Interest rate is a cornerstone variable in the pricing of nearly all derivatives. This blog post aims to give a brief overview on how interest rates are measured and analyzed with the application to the bond pricing.

An interest rate is usually defined as the cost of the borrowed funds, expressed as a percentage of the loan amount. There is a variety of interest rates for any given currency. Credit risk play the key role in determination of the interest rate applicable in the lending event. It is interpreted as the risk that a creditor will not receive a principal and the interest payment of the funds borrowed. One of the main rates used in finance are:

- Treasury rates (Rate an investor earns on Treasury Bills and Treasury Bonds)
- LIBOR rates (A short-term borrowing rate among banks)
- Repurchase Agreement (A rate for dealers in government securities)
- The Fed Funds Rate (Overnight rate among banks)

In the derivatives pricing, the so called* risk-free rate *is used. It is usually argued that a riskless derivatives portfolio should provide the return equal to the risk-free rate. In practice, there is no agreed rate which is used in the context of being risk-free. Depending on the situation, the LIBOR or overnight indexed swap (OIS) rates are used for this purpose. One example of how the OIS is used can be found here.

## Interest Rate Measurements

A compound interest is the main result used in the financial world and has a widely known formula \[A_0(1+\frac{r}{m})^{mn}\] where an amount $A_0$ is invested for $n$ years at the interest rate of $r$ per annum. To generalize a result, the compounding frequency $m$ is used (e.g. monthly).

The limit of the compound interest formula as $m$ tends to infinity is called a *continuous compounding. *It simplifies the formula to \[A_0 e^{rn}\]

This result is the widely used in the derivatives valuation, especially in the form of the discounting factor $e^{-rn}$. This formula is often used in the context of the $n$-year *zero rate* (Total rate of interest earned on the investment lasting $n$ years without additional payments).

## Bond Valuation

On of the simplest securities that are priced via interest rates are bonds. Bond holders periodically receive payments called coupons. The face value (known as principal or par value) is received at the bond's maturity. In theory, the price of the bond is equal to the sum of all future cash flows discounted to present by the separate risk-free zero rate. The most popular way to determine the zero rate called bootstrap method and is described here.

Mathematically, price of the bond $B$ with the cash flows $c_i$ paid to the holder at the time $t_i$, where $i=1..n$ and $r(0,t_i)$ is the continuously compounded zero rate is \[B=\sum_{i=1}^{n}c_ie^{-r(0,t_i)t_i}\] If the instantaneous interest rate curve r(t)is known, the formula can be expanded to \[B=\sum_{i=1}^{n}c_i\space\exp\left ( -\int_{0}^{t_i}r(\tau )d\tau \right )\]

### Bond and Par Yields

Bond Yield the constant rate at which the sum of the discounted future cash flows of the bond is equal to the market price of the bond. This rate is found numerically (e.g. Newton method applied to the non-linear equation $B(r)=0$), similarly to the procedure of founding Internal Rate of Return (IRR) in Finance. Par yield is the coupon rate that makes the par value of the bond equal to its price. Another important rate used in finance is called forward rate and is explained here.

### Duration

The duration of the Bond simply measures how long on average the holder of the bond will wait until receiving cash payments. A zero-coupon bond that lasts for $n$ years has a duration of $n$ years, while a bond with coupons has a duration less then $n$ years. The duration, $D$, of the bond is expressed as \[D=\sum_{i=1}^{n}t_i\space\left [\frac{c_ie^{-rt_i}}{B}\right]\] with the same definitions as in a mathematical extract on bond valuation.

As can be seen, the duration of the bond is simply the weighted average of the series of payments, where the weight is equal to the present value of the cash flow at the given time point (Discount rate is assumed to be the bond yield). The approximation of the relationship between the yield and the bond price can be provided as \[\frac{\Delta B}{B}=-D\Delta r\] where $r$ is the bond yield. It can be further modified to change continuous compounding to any compounding frequency $m$.

One of the application of the duration measure can be the elimination of the exposure to the small changes in the yield curve. By having a portfolio of assets with the same duration as the total liabilities, one can reduce such risk and be vulnerable only to the large shifts in yield curve.

### Convexity

While the approximation formula of the duration given above showing good results with small changes in yield curve, larger moves require more reliable measure to capture bond price changes. When two bond portfolios have the same duration, the large change in the yield curve can result in the different percentage change of the portfolio values. The extent of the curvature of the relationship is known as convexity and is measured as follows \[C=\frac{1}{B}\frac{d^{2}B}{dy^{2}}\] If we approximate the series using Taylor expansion, a more reliable measure can be derived \[\frac{\Delta B}{B}=-D\Delta r+\frac{1}{2}C(\Delta r)^{2}\] Having a portfolio of assets and liabilities with the same convexity and duration measures, one can be safely protected from a large dynamics of a yield curve.

## Conclusion

For trades utilizing derivatives in their trading, Treasury and LIBOR rates can be reliable used as a proxy to the risk-free rate. Two important concepts when dealing with interest rate products are convexity and duration, which basically measure sensitivity of the bond portfolio to the changes in yield curve. Finally, while the topic can be very technical in nature, the basic understanding is required by all derivatives trades, who wish to expand their knowledge on the valuation of the products they trade. Feel free to suggest topics which you want me to cover in the comments section below!