What are the total returns generated by a Bond and how to accurately measure Bond Yield?
When a new Bond is being issued to the public, a Coupon Rate is pre-defined by the Bond Issuer. But this interest rate should not be mistaken as the actual investment returns that the investor will be making on the Bond. This is because the generated returns will depend on many other factors like Coupon Frequency, Coupon amount, Face Value of Bond, acquisition price of Bond etc.
It is important to calculate Bond Yield by considering all the factors, and to understand how the yield can vary in different situations. For example, the yield generated by the Bond can be very different if it is acquired at the Face Value of INR 1,000, and when it is acquired at a price of INR 950 from the market.
So, to make it easy for the investors to understand, we give basic details about how to calculate yield on a bond. On this page, we will not cover the situations where investors can make a profit through Capital Gains, due to the movement in the market price of the Bond.
How Coupons affect Bond Yield calculation?
In case of Bonds, the primary return that is generated for the investors is through Bond Coupons. These are basically the interest payments that the borrower (Bond issuer) makes to the investors, at regular intervals. The principal amount in Bonds is usually returned at the maturity time. (Refer: What is Coupon frequency in bonds?)
The Coupon Rate of the Bond is fixed at the issuance time itself, and it usually does not change during the lifecycle of the Bond. This defined rate and the Coupon payment frequency can be used to calculate the exact amount that will be received by the investors at different time intervals.
Impact of Compounding
The concept of compounding has the potential to generate additional returns for the investors. This is because the Coupon amount received on a particular date, can be reinvested for the remaining duration of the Bond.
Example: Suppose that a Bond has a Face Value of INR 1,000, a Coupon Rate of 9%, and a borrowing duration of 5 years. The INR 90 Coupon received at the end of first year, can be reinvested for the remaining 4 years. At the end of second year, INR 98.10 will be received as the interest for the total invested amount. This can then be reinvested for the remaining 3 years. Etc.
So, at the end of 5 years, the total returns generated through this Bond investment will be higher than the defined Coupon Rate of 9%. These actual returns are also known as the Effective Yield of the Bond.
Acquisition price of Bond
The Coupon Rate, Face Value, and the Coupon amount will usually be fixed in advance. Perhaps one of the most important elements in the calculation of Bond Yield is the price at which the Bond is acquired by the investor. Some common ways to acquire the Bonds are:
- Purchase from open market: Many Bonds trade in the open market, and they can also be purchased from the Stock Exchange, or Over the Counter (OTC). So, the purchase price will become the Acquisition price. However, the Coupon Rate and the Coupon payment schedule does not change for these Bonds.
- Subscription (First Come First Serve): In this case, the investors directly submit a request to the Bond issuer, during a pre-defined subscription window. The investors usually receive the Bonds at the Face Value (or the fixed value) that is defined at the issuance time.
- Subscription (Competitive Bidding): Here, the investors have to Bid for a certain quantity, and define the price that they are willing to pay. Depending on the supply and demand, the Bond issuer chooses the Bids that will be accepted. After that, the Bonds will be issued at the bid price, that was mentioned by the corresponding investor.
Quick indication of Bond Yield
In order to calculate yield of Bond, the first step is to decide whether we want to apply the assumption of compounding of interest/Coupons. The Coupon Rate and the current market price of the Bond can give a high-level indication of the yield of the Bond. Let us look at some cases below, to understand how this can be determined.
Case 1: Market Price = Face Value (At Par)
If the Bond is purchased at Face Value and there is no Compounding/Reinvestment, then the investment returns for the investor will be equal to the Coupon Rate.
If there is Compounding of interest, then the returns will be more than the Coupon Rate.
Case 2: Market Price > Face Value (Above Par)
If the Bond is purchased above the Face Value and there is no Compounding/Reinvestment, then the Bond Yield will be lower than the Coupon Rate.
But if there is Compounding of interest, then the returns could be lower than / higher than / or equal to the Coupon Rate. This would largely depend on the price difference, and deeper analysis is needed to calculate Bond Yield in this case.
Case 3: Market Price < Face Value (Below Par)
If the Bond is purchased below the Face Value, then the Bond Yield will always be higher than the Coupon Rate. In addition, the investor will also get a Capital Gain on Maturity, because the Bond will pay the Face Value at Redemption time.
Different techniques to calculate Bond Yield
The above-mentioned method helps to quickly get a rough indication about the investment returns, without actually doing the calculations of Bond Yield. However, there are more precise ways to get an idea about the Yield that will be generated.
Also, there are primarily 2 ways or purposes for approaching the Bond Yield calculations:
- The acquisition price of the Bond is known, and the investor wants to calculate Bond Yield that will be generated by the investment. This situation usually happens when an investor is acquiring the Bonds from open market.
- The investor is targeting a particular investment yield from the Bond, and he/she wants to calculate an acquisition price at which the particular Bond should be purchased.
Let us now look at some techniques below, that will help to more accurately calculate Bond Yield.
Current Yield
This is one of the fastest ways to calculate yield of a Bond. The Current Yield basically gives a rough idea about the investment returns that will be generated, depending on the current market price of the Bond.
So, it will give a more precise yield, compared to the defined Coupon Rate. However, the Current Yield is not an accurate measurement of the investment returns. This is because the Current Yield will give equal importance to the Coupon received in the first year of the Bond, and 9th year of the Bond (for example). So, it does not respect the basic concept of Time Value of Money, because both such Coupons are given equal weightage.
