I must confess that valuation may not be the right word to use from an investor centric perspective at the least. The term bond ‘valuation’ / bond pricing makes more sense from an issuer’s perspective, which we are not actually concerned about in this article. Probably the word valuation is used since we are discounting cash flows in bonds just like what is done in a Discounted Cash Flow (DCF) valuation of a company. Keep in mind that mere DCF modelling isn’t valuation – the same holds true with bonds. A better term would be ‘pricing’. This basic article on Bond Pricing is going to require some effort due to a larger content though I will try to simplify it.
- Bond Pricing / Bond Valuations
- Bond Pricing and Yields
- Bond Pricing using Bootstrapping
- Bond Valuation Bootstratpping Example
Bond Pricing / Bond valuations
Assuming you know Bond Basics, there are 5 factors that go into influencing a bond’s price fundamentally. They are:
- Par Value or Face Value and Principal (FV): The amount of money lent to the issuer of the bond (borrower). Par Values are generally in 100 or 1000 per bond. The principal is the number of bonds bought multiplied by the par value.
- Tenor or years to maturity (n): The number of time periods it takes for the bond to mature or get redeemed where the issuer pays back the principal or par value (in the case of a single bond).
- Yield to Maturity (YTM): The rate of return on the bond if held till maturity.
- Coupon Rate: This is the interest rate paid by the issuer for the amount borrowed. The fixed returns/cash flows or coupon flows that the issuer pays as interest payments is calculated by multiplying the Coupon Rate with the Par Value (C) for a single bond.
- Time Value of Money (Discount factor): A basic concept in finance where a dollar today is worth more than a dollar tomorrow since interest rates feature into calculation. So, if $1 today equals $1 x (1+5%)^2 = $1.1025 after two years assuming interest compounds, then $1.1025 after two years is basically $1 today discounting it at a 5% interest rate.
The price of a bond is essentially a function of the above. Combining them all, for a bond paying coupons annually and YTM annually compounded we get:
Bond Pricing Formula
Bond Valuation Formula can also be written as:
No spoon-feeding – observe and understand the differences between the two bond pricing equations carefully. In general about the structure of the formula, it should make sense unless you are an earth shattering thinker
Bond Pricing and Yields
A bond is priced at par if the YTM and the Coupon Rate are the same. Don’t believe me? Try it out with your own numbers applying the above formula and determine the price.
In general, bonds are issued at par and redeemed at par. Although practically done very little, bonds can be issued at a discount or a premium or redeemed at a discount or premium. A practical example of bonds redeemed at a discount are those yielding negative (YTM<0) – Germany recently issued a 10year bond at -0.05% YTM; Switzerland issued a 42year bond at -0.02% YTM. If you aren’t aware of negative rates, read up on them!
Bonds are at a discount to par when the YTM is greater than the Coupon Rate and are at a premium to par when the YTM is lesser than the Coupon Rate.
Here are some important assumptions to be kept in mind regarding the formula:
- The above bond valuation formula assumes that the bond is a ‘fixed bond’e., has fixed cash flows or coupon payments. When people talk about bonds, they generally talk about fixed bonds unless mentioned otherwise.
- The YTM on the bond assumes that the Cash Flows/Coupon Flows are reinvested at the YTM rate. (For those who like valuing projects, doesn’t this strike a chord with IRR’s assumption? YTM is simply the IRR of a bond)
- The formula also assumes that the investor holds the bond till maturity and doesn’t sell it before maturity.
Let us look at Bond Pricing Excel. Assume ABC Inc.’s bonds are issued at a par of $100 with a YTM of 5% pa semi-annually compounded for 3 years.
Since it is a par bond, the YTM and coupon rate will be the same. Using the formula, the Price of the bond at issuance equals $100 as expected since it is issued at par.
The first assumption is easy to understand. Here is proof of the second one:
The bond valuation formula assumes that the cash flows gotten through coupon payments will not remain idle or seek returns in a better or worse investment. It rather gets a return equal to the YTM of the bond. This shouldn’t be surprising at all if you look at the working below!
The assumption is clearly questionable since it may not be easy or even possible to continually reinvest cash flows at the same rate, but that is what it is and we can’t change the formula since it is easy to understand on a fundamental level!
This brings us to the end of valuing/pricing a bond in a basic way.
also, look at Yield curve analysis
Bond Pricing using Bootstrapping
The same bond above can be priced differently. Here, we must be aware of the concept of Zero Coupon Bonds (ZCBs). These bonds do not give coupon payments and hence are called Zero Coupon Bonds or Zeros. These are bonds which are either issued at a discount and redeemed at par or, issued at par and redeemed at a premium. A ZCB has only one cash flow throughout its life which is at maturity (unless redeemed before – which isn’t an assumption by the formula)
So, a ZCB yielding 10% in 5 years can either be issued at a price of $62.09 today and redeemed at $100 after 5 years or be issued at $100 today and redeemed at $161.05 after 5 years – calculations? DIY!
Given that a ZCB has only one cash flow throughout its life which is at maturity, you can consider looking at a bond as a collection of multiple ZCBs. In this case, each cash flow would be like a ZCB maturing. Starting today, the first year’s coupon payment is similar to a ZCB maturing in one year; the second year’s coupon payment is similar to a ZCB maturing in two years and so on until the nth year where the principal and coupon are paid which is similar to a ZCB maturing in ‘n’ years.
If you’ve gotten the above concept, then the bond pricing formula making would be easy too!
The yields of fixed coupon paying bonds found using corresponding ZCB yields gives what is known as ‘Spot yields’ and the yield curve constructed using spot yields is called the ‘Spot Curve’ or ‘ZCYC’ (Zero Coupon Yield Curve).
