 = 0.05 + [ 0.6155 ] / (277.54)
P0 = 0.05 + 0.002218
P0 = 0.05221
The YTM comes out to approximately 5.22%
You can use MS Excel to arrive at an exact value for YTM of 5.18% by using this formula
=RATE(20,60,-1100,1000)
YTM=5.18%
Asked:
1,000 coupon bond with 7% annual coupon rate and 20 years to maturity has a price of 1,249.2442.
I know the answer is 5. i just need to figure out how to get ther
Replied:
Hi morgan
There is no direct way to compute YTM, instead we resort to using linear interpolation to approximate the actual YTM
It starts off with an initial guess of the YTM at which we check to see if the computed price of the bond is either above or below the Market Price of the bond. If the calculated price at this rate is higher than the market price, we select a second rate at which the calculated price comes below the market price and vice verca
P = INTEREST x PVIFA(i%, n) + Maturity Value x PVIF(i%,n)
PVIFA(i%, n) = [1 - { 1 / (1+i)^n }] / i
PVIF(i%, n) = 1 / (1+i)^n
INTEREST = Coupon Rate x Maturity Value
YTM Calculation
Let's start off with a rate of 5%
| INTEREST | PVIFA(2.5%, 20) | Present Value |
| 70 | 15.58916229 | $1,091.24 |
| Maturity Value | PVIF(2.5%, 20) | Present Value |
| 1000 | 0.610270943 | $610.27 |
| Bonds Calculated Price | $1,701.51 |
At 2.5% , the calculated bond price is $1,701.51 which is greater than bond's current market price of $1249.2442 Thus we select a higher rate of 7.5%
| INTEREST | PVIFA(7.5%, 20) | Present Value |
| 70 | 10.19449136 | $713.61 |
| Maturity Value | PVIF(7.5%, 20) | Present Value |
| 1000 | 0.497176735 | $235.41 |
| Bonds Calculated Price | $949.03 |
At 7.5% , the calculated bond price is $949.03 which is less than bond's current market price of $1249.2442
P0 = iL + [(iH-iL)(pvL - pvYTM)] /(pvL - pvH)
P0 = 0.025 + [ (0.075 - 0.025) (1,701.51 - 1249.2442) ] / (1,701.51 - 949.03)
P0 = 0.025 + [ (0.05) (452.2658) ] / (752.48)
P0 = 0.025 + [ 22.61329 ] / (752.48)
P0 = 0.025 + 0.0300517
P0 = 0.0550517
The YTM comes out to approximately 5.5%
You can use MS Excel to arrive at an exact value for YTM of 5% by using this formula
=RATE(20,70,-1249.2442,1000)