## Interest Rate Risk

Interest rate risk is the risk that arises for bond owners from fluctuating interest rates. How much interest rate risk a bond has depends on how sensitive its price is to interest rate changes in the market. The sensitivity depends on two things, the bond's time to maturity, and the coupon rate of the bond.

## Calculating Interest Rate Risk

Interest rate risk analysis is almost always based on simulating movements in one or more yield curves using the Heath-Jarrow-Morton framework to ensure that the yield curve movements are both consistent with current market yield curves and such that no riskless arbitrage is possible. The Heath-Jarrow-Morton framework was developed in early 1991 by David Heath of Cornell University, Andrew Morton of Lehman Brothers, and Robert A. Jarrow of Kamakura Corporation and Cornell University.

There are a number of standard calculations for measuring the impact of changing interest rates on a portfolio consisting of various assets and liabilities. The most common techniques include:

1. Marking to market, calculating the net market value of the assets and liabilities, sometimes called the "market value of portfolio equity".
2. Stress testing this market value by shifting the yield curve in a specific way.
3. Calculating the Value at Risk of the portfolio.
4. Calculating the multi-period cash flow or financial accrual income and expense for N periods forward in a deterministic set of future yield curves.
5. Doing step 4 with random yield curve movements and measuring the probability distribution of cash flows and financial accrual income over time.
6. Measuring the mismatch of the interest sensitivity gap of assets and liabilities, by classifying each asset and liability by the timing of interest rate reset or maturity, whichever comes first.
7. Analyzing Duration, Convexity, DV01, and Key Rate Duration.

Duration and Convexity

Definition: The duration of a financial asset that consists of fixed cash flows, for example, a bond, is the weighted average of the times until those fixed cash flows is received. When an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields.

Duration is an estimated measure of the price sensitivity of a bond to a change in interest rates. It can be stated as a percentage or in dollar amounts. It can be helpful to "shock" or analyze what will happen to a bond when market rates increase or decrease.

Types of Duration Calculation:

Macaulay Duration: The weighted average term to maturity of the cash flows from a bond. The weight of each cash flow is determined by dividing the present value of the cash flow by the price and is a measure of bond price volatility with respect to interest rates.

Macaulay duration can be calculated by:

Modified Duration: This is a formula that expresses the measurable change in the value of security in response to a change in interest rates. It is calculated as follows:

In which, n = number of coupon periods per year and YTM = the bond's yield to maturity

Example:

Let's assume that the calculation yields a duration of 6.14, this means that if interest rates change, the value of the bond will change by 6.14%. If there is a 50 basis point change, the value will change by 3.07% and for a 25 basis point change would equal a 1.53% change.

Effective Duration: A duration calculation for bonds with embedded options. Effective duration takes into account that expected cash flows will fluctuate as interest rates change.

Example:

Stone & Co. bonds are selling at 95, yielding 5.25%

Let's assume that yields increase by 25bps, causing the price to decline to 93.

Therefore, the price changes by 2.1%. Now let's assume that yields decrease by 25bps, causing the price to increase to 98. As a final step, just average the two percentage price changes for a 1 basis points move in rates.

Duration = Price if yield decline - Price if yield increase / 2 * (initial price) *change in yield in decimals

As such: 98-93/ 2*95*.0025 = 10.52

Approximate Percentage Price Change of a Bond Given a Change in Duration

Let's continue with the above duration of 10.52. This would equal a percentage price change of 10.52 % for a change of 100 basis points in either direction. If the basis points change were 50, then the percentage price change would be 5.26% (10.52/2). If it were a 25bps change, the value would be 2.63% (10.52 / 4).

Approximate New Price of a Bond Given the Duration and New Yield Level

Let's return once again to working with a duration of 10.52. This time, we'll add a total market value of the Stone & Co bonds of \$10,000,000.

Assume that the rates change by 100 bps. This would cause the value of the bonds to change by \$ 1,052,000 (\$10,000.000 *.1052). This is also known as dollar duration. The price will then range from \$11,052,000 to \$8,948,000.

If rates increase by 50 basis points, however, the dollar change would be \$526,000 giving the bonds a price range of \$ 10,526,000 to \$ 9,474,000.

Convexity is a measure of the sensitivity of the duration of a bond to changes in interest rates, the second derivative of the price of the bond with respect to interest rates (duration is the first derivative). In general, the higher the convexity, the more sensitive the bond price is to the change in interest rates.

## Why bond convexities may differ

The price sensitivity to parallel changes in the term structure of interest rates is highest with a zero-coupon bond and lowest with an amortizing bond (where the payments are front-loaded). Although the amortizing bond and the zero-coupon bond have different sensitivities at the same maturity, if their final maturities differ so that they have identical bond durations they will have identical sensitivities. That is, their prices will be affected equally by small, first-order, (and parallel) yield curve shifts. They will, however, start to change by different amounts with each further incremental parallel rate shift due to their differing payment dates and amounts.

For two bonds with same par value, same coupon and same maturity, convexity may differ depending on at what point on the price yield curve they are located.

Suppose both of them have at present the same price yield (p-y) combination; also you have to take into consideration the profile, rating and more of the issuers: let us suppose they are issued by different entities. Though both bonds have same p-y combination bond A may be located on a more elastic segment of the p-y curve compared to bond B. This means if yield increases further, price of bond A may fall drastically while price of bond B won’t change, that is bond B holders are expecting a price rise any moment and are therefore reluctant to sell it off, while bond A holders are expecting further price-fall and ready to dispose of it.

This means bond B has better rating than bond A. So the higher the rating or credibility of the issuer the less the convexity and the less the gain from risk-return game or strategies; less convexity means less price-volatility or risk; less risk means less return.

Interest Rate Derivatives

1. Interest rate swap (fixed-for-floating): An interest rate swap (IRS) is a popular and highly liquid financial derivative instrument in which two parties agree to exchange interest rate cash flows, based on a specified notional amount from a fixed rate to a floating rate (or vice versa) or from one floating rate to another. Interest rate swaps are commonly used for both hedging and speculating.
2. Interest rate cap or interest rate floor: An interest rate cap is a derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment for each month the LIBOR rate exceeds 2.5%.The interest rate cap can be analyzed as a series of European call options or caplets which exist for each period the cap agreement is in existence.
3. Interest-rate floor are similar to caps in that they consist of a series of European interest put options (called caplets) with a particular interest rate, each of which expire on the date the floating loan rate will be reset. In an interest rate floor, the seller agrees to compensate the buyer for a rate falling below the specified rate during the contract period
4. Interest rate swaption: A Swaption provides you with the right but not the obligation to enter into an Interest Rate Swap at a predetermined interest rate on a fixed date in the future.

There are two types of swaption contracts:

A payer swaption gives the owner of the swaption the right to enter into a swap where they pay the fixed leg and receive the floating leg.

A receiver swaption gives the owner of the swaption the right to enter into a swap in which they will receive the fixed leg and pay the floating leg.

1. Forward rate agreement (FRA): It is a forward contract, an over-the-counter contract between parties that determines the rate of interest, or the currency exchange rate, to be paid or received on an obligation beginning at a future start date. The contract will determine the rates to be used along with the termination date and notional value. On this type of agreement, it is only the differential that is paid on the notional amount of the contract. It is paid on the effective date. The reference rate is fixed one or two days before the effective date, dependent on the market convention for the particular currency. FRAs are over-the counter derivatives. A FRA differs from a swap in that a payment is only made once at maturity. Swaps are considered to be series of FRA's.