The internal rate of return or IRR is the rate that will discount all cash inflows and outflows to a net present value of $0. In other words, the IRR model provides you with the true, effective interest rate being earned on a project after taking into consideration the time periods when the various cash amounts are flowing in or out. If you use present value tables to calculate the internal rate of return, it will require some trial and error or iterations to determine the exact rate the project is earning. Software or some financial calculators will provide a quicker and more accurate answer.

The net present value (NPV) discounts all of the cash inflows and outflows by a specified interest rate. The net amount of all of the discounted amounts is the net present value. If the net present value is $0, the project is expected to earn exactly the specified rate. If the net present value is a positive amount, the project will be earning more than the specified interest rate. A negative net present value means the project is expected to earn less than the specified interest rate.

Learn more about the internal rate of return and the net present value at Evaluating Business Investments.

The bond premium is the present value of both the future interest payments and the maturity amount minus the bond’s undiscounted maturity amount. In short, the bond market is very efficient. If a bond’s actual interest payments will be greater then the interest payments expected by the market, the bond will sell for more than the bond’s maturity amount. If the bond’s interest payments will be lower than the interest payments expected by the market, the bond will sell for less than the bond’s maturity amount. The difference (premium or discount) is computed by discounting all of the future cash amounts.

Paying a small premium is not unusual. Remember that the premium is directly related to the higher interest amounts you will be receiving. If the bond purchase will result in a large premium, you should investigate whether the bond is callable and its call price. The call price may limit the amount of premium that you are willing to pay.

Learn more about discounting future cash amounts at Present Value of a Single Amount and Present Value of an Ordinary Annuity.

Net present value is the present value of the cash inflows minus the present value of the cash outflows. For example, let’s assume that an investment of $5,000 today will result in one cash receipt of $10,000 at the end of 5 years. If the investor requires a 10% annual return compounded annually, the net present value of the investment is $1,620. This is the result of the present value of the cash inflow $6,210 (from above) minus the present value of the $5,000 cash outflow. (Since the $5,000 cash outflow occurred at the present time, its present value is $5,000.)

Present value calculations can be seen at Present Value of a Single Amount and Present Value of an Ordinary Annuity Net present value calculations can be seen at Evaluating Business Investments.

Under accrual accounting, revenues and expenses are reported based on accounting principles. This means that revenues are reported when they are earned, and expenses are matched to the periods of the revenue. In other words, revenues and expenses are not reported on the income statement when the money is received or spent. Further, the revenue and expense amounts are not adjusted for the time value of money because of the monetary unit assumption.

AccountingCoach.com contains explanations, drills, and crossword puzzles of the Present Value of a Single Amount and the Present Value of an Ordinary Annuity.

For example, a corporation might issue 9% $100 Preferred Stock. The stock agreement (indenture) states that the stock is callable by the corporation after three years at $109 per share plus any accrued interest. If in the fourth year, market interest rates decline to say 7%, the corporation can call-in the preferred stock by paying the call price of $109 plus any accrued interest.

The callable feature allows the corporation to get out of the preferred stock agreement requiring it to pay 9% interest. In turn, the stockholders will be deprived of receiving the 9% interest in a 7% market. The call price has the effect of limiting how high the market value of preferred stock will rise.

Learn more about Stockholders’ Equity.

The future cash receipts for a typical bond are the semiannual interest payments (interest rate stated on the bond X face amount of bond X 1/2 year) and the maturity amount of the bond. The interest payments form an ordinary annuity and the maturity amount is a single payment.

If the bond’s stated interest rate is less than the current market interest rate, the bond’s market value is less than the maturity or face amount of the bond. In financial jargon, the bond will sell at a discount.

If the bond’s interest rate is more than the current market interest rate, the bond will have a market value that is more than the maturity or face amount of the bond. In other words, the bond will sell at a premium.

The payback method simply computes the number of years it will take for an investment to return cash equal to the amount invested. For example, if an investment of $100,000 is made in January 2007 and it generates cash of $50,000 for two years followed by $10,000 per year for four additional years, its payback is two years ($50,000 + $50,000). If another investment of $100,000 generates cash of $20,000 per year for two years and then provides cash of $40,000 per year for six additional years, its payback is approximately 3.5 years ($20,000 + $20,000 + $40,000 + 0.5 times $40,000).

As you can see in the examples, payback only answers one question: How long before the cash invested is returned? Payback does not address which investment is more profitable. Payback not only ignored the time value of money, it ignored all of the cash received after the payback period.

The accounting rate of return or return on investment (ROI) are two more examples of methods used in capital budgeting that does not involve discounting future cash amounts.

To overcome the shortcomings of payback, accounting rate of return, and return on investment, capital budgeting should include techniques that consider the time value of money. Two of these methods include (1) the net present value method, and (2) the internal rate of return calculation. Under these techniques, the future cash flows are discounted. This means that each dollar in the distant future will be less valuable than each dollar in the near future, and both of these will have less value than each dollar invested in the present.

The semiannual interest payments are considered an annuity, since the bond’s stated interest rate and the resulting interest payments do not change even though the market rates are changing each hour. The annuity consisting of the semiannual interest payments will be discounted by the current market interest rate (the rate an investor desires). The lump sum at maturity is a single payment and that too will be discounted by the same market rate used to discount the interest payments. The resulting present value of the lump sum plus the present value of the interest annuity will be the present value and the market value of the bond.

When market interest rates increase, the locked-in or fixed interest payments promised in the bond will become less attractive. Hence the present/market value of the bond will decline as the market demands higher interest payments. If market interest rates decrease, the locked-in or fixed interest payments promised in the bond will become more attractive and will result in a higher present/market value.

One caution for potential investors in bonds: Bonds will likely have a call feature. This means the issuer of the bond has the option to call in the bonds at a specified amount, such as 106% of its face value. This could mean a limit to the upside gain for an investor if interest rates decline. You should discuss this further with your financial adviser.

PVOA = PMT x PVOA factor for n=6, i=?;

$5,000 = $1,000 x PVOA factor for n=6, i=?;

$5,000/$1,000 = PVOA factor for n=6, i=?;

5.000 = PVOA factor for n=6, i=?

The factor 5.000 appears in the row n=6 where i = 5.5%. Hence, this loan has an implicit interest rate of 5.5%.

For more information on present value and present value tables, see Present Value of a Single Amount and Present Value of an Ordinary Annuity.