The interest rate is the most important variable in the financial market. Only by understanding the interest rate can we understand the return on investment. This article will briefly introduce some of the most basic questions about interest rates (zero coupon rate, yield to maturity), and briefly introduce some of the most important interest rates in asset pricing (Treasury bond yield curve, OIS zero coupon rate) curve).

## Zero rate vs yield to maturity

If you know anything about bonds, you know that basic bonds are divided into zero-coupon and coupon bonds (along with convertible bonds with embedded options, callable bonds, etc.).

Zero-coupon bonds do not pay interest, and the bid-ask spread constitutes investment income. Zero-coupon bonds are usually issued at a discount. If you buy a zero-coupon bond at the price of Bz, and hold it until maturity (experience from purchase time to bond maturity) After time T (the unit of T is years)), the return on the face value F of the zero-coupon bond is obtained at maturity (without considering transaction costs and taxes), then what is the rate of return R of this investment?

Obviously, Bz*(1+R)^T=F, and R in the equation is the zero coupon rate.

Consider a series of zero-coupon bonds with different maturities (T) from the same issuer (e.g. government, a company), overnight, 1 week, 2 weeks, 1 month, 3 months, 6 months, 9 months, 1 year …to get a zero-coupon curve. The zero coupon rate curve measures the cost of capital (the cost of capital changes over time) over a series of investment cycles.

Coupon bonds promise to pay investors a fixed percentage of the face value at a fixed time (usually twice a year). This fixed percentage is called the coupon rate. reason. How are coupon bonds priced? How to measure the investment yield of coupon bonds?

Since the cash flow of the coupon bond is distributed at multiple points in time, the discounted cash flow (Discount Cash Flow) method can be used to price according to the zero-coupon interest rate curve that matches the risk.

In the above formula, Bc is the coupon bond price, Ci is the cash flow at a certain time ti, and e^(-ri*ti) is the discount factor (e^(-ri*ti) discounted with the zero coupon rate ri at a certain time ti ri*ti) is a continuous interest rate discount factor, which is only different in form from the usual (1+ri)^(-ti) discount factor). The above formula shows that the coupon bond price is equal to the sum of the discounted cash flows.

The price of a coupon bond that is fairly priced should be the result of the above formula. Since cash flows at different time points are discounted at different zero coupon rates, pricing requires a series of discount rates. We can use a discount rate instead of this. A series of discounted interest rates, this is the yield to maturity (yield for short). In the actual market, we can see the price of the coupon bond, and know all the relevant data such as the coupon rate, bond duration, interest frequency, etc., we can get the yield to maturity.

The difference between the above formula and the previous formula is that the yield to maturity y does not change with time during the term T, while ri is a series of zero coupon rates that change with time. In essence, the yield to maturity y can be understood as the geometric mean of a series of zero coupon rates ri in the bond term T.

Similar to the zero-coupon rate, we can also get a yield-to-maturity curve (referred to as the yield curve) through a series of coupon bonds with different maturities T. The yield-to-maturity curve of national government bonds is the most commonly used curve in the market.

The Zero Rate Curve and the Yield-to-Maturity Curve are both discount rates commonly used in Asset Pricing. The two are equivalent for Discount-Cash-Flow Valuation.

## Risk-Free Rates: Treasury Yield Curve vs OIS Zero Coupon Rate Curve

The Treasury bond yield curve is a common tool in the field of stock valuation. For the U.S. market, the 10-year Treasury bond yield is the market-recognized risk-free rate. However, in the field of derivatives valuation, the OIS zero-coupon curve plays a more important role. The yield curve of treasury bonds is selected for stock valuation because the interest rate of treasury bonds reflects the opportunity cost of investing in stocks (the risk premium of stock investment relative to treasury bond investment is an important reference standard for stock investors); the OIS zero-coupon interest rate curve is selected for derivative product valuation. Because OIS roughly reflects the true cost of capital for derivatives market participants.

OIS is the abbreviation of Overnight Index Swap (Overnight Index Swap). The word Swap means exchange, also known as swap. This article uses the word swap to facilitate understanding (“swap” implies “swap period”. mean).

How to understand swap (Swap)? The simplest swap contract is a fixed interest rate-floating interest rate swap. The fixed interest payer pays the floating interest payer the fixed interest of the same principal at a certain time in the future, and the floating interest payer pays the floating interest at the same future time points. The floating interest (which may refer to a market interest rate at that time, such as Libor) on the same principal is paid to the fixed interest payer, which is actually a net settlement. The fixed rate is called the swap rate. If the swap is fairly priced, the swap rate to some extent reflects the corresponding floating rate expectations.

OIS is a swap of a fixed rate with a series of floating overnight rates, that is, the OIS rate reflects expectations of a future series of overnight rates.

There are many options for floating interest rates for OIS. For the U.S. dollar, the most commonly used are the Effective Federal Funds Rate (EFFR) and the Secured Overnight Financing Rate (SOFR). Among them, EFFR is the unsecured overnight rate, slightly higher than SOFR.

Take, for example, the 1-month OIS zero coupon rate based on the effective federal funds rate, which is the expected geometric mean of the EFFR for each day in the next month. Interest rates usually refer to annual interest rates, which is consistent with the unit of term T being in years.

What is the difference between 1-Month OIS Zero Coupon and 1-Month Fed Funds Futures based on the effective fed funds rate? The difference is that the former is the expected rate of return for one month of continuous investment of funds at the effective federal funds rate (the federal funds rate is the overnight rate) from now on, and the latter is the federal funds rate (the nature of interest rate futures) one month later, which can be used to predict the probability of a Fed rate hike.

Similar to the above, the OIS zero-coupon rate can be extended to a zero-coupon rate curve with many tenors. The OIS-EFFR zero-coupon interest rate curve and the OIS-SOFR zero-coupon interest rate curve are the most important reference rates for the pricing of USD derivatives. The former reflects the cost of funding for unsecured loans of most derivatives market participants, and the latter reflects the cost of funding for their secured loans, which is slightly lower than the former. Both curves are available through the Bloomberg terminal.

## Summarize

The treasury bond yield curve is the reference risk-free rate for stock valuation, and the OIS zero-coupon rate curve is the reference risk-free rate for derivative product pricing. We can use these two curves to price most financial assets in order to value an investment (everything in this article is based on nominal returns and does not take into account real returns that are not affected by inflation).

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