Bonds are long-term debt securities that corporations and government entities issue. Purchasers of bonds receive periodic interest payments, called coupon payments, until maturity, when they receive the bond’s face value and the last coupon payment. Most bonds pay interest semiannually. The Bond Indenture or Loan Contract specifies the features of the bond issue.

The following terms are used to describe bonds.

#### Par or Face Value

A bond’s par or face value is the amount of money paid to the bondholders at maturity. For most bonds, the amount is INR 1,000. It also generally represents the amount of money borrowed by the bond issuer.

#### Coupon Rate

The coupon rate, which is generally fixed, determines the periodic coupon or interest payments. It is expressed as a percentage of the bond’s face value. It also represents the interest cost of the bond issued to the issuer.

#### Coupon Payment

The coupon payments represent the periodic interest payments from the bond issuer to the bondholder. The annual coupon payment is calculated by multiplying the coupon rate by the bond’s face value. Since most bonds pay interest semiannually, half of the annual coupon is generally paid to the bondholders every six months.

#### Maturity Date

The maturity date represents the date on which the bond matures, i.e., the date on which the face value is repaid. The last coupon payment is also paid on the maturity date.

#### Original Maturity

The time remaining until the maturity date when the bond was issued.

#### Remaining Maturity

The time currently remaining until the maturity date.

#### Call Date

For callable bonds, i.e., bonds that the issuer can redeem before maturity, the call date represents the date at which the bond can be called.

#### Call Price

The amount of money the issuer has to pay to call a callable bond. When a bond first becomes callable, i.e., on the call date, the call price is often set to equal the face value plus one year’s interest.

#### Required Rate of Return

The rate of return that investors currently require on a bond.

#### Bond Cash Flows

The figure below illustrates the cash flows for a semiannual coupon bond with a face value of INR 1000, a 10% coupon rate, and 15 years remaining until maturity. (Note that the annual coupon is INR 100, calculated by multiplying the 10% coupon rate times the Rs1000 face value. Thus, the periodic coupon payments equal INR 50 every six months.)

= 50(1st Yr) + 50(2nd Yr) + 50(3rd Yr) +——–(50+1000) (15tth Yr)

#### Par Bonds

A bond is considered a par bond when its price equals its face value. This will occur when the coupon rate equals the required return on the bond.

#### Premium Bonds

A bond is considered a premium bond when its price is more significant than its face value. This will occur when the coupon rate exceeds the bond’s required return.

#### Discount Bonds

A bond is considered a discount bond when its price is less than its face value. This will occur when the coupon rate is less than the required return on the bond.

#### Yield

Yield is a general term that relates to the return on the capital you invest in a bond. Price and yield are inversely related: As the price of a bond goes up, its yield goes down, and vice versa.

#### Clean and Dirty Bond Prices

The clean price of a bond is the conventional basis for quoting bond prices, excluding accrued interest. The dirty price of a bond includes accrued interest and is the total amount payable on the sale or purchase of the bond in between interest payment dates.

So the clean price is always less than the dirty price, other than immediately after an interest payment when there is no difference between the clean and dirty prices.

### Bond Yield

Several definitions are essential to understand when talking about yield as it relates to bonds: coupon yield, current yield, yield-to-maturity, yield-to-call, and yield-to-worst.

Let’s start with the basic yield concepts.

#### Coupon Yield

Coupon yield, also known as the coupon rate, is the annual interest rate established when the bond is issued that does not change during the bond’s lifespan.

#### Current Yield

It is the bond’s coupon yield divided by its current market price. If the current market price changes, the current yield will also change.

For example, if you buy an INR 1,000 bond at par (often described as “trading at 100,” meaning 100 percent of its face value) and receive INR 55 in annual interest payments, your coupon yield is 5.5 percent. If the price goes up and the bond trades at 104 (INR 1,040), the coupon yield will fall to 5.28 percent.

Current yield matters if you plan to sell your bond before maturity. But if you buy a new bond at par and hold it to maturity, your current yield will be the same as the coupon yield when it matures.

#### Yield to Maturity (YTM)

YTM is the overall interest rate earned by an investor who buys a bond at the market price and holds it until maturity. Mathematically, it is the discount rate at which the sum of all future cash flows (from coupons and principal repayment) equals the bond’s price. YTM is often quoted as an annual rate and may differ from the bond’s coupon rate. It assumes that coupon and principal payments are made on time. It does not require dividends to be reinvested, but computations of YTM generally make that assumption. Further, it does not consider taxes paid by the investor or brokerage costs associated with the purchase.

