Interest Rate Fixed Mortgage Robot

Www Interestratefixedmortgage Tag Lose Interest Rate Fixed Mortgage 学习Fixed Income的一个重要概念Bootstrapping-FRM-资格证书-天下金融网

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• A discount factor is the decimal form of a zero-coupon bond. That is, if d(1) = 0.95, then a one- year zero-coupon bond with par value of $100 is priced at $95 (if par is $1,000, price is $950)

• A discount factor translates into, and necessarily implies, a spot rate. A spot rate is the yield on a zero-coupon instrument. If we are pricing Treasuries, the spot rates are Treasury spot rates. (And note, since Treasuries pay coupons, we are referring to Treasury STRIPS). 天下金融网

• The set of Treasury spot rates over time (six month Treasury spot, one year Treasury spot, 1.5 year Treasury spot and so on) produces a theoretical spot rate curve or term structure of interest rates. (Note this is sometimes called the Treasury yield curve, but technically the Treasury yield curve plots coupon-bearing yields to maturity [YTM] over time, whereas we are instead talking about zeros here. This distinction points out why we are talking about 'theoretical' spot rates.). 21jrr.com

Bootstrapping is to build out the yield curve one coupon at a time 天下金融网

Per the learning outcome above (LO 21.1), the key to generating the discount function is bootstrapping. First, we compute the discount factor for the near-term six-month bond. Why? Because it has only one cash flow! Now that we have the six-month discount factor, we can use it to compute the discount factor for the one-year coupon-paying bond. In this way, by building out, we only have in each case one unknown to solve for.

An Example

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For example, in the following spreadsheet below, I conjectured five coupon-bearing bonds (yellow background indicates inputs). The first bond is a zero (no coupon) paying par in six months. Since this bonds price is $98, its discount factor is 0.98.

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Now let's bootstrap the one-year (theoretical) spot rate. The price of the 4% coupon one-year bond is $99.00. If d(1) is the one-year discount factor, then the following must be true: $99 = $2(0.98) + $102[d(1)]. Make sure you see how the price of the one year bond is a function of two cash flows: the discounted $2 coupon [$2 x d(0.5)] and the discounted final cash flow [$102 x d(1.0]. 天下金融网

Solving for d(1) gives 0.9514, the one-year discount factor. Now this d(1) discount factor will be used for all coupons that are paid in one year. 21jrr.com

Translate discount factor to spot rate

Below the discount function (i.e., the series of discount factors) are the spot rates. The spot rate is given by:

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..and we could rearrange to solve for the discount rate...

Knowing our d(1) = 0.9514, the one-year theoretical spot rate is 5.05% because 5.05% = 2* [(1/0.9514)^(1/2)-1].

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Term structure of interest rates

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The spreadsheet below repeats this for each maturity. The set of spot rates produces the term structure of interest rates:

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