Prepayment Risk

Study guide: Exam 9

Prepayment Risk is the risk that borrowers repay principal earlier than scheduled — usually because interest rates have fallen and refinancing has become attractive — leaving the investor to reinvest at lower yields and capping the asset's price gain. It dominates the risk of mortgage-backed and other amortizing securities; callable bonds carry the same exposure through the issuer's call. Its mirror image, extension risk, is repayment slowing when rates rise.

$\text{CPR} = 1 - (1 - \text{SMM})^{12}$

$\text{Prepayment}_t = \text{SMM}_t \times \bigl(B_{t-1} - \text{Scheduled principal}_t\bigr)$

SMM (single monthly mortality) is the fraction of the remaining balance prepaid in a month; CPR (conditional prepayment rate) is its annualised equivalent; $B_{t-1}$ is the pool balance at the start of month $t$ . The PSA benchmark ( $100\%$ PSA) assumes a CPR of $0.2\%$ in month 1, rising $0.2\%$ a month to $6\%$ at month 30 and level thereafter; $200\%$ PSA doubles every rate.
The investor is short an option. Each borrower holds a call: the right to repay at par. When rates fall prepayments speed up and principal comes back at par just when it can only be reinvested at lower yields (Reinvestment Risk); when rates rise prepayments slow and the security lengthens just as its value falls. Both moves hurt, which is negative convexity: the price rises less than it falls.
Measure with effective duration. Because the cash flows depend on rates, cash-flow durations mislead. Reprice at $\pm\Delta y$ under a prepayment model: $D_{\text{eff}} = (P_- - P_+)/(2P_0\,\Delta y)$ and $C_{\text{eff}} = (P_- + P_+ - 2P_0)/(P_0\,\Delta y^2)$ .
Why it matters to an insurer. Mortgage-backed securities are widely held for their yield spread, which is partly payment for the option sold. Their duration shortens as rates fall, opening a mismatch against the liabilities, so Interest Rate Risk management must use effective rather than stated durations. Tranching — sequential-pay classes, planned amortization classes — reallocates prepayment risk among investors; see Structured Finance.
Where it sits. The Exam 9 syllabus lists prepayment beside default and reinsurance under Credit Risk: like default, it is the borrower's behaviour changing the cash flows the insurer receives — but it is driven by interest rates, not by inability to pay, so it is usually measured and hedged as part of interest rate risk.

Converting CPR and Projecting One Month's Prepayment A mortgage pool has a balance of $\$200$ M at the start of the month and scheduled principal of $\$400{,}000$ . It is seasoned past month 30 and prepays at $100\%$ PSA. Find the SMM and the month's prepayment.
Answer At $100\%$ PSA after month 30, $\text{CPR} = 6\%$ .
 $\begin{align*} \text{SMM} &= 1 - (1 - 0.06)^{1/12} \\ &= 0.005143 \\ \text{Prepayment} &= 0.005143 \times (200{,}000{,}000 - 400{,}000) \\ &= \$1{,}026{,}545 \end{align*}$ 
The pool returns about $\$1.03$ M of unscheduled principal this month — more than two and a half times the scheduled amount. At $200\%$ PSA (a $12\%$ CPR, as in a refinancing wave) it would be roughly double, and all of it must be reinvested at the lower prevailing yield.

Negative Convexity of a Mortgage-Backed Security An MBS is priced at $100$ . Under the prepayment model it is worth $103$ if yields fall $1\%$ and $95$ if they rise $1\%$ . Find its effective duration and convexity and compare with an option-free bond of the same duration and convexity $+20$ .
Answer
 $\begin{align*} D_{\text{eff}} &= \frac{103 - 95}{2(100)(0.01)} \\ &= 4.0 \\ C_{\text{eff}} &= \frac{103 + 95 - 2(100)}{100(0.01)^2} \\ &= -200 \end{align*}$ 
Using $\Delta P/P \approx -D\,\Delta y + \tfrac{1}{2}C\,(\Delta y)^2$ , the option-free bond gains $4.1\%$ when yields fall and loses $3.9\%$ when they rise. The MBS gains $3\%$ and loses $5\%$ : the borrowers' refinancing option takes the upside away. Matched to liabilities on duration alone, the MBS would leave the insurer short convexity — hurt by a large rate move in either direction.