This yield to maturity, also termed as redemption yield of a bond, is the Internal Rate of Return (IRR) of a bond. Having understood how the price of a bond is derived from its yield, it is helpful to make the reverse calculation, i.e., to calculate yield to maturity or the IRR was given a market price. The calculation for finding the redemption yield follows that for finding the IRR of a set of cash flows, i.e., by interpolation.

#### Yield to Call (YTC)

Many bonds, especially those issued by corporations, are callable. This means that the issuer of the bond can redeem the bond before maturity by paying the call price, which is greater than the face value of the bond, to the bondholder. Often, callable bonds cannot be called until 5 or 10 years after they were issued. When this is the case, the bonds are said to be ‘Call Protected.’ The date when the bonds can be called is called the call date.

The yield to call is the rate of return that an investor would earn if he bought a callable bond at its current market price and held it until the call date, given that the bond was called on the call date. It represents the discount rate which equates the discounted value of a bond’s future cash flows to its current market price, given that the bond is called on the call date.

YTC is calculated the same way as YTM, except instead of plugging in the number of months until a bond matures, you use a call date and the bond’s call price. This calculation considers the impact on a bond’s yield if it is called before maturity and should be performed using the first date on which the issuer could call the bond.

### Valuing a Bond

There are occasions when finance professionals need to know the bond’s market value (i.e., price) in the issue. These occasions include proposals to buy back bonds from the market, perhaps as part of a refinancing, investing in bonds as part of cash and liquidity management, and performing estimates of a company’s overall cost of funds.

Bonds typically pay a fixed coupon (i.e., a fixed interest amount over the bond’s life), so their value fluctuates with interest rates. This is because the cash flows are re-valued according to the prevailing interest rates. As a result, the market value of a bond is equal to the present value of its future cash flows, where the cash flows consist of its face value and the periodic coupons, discounted at the current yield to maturity.

Generally, the market value moves inversely with interest rates. So, for example, if a bond is issued with coupons of 5% when interest rates for its credit standing generally are 5%, then we can expect the bond to be priced around 100. However, if interest rates fall, to say, 4%, then the cash flows remain fixed and to calculate a value requires discounting at 4%, leading to a higher value for the bond.

As the bond nears the end of its life, its value will revert to par, or 100, as that is the final cash flow from the bond.

#### Formula: The present value of a future cash flow is given by the following

PV = FV / (1+ r)^n

**Where,**

PV = present value of the future flow

FV = money amount of the future flow

r = interest rate per period

n = number of the period at the end of which the flow occurs

So the price or market value (P) of a coupon bond is the total of the present values of the future cash flows from the bond:

P = CF/(1+r) + CF/(1+r)^2 +———(CF+F)/(1+r)^n

Where CF is the periodic coupon flow, n is the number of remaining coupon periods to maturity, r is the interest rate per period (here assuming a flat yield curve – or in other words, ‘r’ being the same for all periods), and ‘F’ is the face value of the bond.

The above approach to valuing bonds is useful when we need to drill down into each element. For instance, the coupons CF or interest rates r are not all the same.

But if we know the inputs CF, r and n are constant, then the coupon payments in the form of an annuity, which we can value using the **annuity formula:**

PV(Coupon Flows) = (CF/r) * (1 -(1/(1+r)^n))

**Where:**

CF = periodic coupon payable

r = interest rate per period

n = number of the period at the end of which the flow occurs

The present value of the face value (F) of the bond, PV(face value), is F/ (1+r)^n

Add the present value of the coupon payments to the present value of the face value to give the price of the bond P = PV(coupon flows) + PV(face value), or

P = [(CF/r) * (1 -(1/(1+r)^n)) ] + [F/ (1+r)^n]

### Example: Bond Valuation

#### Example 1: Calculation of Bond Price

Calculate a bond’s price per INR 100 face value with precisely three years to maturity and a 6.0% coupon, assuming a flat yield curve of 5.0% p.a. The bond pays interest annually.

This means there are three annual coupon payments of INR 100 x 6.0% = INR 6, and the periodic yield used for calculating the present values, r, is the yield to maturity of 5% or 0.05.

Using the equation given in the previous lesson, the price of the bond is calculated as follows:

P = 6/ (1+0.05)+6/(1.05)^2+(6+100)/(1+0.05)^3

=INR 5.71+INR 5.44+INR 91.57

= INR 102.72

In the annual bond example above, the price exceeds the face value. This is because the bond is paying a coupon rate (6%) that is higher than the current yield to maturity in the market (of only 5%).