The only problem is that the Treasury does not issue ZCBs. The process of extracting ZCB yields using the Treasury coupon paying bonds
The method of finding ZCB rates using coupon paying bonds is known as ‘Bootstrapping’ the yield curve.
Taking this approach, what we would need to find is the price of a corresponding ZCB and use that to find the ZCB’s yields.
Bond Pricing Bootstrapping Example
Let us look at Bond Pricing Excel using Boostrapping. We have a par curve where bonds are issued by the Treasury with the given maturities in Column I below.
The bond with a 1y maturity can be looked at like a ZCB with a cash flow of $105 at maturity. Thus the YTM of the ZCB is the same as the coupon paying bond @ 5%. Here is the equation:
Solving for Spot Rate1y we get 5% which is the same as the YTM of the 1y coupon paying bond.
The bond with a 2y maturity can be looked at like two ZCBs: one with a 1y maturity giving a cash flow of $6 at maturity and; one with a 2y maturity giving a cash flow of $106 at maturity. The equation goes like this:
Solving for Spot Rate2y we get 6.03% as the yield of the 2y ZCB.
The bond with a 3y maturity can be looked at like three ZCBs: one with a 1y maturity giving a cash flow of $7 at maturity; one with a 2y maturity giving a cash flow of $7 at maturity and one with a 2y maturity giving a cash flow of $107 at maturity; . The equation goes like this:
Solving for Spot Rate3y we get 7.10% as the yield of the 3y ZCB.
Follow this process throughout till the 5 year bond and you arrive at the Spot Curve/ZCYC.
Thus we arrived at the Spot Curve using the Par Curve. Sometimes, a par Curve may not be readily available since the yields on the bonds would’ve changed in the market. You can also use that to construct the ZCYC.
In the above example, we may have something like this:
Notice that the prices of the coupon paying bonds are exactly the same when extracted by ‘Bootstrapping.’
The process of Bootstrapping to find the Spot Curve gives you the arbitrage free prices of bonds since the prices should exactly be the same so that no bond is overpriced or under-priced. If the price using the spot curve is lower than that of the par curve or the yield curve, then the bond with that particular maturity is overpriced. If the price using the spot curve is higher than that of the par curve or the yield curve, then the bond with that particular maturity is under-priced.
There are other methods too of deriving spot curves apart from Bootstrapping which I do not want to get into.
We used the Treasury curve and made it look simple for pricing Treasury Bonds using the ZCB approach. What about corporate bonds? Corporate bonds carry credit risk, some may carry optionality risk, liquidity risk etc. – how do we account for that?
We still use the treasury curve as the benchmark since they generally are the most liquid, carry the least credit risk and do not have embedded options. These 3 major potential risks are taken care of by a treasury bond. To account for these risks, there are ‘spreads’ over the yields on treasury bonds for those bonds which are not issued by the treasury.
These are spreads over the Treasury Par Curve of the same maturity. This spread is added to just one point on the Treasury Par Curve. Therefore it is oblivious to the term structure of interest rates and uses only one point (a maturity point) on the curve and not the entire curve. This spread compensates the buyer for credit risk, liquidity risk and optionality risk only for a single maturity.
Z-Spread (Zero Volatility Spread)
This spread is added to every point on the Treasury Sport Curve and thus uses the entire yield curve unlike the nominal spread which uses only one point. The price of the risky bond is given and the spread over the spot curve (Z-Spread) should be used to discount the bond to arrive at the risky bond’s price.
Z-Spreads also compensate the buyer for credit risk, liquidity risk and optionality risk but for multiple maturities. But it is a constant spread over the yield curve.
OAS (Option Adjusted Spread)
OAS differs from the Z-Spread in that, Z-Spreads include a spread for embedded optionality risk in its calculation along with credit and liquidity risk. OAS removes that spread for embedded optionality and thus compensates the buyer only for credit and liquidity risks.
The OAS minus the Z-Spread gives you the price of the embedded option (in percentage terms else basis point terms).
OAS + Option Price (bps) = Z-Spread
Callable bonds benefit the issuer where the issuer pays the option premium/price:
Price of a Callable bond = Price of an option free bond – Price of the call option
Putable bonds benefit the investor where the investor pays the option premium/price:
Price of a Putable bond = Price of an option free bond + Price of the put option
Both the above are priced from the issuer’s perspective.
In the case of callable bonds, OAS < Z Spread and; for putable bonds, OAS > Z Spread. Why?
The option is bought by the issuer if the bond is callable and thus the benefit to the investor has to come in the form of higher yields and lower bond prices– a higher option price as a proportion of the Z-Spread justifies it to give that benefit to the investor (look at the formula – a higher option price leads to a lower bond price and hence a higher the yield). Thus the Z Spread has a higher option price and a lower OAS exists to adjust.
The option is bought by the investor if the bond is putable and thus the benefit to the investor is brought down in the form of lower yields and higher bond prices– a lower option price as a proportion of the Z-Spread justifies it to reduce the benefit to the investor (look at the formula – a lower option price leads to a higher bond price and hence a lower yield). Thus the Z Spread has a lower option price and a higher OAS exists to adjust i.e., the option price is negative.
This should be it for now. Go over the portion on ‘Spreads’ slowly, especially OAS if you’re doing a CFA. This is one of the harder topics to understand on a generic level itself. I haven’t gotten into the calculation of OASs, but for now it should be sufficient to understand the concept in itself. Another important one: know the difference between a Spot Curve a Par Curve and a Yield Curve which is not trading at Par because in general, when we talk about yield curves it is about the Treasury yield curve which is most often a par curve – the yield curve is generally a par curve.