# Calculation of YTM (Yield to Maturity)

## Calculation of YTM: Jaiib /DBF Paper 2 (Module A) Unit 2:

Dear bankers,

As we all know that  is YTM for JAIIB Exam. JAIIB exam conducted twice in a year. So, here we are providing the YTM (Unit-2), Business Mathematics and Finance (Module A), Accounting Finance for Bankers-Paper 2.

## ♦Meaning of Debt

• Debt means a sum of money due by one party to another. Most business need a mix of debt and equity to run their operations. This is called the capital structure of that firm/company.
• Debts can arise through bank borrowings, fixed deposits, bonds or other instruments. Where the amount is fixed and specific, and does not depend upon any future valuation to settle it.

## ♦Bonds

Debt Capital Consists of mainly bonds and debentures.

#### What are Bonds?

• Bonds are issued by organizations generally for a period of more than one year to raise money by borrowing.
• Organizations in order to raise capital issue bond to investors which is nothing but a financial contract, where the organization promises to pay the principal amount and interest (in the form of coupons) to the holder of the bond after a certain date. (Also called maturity date).Some Bonds do not pay interest to the investors, however it is mandatory for the issuers to pay the principal amount to the investors.

#### Why Investment is Important?

• Every individual needs to put some part of his income into something which would benefit him in the long run. Investment is essential as unavoidable circumstances can arise anytime and anywhere. One needs to invest money into something which would guarantee maximum returns with minimum risks in future. Money saved now will help you overcome tough times in the best possible way.

#### Characteristics of a Bond

• Face value: Also known as, the par value and stated on the face of the bond. It represents the amount borrowed by the firm, which it promises to repay after a specified period.
• Coupon rate: A bond carries a specific rate of interest, which is also called as the coupon rate.
• Market value: A bond may be traded on a stock exchange. Market value is the price at which the bond is usually bought or sold in the market. Market value may be different from the par value or the redemption value.
• Redemption Value: The value, which the bondholders gets on maturity. Is called the redemption value, A bond is generally issued at a discount (less than par value) and redeemed at par.
• Maturity date: Maturity date refers to the final date for the payment of any financial product when the principal along with the interest needs to be paid to the investor by the issuer.

#### Types of Bonds

Following are the types of bonds:

• Fixed Rate Bonds: In Fixed Rate Bonds, the interest remains fixed through out the tenure of the bond. Owing to a constant interest rate, fixed rate bonds are resistant to changes and fluctuations in the market.
• Floating Rate Bonds: Floating rate bonds have a fluctuating interest rate (coupons) as per the current market reference rate.
• Zero Interest Rate Bonds: Zero Interest Rate Bonds do not pay any regular interest to the investors. In such types of bonds, issuers only pay the principal amount to the bond holders.
• Inflation Linked Bonds: Bonds linked to inflation are called inflation linked bonds. The interest rate of Inflation linked bonds is generally lower than fixed rate bonds.
• Perpetual Bonds: Bonds with no maturity dates are called perpetual bonds. Holders of perpetual bonds enjoy interest throughout.
• Subordinated Bonds: Bonds which are given less priority as compared to other bonds of the company in cases of a close down are called subordinated bonds. In cases of liquidation, subordinated bonds are given less importance as compared to senior bonds which are paid first.
• Bearer Bonds: Bearer Bonds do not carry the name of the bond holder and anyone who possesses the bond certificate can claim the amount. If the bond certificate gets stolen or misplaced by the bond holder, anyone else with the paper can claim the bond amount.
• Covered bond: Covered bond are backed by cash flows from mortgages or public sector assets. Contrary to asset-backed securities the assets for such bonds remain on the issuers balance sheet.
• A Government Band: A government band, also called Treasury bond, is issued by a national government and is not exposed to default risk.

