**The need for Market Valuation of your portfolio….** FEDERAL ACCOUNTING STANDARDS ADVISORY BOARD SFFAS 1 – Accounting for Selected Assets and Liabilities 72. Disclosure of market value. For investments in Market-based and marketable Treasury securities, the market valuation should be disclosed.

**From the FedInvest system you can select Prior Days Prices** which takes you to a listing of price files. You can choose the price file for the particular day you wish to value your portfolio.

**Once you choose the price file you use the End of Day price** to calculate your market value.

**Duration**

**We know:** • An increase in interest rates causes bond prices to fall, and a decrease in interest rates causes bond prices to rise. • We also know that longer maturity debt securities tend to be more volatile in price. • For a given change in interest rates, the price of a longer term bond generally changes more than the price of a shorter term bond.

**A 10 year zero coupon bond makes all of its payments at the** end of the term. • Two bonds with the same term to maturity do not have the same interest-rate risk. • A 10 year coupon bond makes payments before the maturity date.

**When interest rates rise, the prices of low coupon** securities tend to fall faster than the prices of high coupon securities. • Similarly, when interest rates decline, the prices of low coupon rate securities tend to rise faster than the prices of high coupon rate securities.

**Knowledge of the impact of varying coupon rates on security** price volatility led to the development of a new index of maturity other than straight calendar time. • The new measure permits analysts to construct a linear relationship between term to maturity and security price volatility, regardless of differing coupon rates.

**Duration…** is the measure of the price sensitivity of a fixed-income security to an interest rate change of 100 basis points. The calculation is based on the weighted average of the present values for all cash flows. Duration is measured in years; however, do not confuse it with a bond’s maturity. For all bonds, duration is shorter than maturity except zero coupon bonds, whose duration is equal to maturity. This is because all cash flows are received at maturity.

**The term “duration,” having a special meaning in the** context of bonds, is a measurement of how long in years it takes for the price of a bond to be repaid by its internal cash flows. It is an important measure for investors to consider, as bonds with higher durations are more risky and have higher price volatility than bonds with lower durations. For each of the two basic types of bonds the duration is the following: 1. Zero-coupon bond – Duration is equal to its time to maturity. 2. Straight bond – Duration will always be less than its time to maturity. Here are some visual models that demonstrate the properties of duration for a zero-coupon bond and a straight bond.

**Duration of a Zero-Coupon Bond** The red lever above represents the four-year time period it takes for a zero coupon to mature. The money bag balancing on the far right represents the future value of the bond, the amount that will be paid to the bondholder at maturity. The fulcrum, or the point holding the lever, represents duration, which must be positioned where the red lever is balanced. The fulcrum balances the red lever at the point on the time line when the amount paid for the bond and the cash flow received from the bond are equal. Since the entire cash flow of a zero-coupon bond occurs at maturity, the fulcrum is located directly below this one payment.

**Duration of a Straight Bond** Consider a straight bond that pays coupons annually and matures in five years. Its cash flows consist of five annual coupon payments and the last payment includes the face value of the bond. The moneybags represent the cash flows you will receive over the five-year period. To balance the red lever (at the point where total cash flows equal the amount paid for the bond), the fulcrum must be further to the left, at a point before maturity. Unlike the zero-coupon bond, the straight bond pays coupon payments throughout its life and therefore repays the full amount paid for the bond sooner.

**Factors Affecting Duration** It is important to note, however, that duration changes as the coupons are paid to the bondholder. As the bondholder receives a coupon payment, the amount of the cash flow is no longer on the timeline, which means it is no longer counted as a future cash flow that goes towards repaying the bondholder. Our model of the fulcrum demonstrates this: as the first coupon payment is removed from the red lever (paid to the bondholder), the lever is no longer in balance (because the coupon payment is no longer counted as a future cash flow).

**The fulcrum must now move to the right in order to balance** the lever again: Duration increases immediately on the day a coupon is paid, but throughout the life of the bond, the duration is continually decreasing as time to the bond’s maturity decreases. The movement of time is represented above as the shortening of the red lever: notice how the first duration had five payment periods and the above diagram has only four. This shortening of the timeline, however, occurs gradually, and as it does, duration continually decreases. So, in summary, duration is decreasing as time moves closer to maturity, but duration also increases momentarily on the day a coupon is paid and removed from the series of future cash flows – all this occurs until duration, as it does for a zero-coupon bond, eventually converges with the bond’s maturity.

**Duartion – Other factors:** Coupon rate and Yield also affect the bond’s duration. Bonds with high coupon rates and in turn high yields will tend to have a lower duration than bonds that pay low coupon rates, or offer a low yield. This makes sense, since when a bond pays a higher coupon rate the holder of the security received repayment for the security at a faster rate. The diagram below summarizes how duration changes with coupon rate and yield.

**Macaulay Duration** The formula usually used to calculate a bond’s basic duration is the Macaulay duration, which was created by Frederick Macaulay in 1938 but not commonly used until the 1970s. Macaulay duration is calculated by adding the results of multiplying the present value of each cash flow by the time it is received, and dividing by the total price of the security. The formula for Macaulay duration is as follows: n = number of cash flows t = time to maturity C = cash flow i = yield to maturity M = maturity par value Let’s go through an example:

**If you hold a five-year bond with a par value of $1,000 and** a coupon rate of 5%. For simplicity, assume that the bond is paid annually and that interest rates are 3% (yield). n = number of cash flows t = time to maturity C = cash flow i = yield to maturity M = maturity par value Fortunately if you are seeking the Macaulay duration of a zero-coupon bond, the duration would be equal to the bond’s maturity, so there is no calculation required.

**Therefore…the lower the coupon rate, the higher the** duration of the bond.

**Coupon Bonds: duration is shorter than maturity** • Discount bonds (yield is greater than coupon): duration increases at a decreasing rate up to a point, after which it declines • Par value bonds: duration increases with maturity. • Premium bonds (yield is less than coupon): duration increases throughout but at a lesser rate than with a par value bond.

**Duration depends on yield-to-maturity.** • The higher the yield the shorter the duration, other things being equal.

**In Treasury bonds, the only source of risk stems from** interest rate changes. • Duration is a measure of this source of risk. • Duration allows bonds of different maturities and coupon rates to be directly compared.

**@DURATION(settlement;maturity;coupon;yield;[frequency];[basis])** calculates the annual duration for a security that pays periodic interest. Example A security has a July 1, 1993, settlement date and a December 1, 1998, maturity date. The semiannual coupon rate is 5.50% and the annual yield is 5.61%. The bond has a 30/360 day-count basis. To determine the security's annual duration: @DURATION(@DATE(93;7;1);@DATE(98;12;1);0.055;0.0561;2;0) = 4.734591

**DURATION(settlement,maturity,coupon yld,frequency,basis)** Example A bond has the following terms: January 1, 1998, settlement dateJanuary 1, 2006, maturity date8 percent coupon9.0 percent yieldFrequency is semiannualActual/actual basis The duration (in the 1900 date system) is: DURATION("1/1/1998","1/1/2006",0.08,0.09,2,1) equals 5.993775

**http://www.investopedia.com/calculator/MDuration.aspx**

**Email Fedinvestor@bpd.treas.gov if you would like to receive** either or both of the reports. Include which report (Market Valuation and/or Duration) you would like to receive, the Account Fund Symbol(s), and a date for which you want the information. Questions?