 # Bonds ~Spot Rate~

Am confused with Spot Rate in context with Bonds. Current yield make sense to me and so do yield-to-maturity(YTM). where YTM = c(1 + r)-1 + c(1 + r)-2 + . . . + c(1 + r)-Y + B(1 + r)-Y = P where c = annual coupon payment (in dollars, not a percent) Y = number of years to maturity B = par value P = purchase price Now how come Spot Rate is coming into picture? Schweser notes says “The appropriate rates for individual future payments are called spot rates” can someone please step in to iron out this confusion? thanks.

A spot rate is a rate starting today and going into the future. For example, a one-year spot rate is a rate starting today for a period of one year; a two-year spot rate is a rate starting today for a period of two years, etc. Now, going back to bonds, the value of a bond today is equal to the present value of all the future cash flows generated by the bond (i.e., the regular coupon payments plus the par value at maturity for a typical level-coupon bond) discounted back to now. At what discount rate? Well, when you calculate the yield-to-maturity, you assume that it is the same discount rate for every single one of these cash flows, whether it happens in a year or in 10 years. Put it another way, you assume that the term structure of interest rates (or yield curve) is flat. But in reality, it is rarely the case; yield curves tend to be upward sloping (except before recessions when they tend to be inverted or downward sloping). What if you want to discount each of the cash flows by the discount rate that reflects the maturity of this cash flow? That’s when your spot rates come into place. The coupon in one year should be discounted at the one-year spot rate, the coupon in two years at the two-year spot rate, etc. Conclusion: Both YTM and spot rates are the discount rates of your bond. YTM assumes that discount rate is constance, spot rates allow you to factor in different discount rates for different maturities. Hope it helps.

When one hears the yields for government bonds such as the 2/5/10 year, with yields of 4.5% etc on financial news reports, are they talking about the secondary market or the primary market…if the gov were to issue new bonds?

First, you have to be careful, because what media report as “yields” are not necessarily YTM (e.g., Financial Times). If it is the YTM, it is very often the average of the YTM of bonds that mature in 2/5/10 years.

Frist : There are different yields(returns) for different time periods. (e.g. you may get 2% per year for 1-year investment but 2.3%per year for 2-year investment). These yields are called spot rates. , Second : When we invest in a 2-year zero coupon bond - we will get all cash flows after two years and therefore our yield should represents spot rate for a 2-year investment. Third : However, when we invest in a 2-year annual coupon paying bond… we will get a small portion of cash flows at end of year one (the coupon) and remaining (coupon + original investment) at end of year 2. Fourth: As we are getting some cash flows after 1 year and some after 2 year… our yield on this bond should be somewhere in between 1-year and 2-year spot rates. This yield is called YTM. Fifth: The relationship between YTM and spot rates goes like this: YTM is a sort of average of spot rates of cash flow dates. Sixth : In case of zero coupon bonds … YTM is equal to spot rate of the year in which bond matures as all cash flows will happen at one point in time. (called bullet payment). Seventh : correct method of valuation of a coupon bond is to calculate present value of each cash flow of bond (each coupon and principle payment) at respective spot rate.

Right, well i’m not sure if i understand, what kind of yield does the media use then, and are these calculated on secondary markets bonds or primary issued bonds? Thanks

Excellent work : frenchriviera Very good explanations.

Very often, what we call a “yield” is the ratio of income over price. For example, for a stock, the dividend yield is the dividend over the stock price. Well, for a bond, what the FT calls yiled in the market data is coupon over bond price - not the YTM.

frenchriviera thanks for the great explanation !

FrenchRiviera is right on as usual. Reineir - the yields quoted are in the secondary market. Treasury notes, bills, bonds, etc are among the most actively traded and liquid debt securities - this is why their price and yields are constantly changing. New notes, bills, bonds are issued twice a month and an auction mechanism determines their price. The results of these auctions are published and closely monitored - you will find information about the primary issues accompanied by discussion of “auction results” MDD

Cheers, thanks.