• <b>United States Federal Reserve</b>

Money and Banking


1. Mathematics of Debt Instruments


In this section, we discuss the mathematics behind present values and future values of debt instruments. A debt instrument is a loan contract that is sold for some dollar amount in the current period, and is repaid with one or more cash flows over time. The present value of a debt instrument is today's value for the promise of some given stream of future cash flows. The present value calculation discounts future cash flows using the interest rate, because money set aside today can earn interest and grow to larger amounts in the future. We also use the related concept of the future value to determine what a savings account or other financial investment will be worth in the future, given some series of cash flows paid to the account over time, and the interest earned on the account.


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Present and Future Values of a Simple Loan

In this Pencast, we illustrate the concepts of present values and future values using the most simple type of debt instrument, a simple loan. [Play Pencast]


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Present and Future Values: Numerical Examples

In this Pencast, we walk through two worked out numerical examples. In the first, we have a scenario that requires us to calculate a future value. In the second, we have a scenario that requires us to calculate a present value. [Play Pencast]


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Fixed Payment Loans

In this Pencast, we introduce a the fixed payment loan debt instrument and show intuitively some of the math behind the present value calculation. [Play Pencast]


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Present Values with a Geometric Series

In this Pencast, we introduce the geometric series which is mathematical shortcut for specific types of long summations, like those involved in the calculation of a net present value with a large number of cash flows occurring over a long time period. We show how the geometric series can be used to calculate the present value of a the fixed payment loan example from the Pencast above. [Play Pencast]


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Example: Computing Present Value for a Fixed Payment Loans

In this Pencast, we walk through a worked out example problem that uses the geometric series to compute the present value of a fixed payment loan. [Play Pencast]


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Future Values with a Geometric Series

In this Pencast, we introduce the meaning and computations for the future value. The future value is often used in applications one pays a series of cash flows into a savings account, or other financial savings instruments, like a retirement account. Given the series of cash flows occurring over time, and the interest that accumulates for the growing balance each period, the future value reveals what the account will be worth at some time in the future. We show how the calculation for the future value is related to the calculation for the present value and we work through an example problem. [Play Pencast]


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Another Example: Future Values with a Geometric Series

In this Pencast, we work through another example problem involving the geometric series and future values. [Play Pencast]


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Introduction to Coupon Bonds

In this Pencast, we introduce the coupon bond, which is another type of debt instrument that is closely related to a fixed payment loan. We walk through a worked-out example calculation for the present value of a coupon bond. [Play Pencast]


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Introduction to Discount Bonds

In this Pencast, we introduce the discount bond, another common type of debt instrument. U.S. Federal Government bonds and many sovereign government bonds issues by countries around the world are discount bonds. These are bonds that promise to make only one cash flow at some date in the future, and are sold present day for a nominal value less than that amount (hence the name, discount). We walk through a worked-out example calculation for the present value of a discount bond. [Play Pencast]


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Yield to Maturity

In this Pencast, we introduce the concept of the yield to maturity on an asset, which is the annual rate of interest which is earned from owning an asset. Mathematically, it is the interest rate that equates the present value of cash flows received from owning an asset with its current day price. The yield to maturity cannot always be computed by hand. The ability to do so depends on the complexity of the present value calculation. Discount bonds have a very easy present value calculation, so we use this debt instrument as an example to compute the yield to maturity. [Play Pencast]


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Rate of Return on Bonds

Because many debt instruments are securities (i.e. financial assets that can be bought and sold on secondary markets), the rate of return from holding a bond is not only equal to the interest earned on the bond, but also includes possible capital gains or capital losses, should the bond be sold on the secondary market for more or less than the purchase price. In this Pencast, we show how to calculate the total return from holding the bond, and how the formula can be decomposed into the interest return and the capital gain/loss. [Play Pencast]


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Secondary Market Arithmetic

Many debt instruments are securities, which means they can be bought and sold on secondary markets, and financial investors can earn capital gains/losses in addition to interest income. The size of the capital gains or capital losses depends on how the price of the debt instrument changes. The price of the security depends on the expected present value of the asset, and therefore depends on the interest rate. In this Pencast, we show some mathematics that illustrates how the price of a bond changes when the interest rate changes. [Play Pencast]


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