Fixed Income Term Structure
 Explained:

The present value of a dollar to be received in a year is less than the present value of that dollar if it were received today. We call this the time value of money. Financial markets use spot curves, forward curves, discount curves and yield curves to describe the time value of money. These are referred to collectively as the fixed income term structure. This article defines these notions.

A cash loan is a loan that commences immediately. A spot loan is a loan that commences spot. A forward loan is one that commences on some date later than spot. For example, in the Eurodollar markets a three-month spot loan commences in two business days (spot) and matures three months after that. A 27 forward loan commences two months from the spot date and lasts for five months. With either type of loan, interest can be paid periodically or it can be accumulated and paid at maturity.

A spot interest rate for maturity m is an interest rate payable on a spot loan of maturity m that accumulates interest to maturity. Spot rates are sometimes called zero-coupon rates because they are the rates of interest payable on obligations that accumulate all interest to maturity. Libor rates for maturities of a week or more are spot rates (GBP Libor is an exception). Exhibit 1 indicates USD Libor rates for monthly maturities as of March 1, 2004.

March 1, 2004
USD Libor

Exhibit 1

 1 month 1.1 2 month 1.11 3 month 1.12 4 month 1.13 5 month 1.15 6 month 1.17 7 month 1.19375 8 month 1.22125 9 month 1.25 10 month 1.29 11 month 1.32875 12 month 1.3675

Source: BBA

A spot curve (or zero-coupon curve) is a graph of spot rates as a function of maturity.

 Spot Curve Exhibit 2 Spot curve constructed from the Libor rates of Exhibit 1.

An n (n + m) forward rate is an interest rate payable on a forward loan that

commences n months from the spot date,

matures m months after that, and

accumulates interest to maturity.

If we have a spot curve, we can calculate forward rates. Suppose we want the 35 forward USD Libor rate for March 1, 2004. We can calculate this from the 3-month and 5-month spot Libor rates. Let r denote the desired forward rate. We use the fact that a 5-month spot loan is financially equivalent to a 3-month spot loan combined with a 35 forward loan. With Libor, simple compounding is used. Based on the 3-month and 5-month spot rates and day counts as of March 1, we conclude

 (1 + .0112(92/360)) (1 + r (61/360)) = (1 + .0115(153/360)) [1]

Solving for r, we obtain the forward rate as 1.19%. Note that this exceeds both the spot rates, which are 1.12% and 1.15%. This makes sense. If there are to be no arbitrage opportunities, the combined interest from the 3-month spot and forward loans must equal the interest earned on the 5-month spot loan. If the rate earned on the 3-month spot loan is lower than that earned on the 5-month spot loan, then the rate earned on the forward loan will have to be greater than that earned on the 5-month spot loan.

A forward curve is a graph of forward rates all for the same maturity but with different forward periods. For example, a forward curve might indicate rates for 03, 14, 25, 36, 47, ... , 120123 forward loans. This would be called a 3-month forward curve. Exhibit 3 indicates a 1-month forward curve calculated from the spot rates of Exhibit 1.

March 1, 2004
USD Forward Curve

Exhibit 3

 0x1 1.1 1x2 1.11927 2x3 1.13754 3x4 1.15735 4x5 1.22402 5x6 1.26254 6x7 1.33145 7x8 1.40115 8x9 1.47255 9x10 1.62928 10x11 1.69269 11x12 1.81135

One-month forward rates calculated from the spot Libor rates of Exhibit 1.

These are graphed in Exhibit 4.

 Forward Curve Exhibit 4 One month forward curve constructed from the spot Libor rates of Exhibit 1. The curve is superimposed over the spot curve of Exhibit 2.

Note that spot and forward curves provide identical information. If you have one, you can construct the other.

A third, also equivalent way to indicate the time value of money is discount factors. When we calculate the present value of some future cash flow, we are said to discount that future cash flow. A discount factor is the factor by which the future cash flow must be multiplied to obtain the present value. For example, if a EUR 100 payment to be made at maturity m has present value EUR 89.4, the EUR discount factor for maturity m is .894. Note that present values are often calculated with a spot value date. If this is the case, discount factors reflect discounting to the spot date as opposed to the current date.

Discount factors can be calculated from spot or forward rates. As an example, from Exhibit 1, the March 1, 2004 USD spot 6-month Libor rate was 1.17%. We calculate the corresponding discount factor as

 1 / (1 + .0117(184/360)) = .99406 [2]

This represents discounting from the date six months after spot back to the spot date.

A discount curve is a graph of discount factors for different maturities. Exhibit 5 indicates discount factors calculated from the spot Libor rates of Exhibit 1. These are graphed in Exhibit 6.

March 1, 2004
USD Discount Curve

Exhibit 5

 1 month 0.99905 2 month 0.99812 3 month 0.99715 4 month 0.99619 5 month 0.99514 6 month 0.99406 7 month 0.99295 8 month 0.99176 9 month 0.99054 10 month 0.98915 11 month 0.98771 12 month 0.98633

Discount factors calculated from the spot Libor rates of Exhibit 1.

 Discount Curve Exhibit 6 Discount curve constructed from the spot Libor rates of Exhibit 1.

The fourth way the time value of money can be described is with a yield curve. This is simply a graph of bond yields for various maturities. The curve is typically fit in some manner to price data for bonds of various maturities trading close to par and generally of the same credit quality. Yield curves are falling out of use today. Widespread use of computers in finance makes spot curves, forward curves and discount curves easier to construct and use in pricing work. Also, while yields continue to be widely quoted for bonds, fixed income markets are increasingly trading instruments other than bonds for which yield is either a meaningless or not useful notion. Today, when people speak of yield curves, they often mean spot curves.

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