**Overview of the content:**

1.Different Sources of Return from a bond

2.Different Durations

3.Price Value of a Basis point

4.Convexity

5.Yield Volatility

**Sources of Return from a FI Security:**

1.Coupon Payments

2.Principal Payment at maturity

3.Reinvestment of Coupon Payments

4.Any potential gain/loss if bond is sold prior to maturity

**Important Conclusions:**

1) If an investor holds the bond till maturity: Annualized Rate of Return = YTM of the Bond

2) If Bond is sold prior to maturity: Annualized Rate of Return = YTM of Bond at Purchase If YTM has not changed since purchase.

3) For a buy and hold Investor**: Realized Return > YTM of Bond** If marketâ€™s YTM increases before first coupon date and vice versa.

4) For a short term: Realized return < YTM of Bond If market YTM increases and Vice Versa

5) For a long term but prior of maturity: Realized Return > YTM of Bond If market YTM increases and vice versa.

**Interest Rate Risk Measures:**

**Interest Rate Risk can be measured by,**

1.Macaulay Duration

2.Effective Duration

3.Properties of Bond Duration

4.Duration of Bond Portfolio

5.Money Duration

6.Present Value of Basis Point

7.Bond Convexity

**Duration:**

The duration of a bond measures the sensitivity of bondâ€™s full price to changes in interest rate.

**Types of Duration:**

1.Yield Duration: Measures the sensitivity of bond price to changes in the bond yield curve.

2.Macaulay Duration

3.Modified Duration

4.Money Duration

5.PVBP

6.Curve Duration: It measures the sensitivity of a bond price to changes in market interest rates.

**Macaulayâ€™s Duration:**

It is used as a measure of a bondâ€™s interest rate risk of a fill price to change in its yield.

It is the weighted average of the time to receipt of the bondâ€™s promised cash flows, where the weights are PV of each cash flow as a % of Bondâ€™s full value.

Macaulayâ€™s Duration is denoted in terms of â€˜Yearsâ€™

Macaulayâ€™s Duration can be best explained with an example illustrated in the next slide.

__Example__: Consider a 6 years 8% Annual Pay Bond at Par.

Macaulay Duration = 0.07462(1)+0.06859(2)+0.0635(3)+0.0588(4)+0.05445(5)+0.6858

Macaulay Duration = 5.024 Years

**Modified Duration:**

Modified Duration provides an approximate % change in Bondâ€™s price for a 1% change in YTM.

__For an annual bond__

ModDur = MacDur / (1 + YTM)

__For a Semi-annual Bond__,

ModDurSemi = MacDurSemi / (1 + YTM/2)

Approximate % change in Bond Price = - ModDur x âˆ†YTM

Approximate Modified Duration = (V- - V+) / (2 * Vo * âˆ†YTM)

Modified Duration is a

*linear estimate*of the relation between*Bondâ€™s Price*and*YTM.*The Actual relation is

*Convex*Modified Duration measures good estimates of bond prices for small changes in Yield.

**Effective Duration:**

Effective Duration is the sensitivity of bondâ€™s price to a change in a benchmark yield curve.

Effective Duration = (PV- - PV+) / (2 * PVo * âˆ†curve)

Effective Duration is a duration calculation for bonds with embedded options.

Effective Duration is used when change in yield may alter the expected cash flows.

For E.g.: Bonds with embedded options like MBS, Callable Bonds etc.

**Importance of Effective Duration:**

The uncertain nature of future cash flows makes modified duration unfit for calculating the interest rate risk for bond with embedded options.

Effective duration is considered the most appropriate measure of interest rate risk for bonds with embedded options.

As the future cash flows not only depend on future the interest rates but also on the path interest rate take over time.

**Factors Affecting Interest Rate Risk:**

__Bond Maturity__: Increase in bondâ€™s maturity will increase its interest rate risk.__Coupon Rate:__Increase in coupon rate will decrease its interest rate risk.__Yield to Maturity__: Increase in bondâ€™s YTM will decrease its interest rate risk.__Embedded Option__: Adding an embedded option to a bond decreases its interest rate risk.

