## Formula to Calculate YTM

Yield to Maturity Formularefers to the formula that is used in order to calculate total return which is anticipated on the bond in case the same is held till its maturity and as per the formula Yield to Maturity is calculated by subtracting the present value of security from face value of security, divide them by number of years for maturity and add them with coupon payment and after that dividing the resultant with sum of present value of security and face value of security divided by 2.

Where,

- C is the Coupon.
- F is the Face Value of the bond.
- P is the current market price.
- n will be the years to maturity.

### Step by Step Calculation of Yield to Maturity (YTM)

**Step 1:**Gathered the information on the bond-like its face value, months remaining to mature, the current market price of the bond, the coupon rate of the bond.**Step 2:**Now calculate the annual income available on the bond which is mostly the coupon and it could be paid annually, semi-annually, quarterly, monthly, etc. and accordingly calculation should be made.**Step 3:**Also, one needs to amortize the discount or premium which is a difference between the face value of the bond and the current market price over the life of the bond.**Step 4:**The numerator of the YTM formula will be the sum of the amount calculated in step 2 and step 3.**Step 5:**The denominator of the YTM formula will be the average of the price and face value.**Step 6:**When one divides step 4 by step 5 value, it shall be the approximate yield on maturity.

### Examples

#### Example #1

**Assume that the price of the bond is $940 with the face value of bond $1000. The annual coupon rate is 8% with a maturity of 12 years. Based on this information, you are required to calculate the approximate yield to maturity.**

**Solution:**

Use the below-given data for calculation of yield to maturity.

We can use the above formula to calculate approximate yield to maturity.

Coupon on the bond will be $1,000 * 8% which is $80.

Yield to Maturity (Approx) = (80 + (1000 – 94) / 12 ) / ((1000 + 940) / 2)

Yield to Maturity will be –

**Yield to Maturity (Approx) = 8.76%**

This is an approximate yield on maturity which shall be 8.76%.

#### Example #2

**FANNIE MAE is one of the famous brands that are trading in the US market. The government of the US now wants to issue 20 year fixed semi-annually paying bond for their project. The price of the bond is $1,101.79 and the face value of the bond is $1,000. The coupon rate is 7.5% on the bond. Based on this information, you are required to calculate approximate yield to maturity on the bond.**

**Solution:**

Use the below-given data for calculation of yield to maturity.

Coupon on the bond will be $1,000 * 7.5% / 2 which is $37.50, since this pays semi-annually.

Yield to Maturity (Approx) = ( 37.50 + (1000 – 1101.79) / (20 * 2) )/ ((1000 + 1101.79) / 2)

Yield to Maturity will be –

** Yield to Maturity (Approx) = 3.33%**

This is an approximate yield on maturity which shall be 3.33% which is semiannual.

Annual Yield to Maturity will be –

Therefore, the annual Yield on maturity shall be 3.33% * 2 which shall be **6.65%.**

#### Example #3

**Mr. Rollins has received the lump sum amount in the form of the lottery. He is a risk-averse person and believes in low risk and high return. He approaches financial advisor and the advisor tells him that he is the wrong myth of low risk and high returns. Then Mr. Rollins accepts that he doesn’t like risk and low-risk investment with a low return will do. The advisor gives him two investment options and the details of them are below:**

**Both the coupons pay semi-annually. Now Mr. Rollins is perplexed which bond to select. He asks Advisor to invest in option 2 as the price of the bond is less and he is ready to sacrifice a 0.50% coupon. However, Advisor tells him instead to invest in option 1.**

**You are required to validate the advice made by the advisor.**

**Solution:**

**Option 1**

Coupon on the bond will be $1,000 * 9% / 2 which is $45, since this pays semi-annually.

Yield to Maturity (Approx) = (45 + (1000 – 1010) / (10 * 2)) / (( 1000 +1010 )/2)

Yield to Maturity will be –

**Yield to Maturity (Approx) = 4.43%**

This is an approximate yield on maturity which shall be 4.43% which is semiannual.

Annual Yield to Maturity will be –

Therefore, the annual Yield on maturity shall be 4.43% * 2 which shall be **8.86%.**

**Option 2**

Coupon on the bond will be $1,000 * 8.50% / 2 which is $42.5, since this pays semi-annually.

** **

Yield to Maturity (Approx) = (42.50 + (1000 – 988) /(10 * 2))/ (( 1000 +988 )/2)

Yield to Maturity will be –

**Yield to Maturity (Approx) = 4.34%**

This is an approximate yield on maturity which shall be 4.34% which is semiannual.

Annual Yield to Maturity will be –

Therefore, the annual Yield on maturity shall be 4.34% * 2 which shall be **8.67%.**

Since the yield on maturity is higher in option 2 hence the advisor is correct in recommending investing in option 2 for Mr. Rollins.

### Relevance and Uses

The approximate yield to maturity formula is almost similar to the current yield that divides cash flows which are coupons and amortize premiums or discounts by price of the bond so as to determine what is the return on the bond if the investor holds the bond for a year. Well, it only approximates the Yield to maturity and if one needs to calculate accurate yield to maturity then one needs to find IRR or the rate at which the coupon and the amortize values along with face value that equals to current bond market price which can be done using trial and error method.

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