#### Example 2: Annual bond valuation using the annuity formula

Calculate the price per INR 100 face value of a bond with precisely three years to maturity and a 6.0% coupon if the three-year yield to maturity is 5.0% p.a. The bond pays interest annually.

P= [6/ (0.05) * (1-(1/(1+0.05)^3))] + [100/(1.05)^3]

= 16.34 + 86.38

= INR 102.72

#### Example 3: Bond Yield

DDG Corporation wants to raise capital for 10 years at a maximum cost (i.e., redemption yield) of 7.5% for its exploration projects. Its investment bank advisers suggest that it can issue a bond with a face value of 100 and an annual coupon of 5% at an issue price of 85. Calculate whether this satisfies DDG Corporation’s target cost.

We need to calculate the IRR of these cash flows by interpolation. We first calculate the present value of the coupon and principal repayment, discounting at a rate of 7%:

Present Value = [5/0.07 *(1-1/(1+0.07)^10) ]+ [100/(1+0.07)^10]

=35.118+50.835

=85.953

If the issue price is 85, the NPV of the bond flows (to the issuer) is 85 – 85.953 = -0.953.

We then calculate the present value of the coupon and principal repayment, discounting at a rate of 8%

Present Value = [5/0.08 *(1-1/(1+0.08)^10)]+ [100/(1+0.08)^10]

= 33.550 + 46.319

= 76.869

If the issue price is 85, the NPV of the bond flows (to the issuer) is 85 – 76.869 = 8.131.

We can then use interpolation to find the rate at which the NPV = 0

A formula for this interpolation is as follows:

IRR% = a% + A / (A-B) * (b%-a%)

**Where,**

- a%, b% = two estimated costs of capital,
- A and B = the NPVs obtained at each such cost of capital.

The IRR is

= 7 +{(-0.953)/ (-0.953-8.131) }*(8 – 7)

=7 +( -0,953/ -9.084)*1

= 7 + (0.1049)*1

=7 + 0.1049

=7.1049,, or

7.105%

Thus the IRR or yield of the bond is 7.105% which is less than the target yield of 7.5%.

Volatility

#### Price and Yield

An essential property of all conventional bonds is that yields rise as bond prices fall or yields fall as bond prices rise. If a bond’s price were measured against its yield for instantaneous changes in yield, it would appear convex.

### Volatility in Bond’s Price

Volatility is measured by the percentage change in the bond’s price for a given change in yield. Assume there are nine semi-annual coupon bonds. Assume all nine bonds were initially traded with a yield of 5%. The table below shows the percentage change in the price of the bonds if the yield instantaneously changes to the new yield in the first column.

Yield |
Coupon Rate |
||||||||

0% | 0% | 0% | 5% | 5% | 5% | 10% | 10% | 10% | |

2 Yrs. | 10 Yrs. | 20 Yrs. | 2 Yrs. | 10 Yrs. | 20 Yrs. | 2 Yrs. | 10 Yrs. | 20 Yrs. | |

4.00% | 1.98 | 10.27 | 21.60 | 1.90 | 8.18 | 13.68 | 1.84 | 7.25 | 11.86 |

4.50% | 0.98 | 5.01 | 10.26 | 0.95 | 3.99 | 6.55 | 0.92 | 3.55 | 5.70 |

4.90% | 0.20 | 0.98 | 1.97 | 0.19 | 0.78 | 1.27 | 0.18 | 0.70 | 1.10 |

4.99% | 0.02 | 0.10 | 0.20 | 0.02 | 0.08 | 0.13 | 0.02 | 0.07 | 0.11 |

5.01% | -0.02 | -0.10 | -0.19 | -0.02 | -0.08 | -0.13 | -0.02 | -0.07 | -0.11 |

5.10% | -0.19 | -0.97 | -1.93 | -0.19 | -0.78 | -1.24 | -0.18 | -0.69 | -1.09 |

5.50% | -0.97 | -4.76 | -9.28 | -0.93 | -3.81 | -6.02 | -0.91 | -3.39 | -5.27 |

6.00% | -1.93 | -9.27 | -17.69 | -1.86 | -7.44 | -11.56 | -1.80 | -6.63 | -10.15 |

#### From the above table, we can conclude that:

- For a given change in market yield, the percentage change in bond prices is more significant for
- longer maturities.
- For more significant changes in yields, a given downward move in yield results in a more significant percentage change in price than the same move in yield upward.
- The higher the coupon, the smaller the percentage price change will be for a given change in
- yield.
- Thus price volatility is dramatically affected by coupon size and term to maturity. The measure which connects these two concepts is duration.