__Call Option__: The value of the call increases as yield fall, so a decrease in yield will have less

effect on the price of the bond.

__Put Option__: The value of put option increases as yields rise, so an increase in yield will have

less effect on the price of the bond.

**Duration of Bond Portfolio:**

Two Methods to estimate the duration of a portfolio

** First Approach**

Weighted Average of time to receipt of the aggregate cash flows.

The yield measure for calculating portfolio duration is the cash flow yield.

Difficult to ascertain the amount and timing of cash flows.

Interest rate risk is usually expressed as a change in benchmark interest rates, not as a change in the cash flow yield.

Change in the cash flow yield is not necessarily the same amount as the change in the yield convexity on the individual bonds.

** Second Approach**

Weighted average of the individual bond duration that comprise the portfolio.

Easy to use a measure of interest rate risk.

More accurate as difference in YTMs of bonds in portfolio becomes smaller

Assumes parallel shift in yield curve.

**Money Duration:**

The

*Money Duration*is a measure of a bondâ€™s price change in units of currency in which it is denominated.Money duration is also called

*dollar*duration and is expressed in currency units.Money Duration =

âˆ†PV Full = Money Duration x âˆ†Yield

Multiplying the money duration of a bond times a given change in YTM will provide the change

in bond value for the change in YTM.

**Price Value of a Basis Point (PVBP):**

The PVBP is an estimate of the change in the full price of a bond, given a 1 basis point change in the YTM.

PVBP = (PV- - PV+) / 2

It is another way to measure interest rate risk.

It does not matter if it is an increase or decrease in rates, because such a small move in rates will be about the same in either direction according to the second property of a bond's price.

**Convexity:**

Convexity is a measure of the curvature of the price yield relation.

The more curved it is , the greater its convexity adjusted to a duration based estimate of a % change in price for a given change in YTM.

For an option-free bond the price-yield curve is convex towards the origin.

Price falls at a decreasing rate as yields increase.

**Effective Convexity:**

*Effective Convexity*measures the secondary effect of a change in benchmark yield curve.It is used for bonds with embedded options

Approximate Effective Convexity = V- + V+- 2V0

(âˆ† curve)2 x V0

Convexity gets affected by:

Time to maturity

Coupon Rate

Yield to Maturity

Dispersion of Cash Flows

**Positive and Negative Convexity:**

**Positive Convexity:**

When shape of a bondâ€™s yield curve is convex.

Gain is greater than the loss for a given change in rates.

For E.g.: Option Free Bonds.

**Negative Convexity:**

When the shape of a bondâ€™s yield curve is concave.

Loss will be greater than the gain.

For E.g.: Callable Bond.

Convexity of callable bond is Negative because at low yields the call option becomes more valuable and the call price puts an effective limit on increases in Bond Value.

**% Change in Price Combining Duration and Convexity:**

Change in full bond price = Annual Modified Duration(âˆ†YTM)+Â½ Annual Convexity (âˆ†YTM)2

By taking account of both Bondâ€™s duration and convexity, we can evaluate the effects of change in yield on a bondâ€™s value.

Duration based estimate of the increase in price is too low for a bond with convexity.

The decrease in price resulting for an increase in yield is larger than the actual decrease.

**Yield Volatility:**

Yield Volatility refers to volatility of a Bondâ€™s YTM.

Typically short term bonds have greater yield volatility than long term bonds.

Short term bonds tend to get affected more by the monetary policy.

The volatility of a bondâ€™s price has two components :

The sensitivity of the bondâ€™s price to a given change in yield.

The volatility of the bondâ€™s yield.

**Investment Horizon, Macaulay Duration and Interest Rate Risk:**

With larger term horizon, we are concerned with Price Risk and Reinvestment Risk.

Over a short term horizon, a change in YTM affects market price more than it affects market price.

