# How to calculate the interest rate risk of bonds?

The potential for investment losses that arise due to a change in interest rate is categorized as Interest rate risk. If the interest rate rises, the bond value or the value of other fixed-income investments tends to decline. Additionally, this change in the price of the bond due to a change in interest rates, is known as its duration.

The impact of interest rate risk can be reduced by holding bonds of different durations, and investors can also hedge fixed-income investments with interest rate swaps, options, or other interest rate derivatives to allay interest rate risk.

## How to calculate the interest rate risk of bonds?

The interest rate risk analysis is based on simulating movements in one or more yield curves using the approach of Heath-Jarrow-Morton. The framework is used to ensure that the yield curve movements are both consistent with current market yield curves, in a manner that no riskless arbitrage is possible.

The most common techniques to calculate interest rate risk to measure the impact of fluctuating interest rates on a portfolio, comprising various assets and liabilities include the following:

1. Calculating the net market value of the assets and liabilities also referred to as the market value of the portfolio equity or Marking to market (MTM)
2. Shifting the yield curve in a specific way to stress test this market value.
3. Calculating the Value at Risk of the portfolio
4. Calculating the multi-period cash flow or financial accrual income and expense for the N period forward in a deterministic set of future yield curves.
5. In addition to the previous approach with random yield curve movements and measuring the probability distribution of cash flow and the financial accrual over time.
6. The mismatch of the interest sensitivity gap of assets and liabilities is also measured, by classifying each asset and liability on the criteria of the timing of the interest rate reset or maturity, whichever comes first.
7. Characteristics such as duration, convexity, DV01, and Key Rate Duration are also analysed as an approach. These are characteristic of a bond when it comes to the measuring of the interest rate risk impact on the bond

The duration of a financial asset consists of fixed cash flows, like in the case of bonds, the weighted average of the times until the fixed cash flows are received. For assets that are considered as a function of yield, the duration also measures the price sensitivity to yield. Price sensitivity is the rate of change of price for the yield or the percentage change in price for a parallel shift in yields.

Duration is considered to be an estimated measure of the price sensitivity of a bond to a fluctuation in interest rates. It is commonly expressed as a percentage or in dollar amounts and is useful in analysing the impact of the increase or decrease of market rates on the bond price. There are two types of durations:

1. Macaulay Duration: It is defined as 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. It is a measure of the volatility of the bond price concerning changes in interest rates.

2. Modified Duration: The formula for modified duration expresses the measurable change in the value of the security with the change in interest rates.

## Effects of Interest Rate

Bond prices have an inverse relationship to interest rates. With an increase in the cost of borrowing (money), the bond prices tend to fall and vice versa. The increased cost of borrowing implies an increase in interest rates.

Regarding the effect of interest rates, the following are key conclusions:

• Many bonds pay a fixed interest rate. These tend to make an attractive investment option, in events when interest rates fall, driving up their demand and consequently the price of the bond.
• Consequently, during scenarios of rising interest rates, investors show no preference for lower fixed rates paid by the bonds, causing bond prices to decline with reduced investor demand.
• A zero-coupon bond does not pay regular interest. It derives all of its value from the difference between the purchase price and the par value of the bond, paid at maturity. They are issued at a price discounted to the par value, with yields as a function of the purchase price, the par value, and the time left till maturity.
• Zero-coupon bonds tend to have a lock in the bond’s yield, making them attractive as an investment option to certain investors.

## Some Things to Keep in Mind When Calculating Interest rate risk

Convexity is an indicator of the sensitivity of the duration of a bond to the changes in the interest rates, becoming the second derivative of the price of the bond concerning the interest rates. Duration is the first derivative.

A high convexity indicates higher sensitivity of the bond price to changes in interest rates.

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

Considering two bonds present at the same price to yield combination, it is important to look at the profile, the rating, and the issuer of the bond. Though both bonds may have the same price yield combination, one bond, say A might be located on a more elastic segment of the p-y curve compared to the other bond B. This implies, that if the yield increases further, the price of bond A may fall, while the price of bond B may not change. In such a case, holders of bond B expect an increase in the price and therefore show reluctance to sell it off, whereas holders of bond A, expect a further fall in the price and prefer to dispose of it.

This indicates that bond B has a better rating than bond A. Thus, a higher rating or credibility of the issuer means the less the convexity and the less the gain from the risk-return game or strategies. Less convexity means lower price volatility or risk; less risk also implies lower returns.

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