#### Optionality In Bonds

Occasionally a bond may contain an embedded option; that is, it grants option-like features to the holder or the issuer:

• Callability: Some bonds give the issuer the right to repay the bond before the maturity date on the call dates. This is call option. These bonds are referred to as callable bonds. Most callable bonds allow the issuer to repay the bond at par. With some bonds, the issuer has to pay a premium, the so-called call premium.
• Putability- Some Bonds give the holder the right to force the issuer to repay the bond before the maturity date on the put dates. This is put option. These are referred to as retractable or putable bonds.

## ♦Valuation of Bonds

• A security/Bond can be regarded simply as an asset that pay a series of dividends or interests over a period. Therefore, the value of any security can be defined as the present value of these future cash streams, i.e, the intrinsic value of an assets is equal to the present value of the benefits associated with it. It is quite clear that the holder of a bond receives a fixed annual interest payment for a certain value (equal to par value) at the time of maturity. Therefore the intrinsic value of the present value of a bond is Vo=intrinsic value of the bond

I= Annual interest payable on the bond

F= Redeemable value of the bond

n= Maturity period of the bond

kd= Cost of capital

Note: Solving the problems related to bond valuation, usually Present value Interest Factor of Annuity pertaining to the applicable interest rate are provided. PVIF represents the discount value of one Rupee for the period concerned and interest rate while PVIFA represents the present value of an ordinary annuity for the period concerned and interest rate. Example- PVIF (10%, 6) means present value of one Rupee to be received after 6 periods at the interest rate of 10% period. PVIFA (10%,6) means present value of an ordinary annuity one Rupee per period for 6 period at the interest rate of 10% per period.

Example:

A bond, whose par value is Rs 1000, bears a coupon rate of 12% and has a maturity period of 3 years. The required rate of return on the bond is 10%. What is the value of this bond?

Solution-

Annual interest payable= 1000* 12%=120

Principle repayment at the end of 3 years= Rs 1000

The value of the bond

120(PVIFA 10%, 3yrs) + 1000 (PVIF 10%, 3 yrs)

=120(2.487)+1000(0.751)

=298.44+ 751

=1049.44

## ♦Bond Value with Semi- Annual Interest

• If the Bond carries a semi-annual, as the amount of the half-yearly interest can be reinvested, the value of such bonds would be more the value of bonds with an annual interest payment. Hence, by multiplying the numbers of years to maturity by two and dividing the (i) annual interest payment, (ii) discount rate by two we can modify bond valuation formula as follows: Example:

A bond, whose par value is Rs 1000 bears a coupon rate of 12% payable semiannually and has a maturity period of 3 years. The required rate of return on bonds is 10%. What is the value of this bond?

Solution-

Semi-annual interest payable= 1000*12%/2=60

Principal repayment at the end or 3 years=1000

The value of the bond

=60(PVIFA 10%/2, 6dps)+ 1000(PVIF 10%/2, 6pds)

=60 (5.0746)+ 1000 (0.746)

=304.48+ 746

=1050.48

## ♦Current Yield on Bond

Current yield represents the prevailing interest rate that a bond or fixed income security is delivering to its owners.

The formula for current yield is defined as follows:

CY = Annual interest payment / Current Bond Price

For example, let’s assume a particular bond is trading at par, or 100 cents on the dollar, and that it pays a coupon rate of 3%. In this case, the bond’s current yield will also be 3% (as shown below).

CY = 3 / 100 = 3.00%

However, let’s now assume that the same bond is trading at a discount to its par value. For the sake of example, let’s say investors can now purchase the bond for just 95 cents on the dollar. In this case, even though the bond will still be paying a 3% coupon, its current yield will actually be slightly higher (as shown below):

CY = 3 / 95 = 3.16%

As another example, let’s say the bond is trading at a premium to its face value — 110 cents on the dollar. In this case, even though the bond will still be paying a 3% coupon, its current yield will actually be quite a bit lower (as shown below):

CY = 3 / 110 = 2.73%

Use our Yield to Call (YTC) Calculator to measure your annual return if you hold a particular bond until its first call date.

