Interest rate risk calculation LEARNOVITA

Interest Rate Risk – How to Mitigate the Risk | A Complete Guide For Beginners

Last updated on 02nd Nov 2022, Artciles, Blog

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    • In this article you will get
    • 1.Interest Rate Risk
    • 2.Calculating Interest Rate Risk
    • 3.Why bond convexities may differ?
    • 4.Interest Rate Derivatives
    • 5.Conclusion

Interest Rate Risk

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

Elements of Interest Rate Risk

Calculating Interest Rate Risk

Interest rate risk analysis is almost always based on a simulating movements in one or more yield curves using Heath-Jarrow-Morton framework to ensure that are yield curve movements are both consistent with a current market yield curves and such that no riskless arbitrage is possible. The Heath-Jarrow-Morton framework was developed in an early 1991 by David Heath of Cornell University, Andrew Morton of Lehman Brothers, and Robert A.Jarrow of a Kamakura Corporation and Cornell University. There are a number of standard calculations for measuring an impact of changing interest rates on portfolio consisting of different assets and liabilities. The most general techniques include:

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

Duration and Convexity

Definition: The duration of the financial asset that consists of fixed cash flows, for example, a bond, is weighted average of times until those fixed cash flows is be received. When an asset is considered as function of yield, duration also measures a price sensitivity to yield, the rate of change of price with respect to the yield or the percentage change in price for the parallel shift in yields.Duration is estimated measure of price sensitivity of a bond to a change in interest rates. It can be stated as the percentage or in dollar amounts. It can be helpful to “shock” or analyse what will happen to a bond when market rates may increase or decrease.

Types of Duration Calculation:

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

Macaulay duration can be calculated by:

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

Macaulay Duration
  • In which, n = number of a coupon periods per year and YTM = bond’s yield to maturity


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

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


  • A Stone & Co. bonds are selling at a 95, yielding 5.25%.
  • Let’s assume that yields increase by a 25bps, causing a price to decline to 93.
  • Therefore, price changes by 2.1%. Now let’s assume that yields decrease by a 25bps, causing price to increase to 98. As a final step, just average two percentage price changes for a 1 basis points move in a rates.


  • Duration = Price if a yield decline – Price if yield increase / 2 * (initial price) *change in yield in a decimals
  • such : 98-93/ 2*95*.0025 = 10.52

Approximate Percentage Price Change of Bond Given a Change in the Duration.Let’s continue with above duration of 10.52. This would equal to percentage price change of 10.52 % for a change of 100 basis points in either direction. If basis points change are 50, then percentage price change would be a 5.26% (10.52/2). If it were a 25bps change, a value would be 2.63% (10.52 / 4).

Approximate New Price of Bond Given Duration and New Yield Level.Let’s return once again to the working with a duration of 10.52. This time, add a total market value of Stone & Co bonds of $10,000,000.

Assume that rates change by a 100 bps. This would cause a value of the bonds to change by a $ 1,052,000 ($10,000.000 *.1052). This is also known as a dollar duration. The price will then range from the $11,052,000 to $8,948,000.If rates increase by a 50 basis points, however, 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 a interest rates, the second derivative of price of the bond with respect to the interest rates (duration is first derivative). In general, higher the convexity, the more sensitive bond price is to the change in interest rates.

Why bond convexities may differ?

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

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

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

Interest Rate Derivatives

1.Interest rate swap (fixed-for-floating): An interest rate swap (IRS) is the famolus and highly liquid financial derivative instrument in which two parties agree to be exchange interest rate cash flows, based on the specified notional amount from fixed rate to a floating rate (or vice versa) or from a one floating rate to another. Interest rate swaps are generally used for both the hedging and speculating.

2.Interest rate cap or interest rate floor: An interest rate cap is the derivative in which the buyer receives payments at end of each period in which the interest rate exceeds agreed strike price. An example of acap would be an agreement to receive the payment for each month the LIBOR rate exceeds a 2.5%.The interest rate cap can be analyzed as the series of European call options or caplets which exist for the beach period the cap agreement is in existence.

3.Interest-rate floor are similar to the caps in that they consist of series of European interest put options (called caplets) with a specific interest rate, each of which expire on a date floating loan rate will be reset. In interest rate floor, the seller agrees to compensate buyer for a rate falling below specified rate during contract period.

4.Interest rate swaption: A Swaption provides with a right but not the obligation to enter into Interest Rate Swap at the predetermined interest rate on a fixed date in a future.

There are two types of a swaption contracts:

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

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


Interest rate risk is linked to a term structure of interest rates. Our analysis of term structure both under the certainty and uncertainty shows how yield and maturity are related. In both certainty and uncertainty cases, concept of riskless arbitrage plays the key role.

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