Bond Formula Explained with Examples for Indian Investors

Bond formulas look scary at first glance, but each one answers a plain question. What is this bond worth today? What will I earn if I hold it? How much will the price move if interest rates change? Once you see the pattern, the symbols stop being a puzzle.

This article explains the main bond formulas with rupee examples from the Indian market. We will go one formula at a time, work through numbers by hand and show where a tool like the bond calculator can take over the heavy lifting.

The Symbols You Need to Know

Every bond formula uses the same few letters. Learn them once and the rest is easy. Here is the key to the symbols used below.

Symbol	Meaning	Example
F	Face value (repaid at maturity)	Rs 1,000
C	Coupon paid each period in rupees	Rs 90
y	Yield (discount rate) per period	8% = 0.08
n	Number of periods left	3 years
P	Price of the bond today	to be found

For half-yearly bonds, such as Indian government securities, divide the coupon and the yield by two and double n. A 3-year bond then has 6 periods.

The Bond Price Formula

The idea behind the formula is the time value of money. A rupee received three years from now is worth less than a rupee today, because you could earn interest on it in the meantime. So we shrink each future payment back to today’s value, and then add them up.

The formula is: P = C/(1+y) + C/(1+y)^2 + … + (C + F)/(1+y)^n. The shortcut for the same thing is P = C x [1 – (1+y)^-n] / y + F x (1+y)^-n. The first part is the value of the coupons, and the second part is the value of the face value you get at the end.

Worked Example 1: Price of a 9 Percent Bond When Yield Is 8 Percent

A company bond has a face value of Rs 1,000, a coupon of 9 percent (Rs 90 a year) and 3 years left. Similar bonds in the market now give 8 percent. What should this bond cost?

1. Annuity factor at 8 percent for 3 years: [1 – 1.08^-3] / 0.08 = 2.5771.
2. Present value of coupons: 90 x 2.5771 = Rs 231.94.
3. Discount factor for the face value: 1 / 1.08^3 = 0.79383.
4. Present value of face value: 1,000 x 0.79383 = Rs 793.83.
5. Price = 231.94 + 793.83 = Rs 1,025.77.

The bond sells at a premium because its coupon is higher than the market yield. Buyers are willing to pay about Rs 25.77 extra for the better interest.

Worked Example 2: The Same Bond When Yield Rises to 10 Percent

Now suppose the market yield jumps to 10 percent. The annuity factor becomes 2.4869 and the discount factor becomes 0.75131. Coupons are worth 90 x 2.4869 = Rs 223.82, and the face value is worth Rs 751.31. The price is Rs 975.13.

The same bond now trades at a discount. Nothing changed about the bond itself, only the market’s required return. This is exactly why bond funds can show losses when RBI raises rates.

Expert insight: If the yield equals the coupon rate, the bond prices at exactly face value. In our example, at a 9 percent yield the price is Rs 1,000. Premium and discount appear only when yield and coupon differ.

The Current Yield Formula

Current yield is the simplest formula: Current yield = Annual coupon / Market price. For the premium bond above, that is 90 / 1,025.77 = 8.77 percent. For the discount bond, it is 90 / 975.13 = 9.23 percent.

Current yield is easy to compute, but it ignores the gain or loss at maturity. It works best for long bonds, where that gain or loss is small per year. For short bonds it can mislead, so treat it as a first filter rather than a final answer.

The Yield to Maturity Formula

Yield to maturity (YTM) is the rate y that makes the price formula match the market price. There is no neat algebra to solve it directly, so people use trial and error, a financial calculator or software. The approximate shortcut is: YTM = [C + (F – P) / n] / [(F + P) / 2].

For the premium bond at Rs 1,025.77, the shortcut gives [90 + (1,000 – 1,025.77) / 3] / [(1,000 + 1,025.77) / 2] = 81.41 / 1,012.89 = 8.04 percent. That is very close to the true 8 percent. For the exact answer, a calculator tests many yields until the price matches.

Accrued Interest and Clean Price

On the secondary market, you rarely buy a bond right on a coupon date. The seller has earned interest from the last coupon date up to the settlement day, and you reimburse them. This is called accrued interest.

The formula is: Accrued interest = Face value x Coupon rate x Days since last coupon / Days in the year. The quoted market price, called the clean price, does not include it. The amount you pay, called the dirty price, equals clean price plus accrued interest.

For example, take a Rs 1,00,000 face value bond with a 9 percent coupon, 73 days after the last coupon date, on an actual 365-day basis. Accrued interest = 1,00,000 x 0.09 x 73 / 365 = Rs 1,800. If the clean price is Rs 1,02,000, you pay Rs 1,03,800. On your next coupon date, you receive the full coupon, so the Rs 1,800 comes back to you.