Consider a 10-year, 8% Annual Coupon Bond priced at $88.5, YTM = 10.40%

If investment Horizon = 10 years - The major risk is Reinvestment Risk.

If investment Horizon = 4 years - The major risk is Price Risk

**Investment Horizon, Macaulay Duration and Interest Rate Risk:**

Macaulay Duration indicates that the investment horizon for which coupon reinvestment risk and market price risk offset each other.

__Assumption__: One time parallel shift in the yield.

MacDur < Investment Horizon : Coupon Reinvestment Risk Dominates

MacDur = Investment Horizon : Coupon Reinvestment Risk offsets the market price risk.

MacDur > Investment Horizon : Market Price Risk Dominates

**Duration Gap = Macaulayâ€™s Duration â€“ Investment Horizon**

**Credit and Liquidity Spread:**

Credit Risk involves the probability of default and degree of default if default occurs.

Liquidity Risk refer to the transaction cost associated with selling a bond.

For a traditional (option free) fixed rate bond , the same duration and convexity statistics apply if a change occurs in the benchmark yield or a change occurs in the spreads.

The change in the spread can result from a change in credit risk or liquid risk.

In practice, there often is interaction between changes in benchmark yields and in the spread over the benchmark.

**Bond Yield to Call:**

A callable bond gives the issuer the right to buy back a bond from the investor at a specified price after the protection period

__Yield-to-Call__

The discount rate which makes the PV of cash flow up to call date equal to the current price

Cash flow includes coupon payment and call price

Often used for premium bond

**Example: Yield to Call:**

Find the yield to call on a semi-annual coupon bond with a face value of $1000, a 10% coupon rate, 15 years remaining until maturity given that the bond price is $1175 and it can be called 5 years from now at a call price of $1100.

__The following values are known__

Bond Price: $ 1175 Face Value: $ 1000 Coupon rate: 10% Years to Maturity: 15 Call Price: $ 1100 Years until Call Date: 5 Solution : Slide Note

**Yield to Maturity:**

Yield to Maturity (

**YTM**) . Yield to maturity (**YTM**) measures the annual return an investor would receive if he or she held a particular bond until maturity. . To understand**YTM**, one must first understand that the price of a bond is equal to the present value of its future cash flows.Although far from perfect, the

*yield-to-maturity*is a relatively accurate

**Example : Yield to Maturity:**

You buy ABC Company bond which matures in 1 year and has a 5% interest rate (coupon) and has a par value of $100. You pay $90 for the bond.

The current yield is 5.56% (5/90).

If you hold the bond until maturity, ABC Company will pay you $5 as interest and $100 par value for the matured bond.

Now for your $90 investment, you get $105,

**so your yield to maturity is 16.67% **

**[= (105/90)-1] or [=(105-90)/90].**

**Yield to Put:**

**Putable bonds**are bonds that give the holder the right to sell his or her bond to the issuer prior to the bond's maturity date.

** How it works/Example:**

The bond indenture will stipulate when and how the bond can be sold, and there are often multiple sell dates throughout the life of a putable bond. Many corporate and municipal securities have one- to five-year put provisions.

**Example â€“ Yield to Put:**

To understand how the put works, let's consider an XYZ bond issued in 2000 and maturing in 2020. The indenture stipulates that the holder may put (sell) the bond after four years. The put provision in the indenture also sets forth the put price, which is what the issuer must pay to redeem the bond. Usually the put price is 100% of face value. In our example, the indenture might say, "The XYZ bond due June 1, 2020 is putable on June 1, 2004 at 100% of par." (The indenture typically provides a table of put dates and prices as well if applicable.)

Recall that when interest rates rise, prices of bonds issued at older, lower rates fall and vice versa. Thus, if interest rates are at 3% on June 1, 2004, but the XYZ bond is only returning 2%, the holder will probably put the bond, get his money back, and reinvest it in something with a higher return. If, however, market returns for similar bonds have fallen to 1%, the XYZ bond is worth more, and the investor will probably be content to hang on to his bond and get above-average market returns.

**Thank you**