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In finance, a perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence. For example, the United Kingdom (UK) government issued them in the past; these were known as consols and were all finally redeemed in 2015.
Real estate and preferred stock are among some types of investments that affect the results of a perpetuity, and prices can be established using techniques for valuing a perpetuity. [1] Perpetuities are but one of the time value of money methods for valuing financial assets.
Perpetuities can be structured as a perpetual bond and are a form of ordinary annuities. The concept is closely linked to terminal value and terminal growth rate in valuation.
A perpetuity is an annuity in which the periodic payments begin on a fixed date and continue indefinitely. It is sometimes referred to as a perpetual annuity. Fixed coupon payments on permanently invested (irredeemable) sums of money are prime examples of perpetuities. Scholarships paid perpetually from an endowment fit the definition of perpetuity.
The value of the perpetuity is finite because receipts that are anticipated far in the future have extremely low present value (present value of the future cash flows). Unlike a typical bond, because the principal is never repaid, there is no present value for the principal. Assuming that payments begin at the end of the current period, the price of a perpetuity is simply the coupon amount over the appropriate discount rate or yield; that is,
where PV = present value of the perpetuity, A = the amount of the periodic payment, and r = yield, discount rate or interest rate. [2]
To give a numerical example, a 3% UK government war loan will trade at 50 pence per pound in a yield environment of 6%, while at 3% yield it is trading at par. That is, if the face value of the loan is £100 and the annual payment £3, the value of the loan is £50 when market interest rates are 6%, and £100 when they are 3%.
The duration, or the price-sensitivity to a small change in the interest rate r, of a perpetuity is given by the following formula: [3]
This of course follows the fact that for bigger changes the new price must be calculated with the present-value formula given that for changes greater than a few basis-points the calculated duration is not reflective of the true-change in price.
Valuing real estate with a capitalization rate or cap rate (the convention used in real estate finance) is a more current example. Using a cap rate, the value of a particular real estate asset is either the net income or the net cash flow of the property, divided by the cap rate. Effectively, the use of a cap rate to value a piece of real estate assumes that the current income from the property continues in perpetuity. Underlying this valuation is the assumption that rents will rise at the same rate as inflation. Although the property may be sold in future (or even the very near future), the assumption is that other investors will apply the same valuation approach to the property.
UK government perpetuities (called consols) were undated as well as irredeemable except by act of Parliament. As with war bonds, they paid fixed coupons (interest payments), and traded actively in the bond market until the government redeemed them in 2015. Very long dated bonds have financial characteristics that can appeal to some investors and in some circumstances: e.g. long-dated bonds have prices that change rapidly (either up or down) when yields change (fall or rise) in the financial markets.
The constant growth dividend discount model for the valuation of the common stock of a corporation is another example. This model assumes that the market price per share is equal to the discounted stream of all future dividends, which is assumed to be perpetual. If the discount rate for stocks (shares) with this level of systematic risk is 12.50%, then a constant perpetuity of dividend income per dollar is eight dollars. However, if the future dividends represent a perpetuity increasing at 5.00% per year, then the dividend discount model, in effect, subtracts 5.00% off the discount rate of 12.50% for 7.50% implying that the price per dollar of income is $13.33.
In finance, a bond is a type of security under which the issuer (debtor) owes the holder (creditor) a debt, and is obliged – depending on the terms – to provide cash flow to the creditor. The timing and the amount of cash flow provided varies, depending on the economic value that is emphasized upon, thus giving rise to different types of bonds. The interest is usually payable at fixed intervals: semiannual, annual, and less often at other periods. Thus, a bond is a form of loan or IOU. Bonds provide the borrower with external funds to finance long-term investments or, in the case of government bonds, to finance current expenditure.
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be equal or more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater than tomorrow. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of borrowed funds is less than the total amount of money paid to the lender.
The time value of money refers to the fact that there is normally a greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference.
In finance, a convertible bond, convertible note, or convertible debt is a type of bond that the holder can convert into a specified number of shares of common stock in the issuing company or cash of equal value. It is a hybrid security with debt- and equity-like features. It originated in the mid-19th century, and was used by early speculators such as Jacob Little and Daniel Drew to counter market cornering.
