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From the MBS-CMOs brochure of Fidelity Investments, I read:

For securities purchased at a discount to face value, faster prepayment rates will increase the yield-to-maturity, while slower prepayment rates will reduce it. For securities purchased at a premium, faster prepayment rates will reduce the yield-to-maturity, while slower rates will increase it. For securities purchased at par, these effects should be lessened

What is the algebra behind those results?

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I can give you the simple intuition using an example, based on the idea that fixed rate mortgage bonds are like regular bonds except with uncertain maturities.

Suppose you have a 2% coupon bond which has an expected maturity of 2yrs. Let’s say you buy this for 98%. Then if it indeed matures after 2yrs, the yield will be about 3% (the coupon plus the 2% gain on investment spread out over 2years). If on the other hand it matures after 1year, the yield will be about 4% (the 2% coupon plus the 2% gain spread out over only 1year). If it matures after 4 years , the yield will be about 2.5%, as the investment gain is spread out further.

If on the other hand the bond is bought for 102%, the yield for a 2yr maturity will be about 1% (coupon minus the 2% loss spread out over 2years). If the maturity is only one year , then the yield will be approximately zero, since the investment loss offsets the coupon income completely.

I’m sure this can be generalized with some bond math algebra but the principle should be clear.

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I cringe a little at the scenario in which the prepayment speed changes, but the observable market price of the security does not change (hence, only the yield reacts).

I will construct an analogy with a simple example bond.

Imagine a 3-year bond that pays annual fixed coupon $C$, and the bond issuer has the right to repay the entire principal (no partial prepayments; Bermudan call) after year 1 or year 2. There are 3 possible cash flow scenarios:

prepay in year 1:

  • coupon $C$ and entire principal in 1 year

prepay in year 2:

  • coupon $C$ in 1 year

  • coupon $C$ and principal in 2 years

pay at 3-year maturity, no prepayment:

  • coupon $C$ in 1 year

  • coupon $C$ in 2 years

  • coupon $C$ and principal in 3 years

Suppose first that the example bond is trading in the market exactly at par (i.e. your initial cash flow is to pay exactly -1). Then when you solve for yield (internal rate of return) under each of the 3 prepayment scenarios, you should get yield $y =$ coupon $C$ each time.

$$0 = -1 + \frac{1+C}{(1+y)^1}$$

$$ 0 = -1 (1+y) + (1+C)$$

$$y = C$$

and you get the same solution for

$$0 = -1 + \frac{C}{(1+y)^1} + \frac{1+C}{(1+y)^2}$$

$$0 = -1 + \frac{C}{(1+y)^1} + \frac{C}{(1+y)^2} + \frac{1+C}{(1+y)^3}$$

Suppose instead that the bond is trading below par (investor pays $1-\epsilon$ initially for some small positive $\epsilon$). For the 1-year scenario,

$$0 = -(1-\epsilon) + \frac{1+C}{(1+y)^1}$$

$$ 0 = (-1+\epsilon) (1+y) + (1+C)$$

$$ 0 = -1-y+\epsilon+\epsilon y + 1+C$$

$$y= \frac{C+\epsilon}{1-\epsilon}$$

Then the yields under the 3 prepaymenr scenarios are all higher than $C$ (in order to increase the denominator and discount the received flows more, in order to still match the lower price), but no longer equal to each other; and the sooner the issuer repays the principal, the higher the yield $y$.

Suppose instead that the bond is trading above par (investor pays $1+\epsilon$ initially). Then the yields under the 3 scenarios are all lower than C; and the longer the issuer waits to repay the principal, the higher the yield $y$.

That's all the algebra that there is to it. We made the prepayment scenarios discrete here, but we could make them continuous instead (i.e., make the call Ameeican rather than Bermudan), and/or allow partial prepayment rather than bullet, and then consider the instantaneous sensitivity of the yield to a small change in prepayment speed or to weighted average life ceteris paribus, assuming that prices remain constant.

But I really don't think that this approach conveys the right intuition about the market behavior. This is like assuming that prices of investment-grade corporate bonds would not move when treasury bonds move (or somewhat like the proverbial spherical cow in a vacuum). The price of our example bond is unlikely to remain constant if the market participants change their assumptions about the likelyhood of each prepayment scenario.

If, while everyone assumes that the example bond would be fully called in 2 years, and it is trading at par (at yield $C$, equal to some spread $s$ over the 2 year treasury debt yield), and then suddenly everyone decided that the principal would rather be repaid in 3 years, then the example bond's new yield observed in the market would approximately equal to almost unchanged spread $s$ over 3 year treasury debt yield (not the same as 2 year yield); and one would back out the price from the new yield.

Likewise, prepayment assumptions react to interest rate assumptions in complicated ways.

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