Use our Yield to Maturity (YTM) Calculator to measure your annual return if you plan to hold a particular bond until maturity.

## ♦Yield-To- Maturity of Bond

It is the rate of return earned by an investor, who purchases a bond and holds it until the maturity. The YTM is the discount rate, which equals the present value of promised cash flows to the current market price/ Purchase price.

Example:

Consider a Rs 1000 par value bond,  whose current market price is Rs 850/-, The bond carries a coupon rate of 8% and has the maturity period of 9 yrs. What would be the rate of return that an investor earns if he purchase the bond and holds until maturity?

Solution:

If kd is the yield to maturity then,

850=80 (PVIFA kd %, 9 yrs)+1000 (PVIF kd, 9yrs)

To calculate the value of kd, we have to try several values:

=80(PVIFA 12%, 9)+1000(PVIF 12% , 9)

=80*5.328+1000*(0.361)

=426.24+361=787.24

Since, the above value is less than 850, we have to try with value less than 12%. Let us try with kd=10%

=80(PVIFA 10%,9)+ 1000(PVIF 10%,9)

=80*5.759+1000*0.424

=884.72

Form the above it is clear that kd lies between 10% and 12%. We have to use linear interpolation in the range of 10% and 12%. Using it, we find that kd is equal to the following:

=10%+(12%-10%)*884.72-850/884.72-787.24

=10%+2%*34.72/97.48

=10.71%

Therefore, the yield to maturity is 10.71%

## ♦Duration of Bond

WHAT IS THE DURATION OF A BOND?

• The duration of a bond expresses the sensitivity of the bond price to changes in the interest rate. In other words, the bond duration measures the movement in the price of the bond for every 1% change in the interest rate.
• The unit of bond duration is expressed in years. Also, the price of the bond and the interest rates are inversely related. Therefore, if a bond has a duration of 5 years, it signifies that for every 1% increase in the interest rate, the price of the bond will fall by 5% and vice-a-versa. The greater is the bond duration, the greater will be the amplification in the movement of bond price for every single unit of change of the interest rates.

There is a simple way of computing the desired duration period:

• Determine the cash flows from holding the bond.
• Determine the present value of these cash flows by discounting the flows with discount rate. (YTM)
• Multiply each of the present values by respective numbers of years left before the present value is received.
• Sum these products up and divide by the present value to get the duration of the bond.

## ♦Properties of Duration

• Duration is less than the term to maturity
• Bond’s duration will be equal to its term to maturity if and only if it is a zero coupon bond
• The duration of perpetual bond is equal to (1+r)/r, where r=current yield of the bond’
• Longer a coupon paying bond’s term to maturity, the greater the difference between its term to maturity and duration.
• Duration and YTM are inversely related.
• Lager the coupon rate, smaller the duration of a bond
• An increase in the frequency of coupon payments decrease the duration, while a decrease in frequency of coupons increases it. Duration of a bond declines as the bond approaches maturity.

## ♦Bond Price Volatility

• The sensitivity of the bond price to changes in the interest rates is called “Bond Volatility”. Bond prices and YTM are inversely related. Therefore, instantaneous changes in market yields cause prices to changes in the opposite direction. The extent of change in the bond princes for a change in YTM measures the interest rate risk of a bond. The interest rate risk is a function of the interest rate elasticity. Interest rate elasticity (IE) can be defined as:

IE= Percentage change in price for bond In period t/Percentage change in yield to maturity for bond

Interest rate elasticity is always a negative number, due to the inverse relationship between YTM and bond prices.

Bond price elasticity can also be computed with the help of following mathematical formula:

IE= D* YTM/1+YTM

The above equation suggests that the duration and interest rate elasticity of a bond are directly related. Anything that causes the duration of a bond to increase will also increase the bond’s interest rate elasticity.

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