reading 37 : INTEREST RATES

Types Of Interest Rates

        Three types of interest rates are particularly relevant in the interest rate derivative markets: Treasury rates, London Interbank Offered Rate (LIBOR) , and repo rates.
        Treasury rates (such as T-bill and T-bond rates) are often considered the benchmark for nominal risk-free rates.
         An overnight indexed swap (OIS) is an interest rate swap where a fixed rate is exchanged for a floating rate and where the floating rate is the geometric average of the overnight federal funds rates during the period. The fixed rate in the OIS is known as the OIS rate.

Compounding Frequencies

        If we have an initial investment of A that earns an annual rate R, compounded m times a year for n years, then it has a future value of:

        In most circumstances, rates are discretely compounded so we need to find the continuously compounded rate that gives the same future value. The continuous rate can be solved as follows:

Spot (Zero) Rates

        Zero (spot) rates correspond to the interest earned on a single cash flow at a single point in time. Bond prices are computed using the spot curve by discounting each cash flow at the appropriate spot rate.
         The yield of a bond is the single discount rate that equates the present value of a bond to its market price.
         Zero rates are computed using the bootstrapping methodology.

Forward Rates

        Forward rates are computed from spot rates. When the spot curve is upward-sloping, the corresponding forward rate curve is upward-sloping and above the spot curve. When the spot curve is downward-sloping, the corresponding forward rate curve is downward-sloping and below the spot curve.

Forward Rate Agreements (FRA)

        A forward-rate agreement is a contract between two parties that an interest rate will apply to a specific principal during some future time period.

Duration

        Duration and modified duration are the same when continuously compounded yields are used, and they both estimate the percentage price change of a bond from an absolute change in yield. Dollar duration is modified duration multiplied by the price of the bond.

Limitations Of Duration

        Duration is only good for relatively small changes in interest rates. As rate changes grow larger, the curvature of the bond price/yield relationship becomes more important, meaning that a linear estimate of price changes, such as duration, will contain errors. The amount of convexity in a bond can be measured and used to supplement duration in order to achieve a more accurate estimate of the change in price.

Calculation

        The approximate change in a bond’s price, B, for a parallel shift in the yield curve of Δy is:

         In order to obtain an estimate of the percentage change in price due to convexity, the following calculation will need to be made:

         Combining duration and convexity creates a more accurate estimate of the percentage change in the price of a bond:
                 percentage bond price change ≈ duration effect + convexity effect

Term Structure Theories

        The expectations theory suggests that forward rates correspond to expected future spot rates. The market segmentation theory states that bonds are segmented into different maturity sectors and that supply and demand dictate rates in the segmented maturity sectors. The liquidity preference theory suggests that longer-term rates incorporate a liquidity premium.