# Perpetuities and Interest Rate Risk

Perpetuities and interest rate risk is one of the top concerns for investors and issuers of perpetuities. This is because perpetuities are highly affected by changes in interest rates.

The change in the value of perpetuities due to fluctuations in interest rates is referred to as the* interest rate risks*. Generally, the higher the interest rates, the lower the value of perpetuities and vice versa.

We shall explore this inverse relationship between perpetuities and interest rates in detail in this article.

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## What is perpetuity in finance?

This type of investment exists in both the private and public sectors, hence all kinds of business corporations and governments may provide investment opportunities that are structured as perpetuities.

Common examples of investments that could be structured as perpetuities include common or preferred stocks, endowment funds, real estate, and bonds that do not have a maturity date.

Stocks, whether preferred or common could serve as a perpetual source of income to the stockholders if they pay dividends to their owners. Since businesses are generally expected to exist *ad infinitum* unless they go bankrupt, shareholders are equally expected to continually benefit from dividend payments.

Thus, stockholders have an infinite source from their stocks provided the company that issued the stocks is operational, does not go bankrupt, continues declaring dividends, and does not buy back its shares.

Endowment funds are an additional example of perpetuity because the interest payments on such funds are mostly used to fund education, research, charity, and other activities that may require a consistent income over a long period. The perpetual nature of endowment funds makes them beneficial to the cause for which they are focused for a long time.

Rental buildings and land are often considered perpetuities due to the infinite possibility arising from rental payments. These payments are mostly received every month and continue coming in for as long as the property exists. The cash flow from rental properties is a good example of a growing perpetuity.

Some bonds issued by the British government in 1751 were also designed as perpetuities. They provided the bondholders with a fixed payment annually and continued paying their holders without fail until 2015 when the government discontinued them.

Despite these differences in form, one common feature of all perpetuities is the unending cash flow that it generate for its owners.

Although the stream of income from perpetuities is infinite, the estimated valuation of the total potential stream of cash flows from the investment can be calculated using the present value of a perpetuity formula.

Determining the present value of a perpetuity often aids investors in deciding if the amount they are investing in a perpetuity is worth it or not.

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## What is interest rate risk?

For investments, the interest rate risk is the possibility that the value of investments will either rise or fall depending on the changes in interest rates. It is the probability of a reduction in the value of an asset due to unexpected fluctuations in interest rates.

For companies, interest rate risk is the potential for higher debt servicing costs if interest rates increase or lower return on investments (ROI) if interest rates decrease. Hence, the income statement of companies will record higher debt servicing costs in the former and lower financial income in the latter.

Interest rate risk also affects a company’s balance sheet by a reduction or increase in the value of the company’s assets or liabilities.

For investors in financial markets, interest rate risks are the probability that the value of their investment will change as interest rates rise or fall. Most investors are typically more concerned about increasing rates as higher rates lower the value of fixed-income securities such as government and corporate bonds.

This implies that if the value of a bond is $100 when interest rates are 3%, the value of the bond will reduce below $100 if interest rates rise to 5%.

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## What is perpetuity interest rate?

## Perpetuities and interest rate risk

Perpetuities such as bonds have interest rates as one of the primary drivers of price, hence movements in interest rates pose a risk to the value of such perpetuities, this is known as the interest rate risk.

The relationship between interest rates and bond prices is inverse. This means that when interest rates rise, the price of bonds falls. Conversely, when interest rates fall, bond prices rise.

For instance, if Mr. James made a perpetuity investment by buying a bond whose present value as of 2024 is $30,000 when interest rates are 4%. If by 2025 a similar bond has a higher interest rate of 6%, the present value of Mr. James’ bond will reduce.

The reduction in his bond’s value is because it has a lower interest rate than the current interest rate, hence, if he chooses to sell his bond in 2025, he will have to sell it at a lower price.

This is because the bond’s present value in 2025 when interest rates are at 6% will be lower than its present value in 2024 when interest rates were 4%.

Continuing the example above, if conversely, the interest rates were to decrease to 2% in 2025, the present value of Mr. James’ bond would increase above $30,000. This is because the interest rate of the bond is higher than the current interest rate.

The inverse relationship that exists between bond prices and interest rates can be explained by opportunity risk.

This is because when an investor purchases a bond today, that investment implies that the investor is likely to forgo the opportunity of purchasing other bonds in the future with better returns if interest rates were to rise.

Generally, when interest rates rise, the demand for existing bonds that have a lower return rate declines as new bonds with higher return rates are issued and sold.

Although bond prices are affected by fluctuations in interest rates, the extent of influence varies based on bond coupon rate and maturity date. Bonds with higher coupon rates are less affected by a rise in interest rates while those with low coupon rates are easily affected by a rise in interest rates.

The longer the maturity date of a bond the higher the probability of interest rate changes and the shorter the maturity date of a bond, the lesser its chances of being affected by changes in interest rates.

Since perpetual bonds do not have a maturity date, it means they face higher interest rate risk as these rates are bound to change several times throughout the existence of the perpetuity.

A case in point is the Consolidated Annuities issued by the British government in 1751. When this perpetuity was issued, interest rates were 3% and later reduced to 2.75% and 2.5% in 1888 and 1902 respectively.

Although the above reflects a positive change in the value of the consol due to a decrease in interest rates, the consol reduced in value between 1926 and 1932 when the government issued consols that had a 4% interest rate.

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## What happens to the value of a perpetuity as the interest rate falls?

For instance, if an investor has a perpetuity investment that pays him $300 annually when the interest rate is 7%, we can calculate its present value (PV) as PV = Cash flow / Interest rate.

PV = 300 / 7 = 42.857.

If the interest rates fall to 5%, the present value of the above perpetuity will be PV = $300 / 5 = 60

This means the value of the perpetuity increases from $42.857 to $60 when the interest rates fall.

## What happens to the future value of a perpetuity if interest rates increase?

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## What is the risk of perpetuity?

Issuer risk may arise when the perpetuity issuer faces financial difficulties which may result in being unable to meet its obligations to investors. For perpetuities issued by companies, there is a possibility of the company going bankrupt. Governments on the other hand may choose to discontinue the perpetuity payments.

Inflation risks may arise when inflation rates keep rising to the point that they erode the purchasing power of the cash flows accruing from perpetuities.

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## Conclusion

The indefinite existence of perpetuities makes them liable to interest rate risks. As years and market conditions change, the probability of changes in interest rates becomes more prevalent.

When the interest rates rise, the value of the perpetuity falls; on the other hand, a decrease in interest rates increases the value of perpetuities. These fluctuations in a perpetuity’s value due to changes in interest rates are the interest rate risk.

To be on the safe side, investors and issuers of perpetuities should factor in the probable changes in interest rates that may occur in the future before investing in or issuing perpetuities.