In finance, the yield on a security is a measure of the ex-ante return to a holder of the security. It is one component of return on an investment, the other component being the change in the market price of the security. It is a measure applied to fixed income securities, common stocks, preferred stocks, convertible stocks and bonds, annuities and real estate investments.
The yield to maturity (YTM), book yield or redemption yield of a fixed-interest security is an estimate of the total rate of return anticipated to be earned by an investor who buys it at a given market price, holds it to maturity, and receives all interest payments and the capital redemption on schedule.
Preferred stock is a component of share capital that may have any combination of features not possessed by common stock, including properties of both an equity and a debt instrument, and is generally considered a hybrid instrument. Preferred stocks are senior to common stock but subordinate to bonds in terms of claim and may have priority over common stock in the payment of dividends and upon liquidation. Terms of the preferred stock are described in the issuing company's articles of association or articles of incorporation.
Fixed income refers to any type of investment under which the borrower or issuer is obliged to make payments of a fixed amount on a fixed schedule. For example, the borrower may have to pay interest at a fixed rate once a year and repay the principal amount on maturity. Fixed-income securities can be contrasted with equity securities that create no obligation to pay dividends or any other form of income. Bonds carry a level of legal protections for investors that equity securities do not: in the event of a bankruptcy, bond holders would be repaid after liquidation of assets, whereas shareholders with stock often receive nothing.
Stock valuation is the method of calculating theoretical values of companies and their stocks. The main use of these methods is to predict future market prices, or more generally, potential market prices, and thus to profit from price movement – stocks that are judged undervalued are bought, while stocks that are judged overvalued are sold, in the expectation that undervalued stocks will overall rise in value, while overvalued stocks will generally decrease in value. A target price is a price at which an analyst believes a stock to be fairly valued relative to its projected and historical earnings.
Rational pricing is the assumption in financial economics that asset prices – and hence asset pricing models – will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.
Bond valuation is the process by which an investor arrives at an estimate of the theoretical fair value, or intrinsic worth, of a bond. As with any security or capital investment, the theoretical fair value of a bond is the present value of the stream of cash flows it is expected to generate. Hence, the value of a bond is obtained by discounting the bond's expected cash flows to the present using an appropriate discount rate.
In finance, the duration of a financial asset that consists of fixed cash flows, such as a bond, is the weighted average of the times until those fixed cash flows are received. When the price of 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.
A collateralized mortgage obligation (CMO) is a type of complex debt security that repackages and directs the payments of principal and interest from a collateral pool to different types and maturities of securities, thereby meeting investor needs.
The current yield, interest yield, income yield, flat yield, market yield, mark to market yield or running yield is a financial term used in reference to bonds and other fixed-interest securities such as gilts. It is the ratio of the annual interest (coupon) payment and the bond's price:
Floating rate notes (FRNs) are bonds that have a variable coupon, equal to a money market reference rate, like SOFR or federal funds rate, plus a quoted spread. The spread is a rate that remains constant. Almost all FRNs have quarterly coupons, i.e. they pay out interest every three months. At the beginning of each coupon period, the coupon is calculated by taking the fixing of the reference rate for that day and adding the spread. A typical coupon would look like 3 months USD SOFR +0.20%.
The duration of a stock is the average of the times until its cash flows are received, weighted by their present values. The most popular model of duration uses dividends as the cash flows. In vernacular, the duration of a stock is how long we need to receive dividends to be repaid the purchase price of the stock. If a stock doesn't pay dividends, other methods using distributable cash flows, may be utilized.
In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC.
A perpetual bond, also known colloquially as a perpetual or perp, is a bond with no maturity date, therefore allowing it to be treated as equity, not as debt. Issuers pay coupons on perpetual bonds forever, and they do not have to redeem the principal. Perpetual bond cash flows are, therefore, those of a perpetuity.
The following outline is provided as an overview of and topical guide to finance:
In finance, bootstrapping is a method for constructing a (zero-coupon) fixed-income yield curve from the prices of a set of coupon-bearing products, e.g. bonds and swaps.