The Duration Formula

Duration tells you how sensitive a bond is to a change in interest rates. Macaulay duration is the weighted average time you wait to receive the bond’s cash flows, with weights equal to the present value of each payment. Modified duration equals Macaulay duration divided by (1 + y).

Worked Example 3: Duration of the 3-Year 9 Percent Bond

Use the earlier bond at an 8 percent yield, with price Rs 1,025.77. The present values of the cash flows are Rs 83.33 in year 1, Rs 77.16 in year 2 and Rs 865.28 in year 3 (coupon plus face value).

1. Multiply each present value by its year: 83.33 x 1 + 77.16 x 2 + 865.28 x 3 = 2,833.49.
2. Divide by the price: 2,833.49 / 1,025.77 = 2.76 years (Macaulay duration).
3. Modified duration = 2.76 / 1.08 = 2.56.

A modified duration of 2.56 means that if yield rises by 1 percentage point, the price falls by about 2.56 percent. Our earlier calculation agrees. At a 9 percent yield the price is Rs 1,000, a fall of 2.51 percent from Rs 1,025.77. Longer bonds have higher duration and swing more when RBI changes the repo rate.

How Maturity and Coupon Change the Numbers

A longer maturity means a larger duration. A lower coupon also means a larger duration, because more of your return comes late. Zero-coupon bonds, like Treasury bills in India, have a duration equal to their maturity.

This is why a 30-year G-Sec moves far more than a 2-year G-Sec when yields change. If you cannot stomach price swings, choose shorter maturities or plan to hold to maturity. You can model other long-term products with the compound interest calculator, which uses a related time-value idea.

Benefits of Understanding Bond Formulas

Knowing the formulas helps you read a bond screen like a pro. You can tell why a bond is priced above face value and whether the yield justifies the risk. You also learn why interest rate news matters to your portfolio.

• You can estimate price changes before RBI policy announcements.
• You can compare bonds of different maturities fairly.
• You can verify any calculator or broker figure for yourself.
• You understand why debt mutual funds rise or fall with yields.
• You can plan how long to hold a bond to match a goal.

Challenges and Limitations

These formulas assume fixed cash flows and no default. Real bonds may have call options, where the issuer repays early, or put options. Floating rate bonds change their coupon, so the simple price formula does not apply directly.

Duration is also only a linear estimate. For large yield moves it understates the price rise and overstates the fall. Tax is not part of the formulas either, so after-tax returns need a separate step, using the slab rules on incometax.gov.in.

Common Mistakes to Avoid

1. Using the annual yield with half-yearly periods. Halve the rate and double the periods, or the price will be wrong.
2. Mixing percent and decimals. Use 0.08 in the formula, not 8.
3. Forgetting the face value in the last period. The final payment is the coupon plus the face value.
4. Treating clean price as the cash you pay. Add accrued interest to find the real outflow.
5. Trusting the shortcut YTM for long bonds. It drifts from the true yield as maturity grows.
6. Assuming duration is exact. It is a first-order estimate that works best for small yield changes.

Best Practices

1. Write the cash flow timeline on paper before using any formula.
2. Check that the period of the rate matches the period of the payments.
3. Test your answer: at yield equal to coupon, price should equal face value.
4. Compute duration for every bond you plan to hold, not only the yield.
5. Use a calculator for exact YTM and keep the manual formula for a sanity check.
6. Confirm rating and issuer details from SEBI-registered rating agency reports.

Conclusion

Bond formulas are one idea repeated: shrink future rupees back to today and add them up. From that comes price, yield, accrued interest and duration. Each answers a question you will meet as an Indian bond investor.

Practise with one bond and the figures above, then let a calculator handle the repeat work. You will understand every number it shows.

FAQs

What is the formula for bond price?

Price equals the present value of all coupons plus the present value of the face value, each discounted at the yield. In short: P = C x [1 – (1+y)^-n] / y + F / (1+y)^n.

Why does bond price fall when yield rises?

Existing coupons stay fixed while new bonds offer higher rates. To match the new yield, the old bond must sell at a lower price. That is why rate hikes by RBI can hurt bond prices.

What is the difference between clean and dirty price?

Clean price is the quoted price without accrued interest. Dirty price is clean price plus accrued interest, and it is what you actually pay on the settlement date.

What does duration tell me?

Duration estimates how much a bond price changes for a 1 percentage point change in yield. A modified duration of 2.56 means a price drop of about 2.56 percent if yield rises by 1 percent.

Can I use the same formulas for G-Secs?

Yes. Indian G-Secs pay interest every six months, so use half the coupon, half the yield and double the number of years as periods.