# Calculating the Yield of an Annuity

## The yield of an annuity is commonly found using either the percent change in the value from PV to FV, or the internal rate of return.

#### Key Points

• The yield of an annuity may be found by discounting to find the PV, and then finding the percentage change from the PV to the FV.

• The Internal Rate of Return (IRR) is the discount rate at which the NPV of an investment equals 0.

• The IRR calculates an annualized yield of an annuity.

#### Terms

• The discount rate that will cause the NPV of an investment to equal 0.

• The present value of a project or an investment decision determined by summing the discounted incoming and outgoing future cash flows resulting from the decision.

• In finance, the term yield describes the amount in cash that returns to the owners of a security. Normally it does not include the price variations, at the difference of the total return. Yield applies to various stated rates of return on stocks (common and preferred, and convertible), fixed income instruments (bonds, notes, bills, strips, zero coupon), and some other investment type insurance products

#### Figures

1. ##### Percent Change

The PV of the annuity is V1 and the FV is V2. This measured by what percentage the FV is different from the PV.

2. ##### Internal Rate of Return

The internal rate of return (IRR) is the interest rate that will cause the NPV to be 0.

3. ##### IRR Example

The setup to find the IRR of the investment with cash flows of -4000, 1200, 1410, 1875, and 1050. By setting NPV = 0 and solving for r, you can find the IRR of this investment.

The yield of annuity can be calculated in similar ways to the yield for a single payment, but two methods are most common.

The first is the standard percentage-change method. Figure 1 Just as for a single payment, this method calculated the percentage difference between the FV and the PV. Since annuities include multiple payments over the lifetime of the investment, the PV (or V1 in Figure 1 is the present value of the entire investment, not just the first payment.

The second popular method is called the internal rate of return (IRR). The IRR is the interest rate (or discount rate) that causes the Net Present Value (NPV) of the annuity to equal 0.Figure 2 That means that the PV of the cash outflows equals the PV of the cash inflows. The higher the IRR, the more desirable is the investment. In theory, you should make investment with an IRR greater than the cost of capital.

Let's take an example investment: It is not technically an annuity because the payments vary, but still is a good example for how to find IRR:

Suppose you have a potential investment that would require you to make a $4,000 investment today, but would return cash flows of$1,200, $1,410,$1,875, and \$1,050 in the four successive years. This investment has an implicit rate of return, but you don't know what it is. You plug the numbers into the NPV formula and set NPV equal to 0. Figure 3 You then solve for r, which is your IRR (it's not easy to solve this problem by hand. You will likely need to use a business calculator or Excel). When r = 14.3%, NPV = 0, so therefore the IRR of the investment is 14.3%.

#### Key Term Glossary

annuity
A specified income payable at stated intervals for a fixed or a contingent period, often for the recipient’s life, in consideration of a stipulated premium paid either in prior installment payments or in a single payment. For example, a retirement annuity paid to a public officer following his or her retirement.
##### Appears in these related concepts:
capital
Money and wealth; the means to acquire goods and services, especially in a non-barter system.
##### Appears in these related concepts:
cash flow
The sum of cash revenues and expenditures over a period of time.
##### Appears in these related concepts:
cash inflow
Cash that is received by the investor. For example, dividends paid on a stock owned by the investor is a cash inflow.
##### Appears in these related concepts:
cash outflow
Any cash that is spent or invested by the investor.
##### Appears in these related concepts:
cost of capital
the rate of return that capital could be expected to earn in an alternative investment of equivalent risk
##### Appears in these related concepts:
Cost of Capital
The rate of return that capital could be expected to earn in an alternative investment of equivalent risk.
##### Appears in these related concepts:
discount
To find the value of a sum of money at some earlier point in time. To find the present value.
##### Appears in these related concepts:
discount rate
The interest rate used to discount future cash flows of a financial instrument; the annual interest rate used to decrease the amounts of future cash flow to yield their present value.
##### Appears in these related concepts:
interest
The price paid for obtaining, or price received for providing, money or goods in a credit transaction, calculated as a fraction of the amount or value of what was borrowed.
##### Appears in these related concepts:
interest rate
The percentage of an amount of money charged for its use per some period of time. It can also be thought of as the cost of not having money for one period, or the amount paid on an investment per year.
##### Appears in these related concepts:
investment
A placement of capital in expectation of deriving income or profit from its use.
##### Appears in these related concepts:
net present value
the present value of a project or an investment decision determined by summing the discounted incoming and outgoing future cash flows resulting from the decision
##### Appears in these related concepts:
Present value
Present value, also known as present discounted value, is the value on a given date of a payment or series of payments made at other times.
##### Appears in these related concepts:
Present Value
Also known as present discounted value, is the value on a given date of a payment or series of payments made at other times. If the payments are in the future, they are discounted to reflect the time value of money and other factors such as investment risk. If they are in the past, their value is correspondingly enhanced to reflect that those payments have been (or could have been) earning interest in the intervening time. Present value calculations are widely used in business and economics to provide a means to compare cash flows at different times on a meaningful "like to like" basis.
##### Appears in these related concepts:
return
Gain or loss from an investment.
##### Appears in these related concepts:
Yield
In finance, the term yield describes the amount in cash that returns to the owners of a security. Normally it does not include the price variations, at the difference of the total return. Yield applies to various stated rates of return on stocks (common and preferred, and convertible), fixed income instruments (bonds, notes, bills, strips, zero coupon), and some other investment type insurance products