Also, if we are calculating Bond Yield in this way, then we do not assume that the received interest / Coupon will be reinvested. So, Compounding of Interest is not applicable in this case. The following formula can be used to calculate Current Bond Yield.
\( CY = \frac{C}{MP}*100\)Where,
\(C\) : Annual Coupon amount paid by the bond
\(MP\) : Market Price of the bond
Example: Suppose that a Bond pays an annual Coupon of INR 7, and it is purchased at a price of INR 95 from the open market. So, the Current Yield of this Bond will be:
INR 7 / INR 95 = 0.0737 or 7.37%
Yield to Maturity (YTM)
The concept of Yield to Maturity is perhaps the most precise method to measure the investment returns that will be generated by the Bond. As the name suggests, it gives the annual rate of return that an investor can hope to achieve, by holding the Bond till maturity date.
On a high-level, this method works by first predicting all the Cash Flows of the Bond, and then Discounting each of them to the current date. The below-mentioned formula can be used to calculate Bond Yield to Maturity.
\(P(0, T) = \sum\limits_{n=1}^{T} (c_n \times Z(0, n))\)Where,
\(P(0, T)\): Price of the bond at time 0, with maturity at time T
\(C_n\): Cash flow at time ‘n’
\(Z(0,n)\): Discount Factor, for discounting the \(C_n\) (cash flow at time ‘n’) to time 0
The Discount Factor depends on the compounding frequency, which in this case is the frequency of interest payments. For example, if we assume annual compounding / annual Coupon, then the Discount Factor will be given by the below formula,
\(Z(0, n) = \frac{1}{(1 + \text{YTM})^n}\)Where,
\(YTM:\) Yield to Maturity of the Bond
The above formula can be used in 2 ways:
- If we know the Yield/YTM, the formula could be used to get the Fair Value of the Bond on the current date: \(P(0, T)\)
- Whereas if we know the current market price of the bond: \(P(0, T)\), then we can solve the above equations to calculate the Yield to Maturity (YTM).
Sample Calculation of Yield to Maturity
Suppose that a Sovereign Bond has been issued with the below-mentioned borrowing terms:
Tenure: 2 years
Face Value: INR 1,000
Coupon: 10%, paid semi-annually
Let us assume that the current trading price of the Bond is INR 960, and we want to calculate Bond Yield to Maturity, if the bond is purchased at this price.
The cash flows for this Sovereign Bond will be: INR 50 after 6 months, INR 50 after 1 year, INR 50 after 18 months, and INR 1,050 after 2 years. Assuming semi-annual compounding, we would need to solve the below equation to find the annualized Yield to Maturity (\(Y\) in the below formula).
\(960 = \frac{50}{\left(1 + \frac{Y}{2}\right)} + \frac{50}{\left(1 + \frac{Y}{2}\right)^2} + \frac{50}{\left(1 + \frac{Y}{2}\right)^3} + \frac{1,050}{\left(1 + \frac{Y}{2}\right)^4}\)
To measure Bond Yield from the above equation, the investors would need to use an advanced calculator, or use tools like ‘Solver’ feature in Microsoft Excel (Refer: How to use Solver in Excel?).
Solving the equation gives us a YTM of ~ 0.12317, which indicates ~ 12.32% return. In simple words, this means that if an investor buys the above Sovereign Bond at INR 960 and holds it till maturity date, then he/she will get a yield of ~ 12.32% on the invested amount.
Calculate Yield for Zero Coupon Bonds
The concepts mentioned above are primarily used to calculate Bond yield for Coupon paying Bonds. But for very short-term Bonds and Zero-Coupon Bonds, there is usually no coupon payment made. Instead, there is a single repayment made at maturity time. So, we need to use a different approach to measure yield of bond in this case.
For Zero Coupon Bonds, a pre-defined Face Value is paid on the Maturity Date. To generate returns for the investors, these Bonds are issued at a Discounted price from the Face Value. So, the investors will make a Capital Gain, due to the appreciation in value of the Bond.
The below-mentioned formula can be used to calculate yield to maturity of a bond with Zero Coupons. (Assuming annual compounding. Formula will be different if the compounding frequency is changed.)
\(\text{YTM} = \left(\sqrt[t]{\frac{\text{FV}}{\text{PV}}}\right) – 1\)Where,
\(t\): the duration of the investment (In years. Since we are calculating annual yield, and assuming annual compounding)
\(FV\): Face Value or Maturity price
\(PV\): Present Value or Initial Value or Purchase Price
Sample Calculation
Suppose that a company is offering a Zero-Coupon Bond with the below borrowing terms:
Tenure: 6 years
Face value: INR 1,000
Issue price: INR 600
Then, we can calculate Zero Coupon Bond Yield as follows:
\(\text{YTM} = \left(\sqrt[6]{\frac{1000}{600}}\right) – 1\)This gives us a Yield to Maturity for this Zero Coupon Bond as ~ 0.0889, which means ~ 8.89% compounded annual return.
Impact of taxation on Bond Yield calculation
The basic parameters of the Bond like Coupon Rate, Coupon frequency, Face Value etc. can only help in getting the direct yield from the investment. However, another big factor at play is the taxation of the interest amount that is received from the Bond.
The applicable tax laws and tax rates will obviously reduce the final returns that will be generated from the Bond. So, the calculated Bond yield will have to be adjusted accordingly, as per the applicable taxes in the region.
This will help in getting a more accurate picture of the final investment returns, because different financial instruments like Stocks, Bonds, Fixed Deposits, Mutual Funds etc. might have different tax rates/rules. (For more details, refer: How to compare yield of tax free bonds and Taxable bonds?)
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