# Fast Formulas

The Fast Formulas posts on this blog provide shortcuts to laborious financial calculations. They were inspired by the Financial Math Puzzles. Some of the formulas are ancient, and some of them might be new.

Here is a guide to the Fast Formulas (“FF”) posts and the key formulas they contain:

 FF# Original Post Key Formulas Follow-up Posts 1 Average Life of Mortgage (As Scheduled) (1.3) $L=rPD_{n}$ (1.4) $W=nD_{n}-\dfrac{1}{r}$ New Insight into an Old Formula 2 Average Life of Mortgage (Amortizing with Balloon) (2.2) $W=D_{n} \left( m+\dfrac {1}{rD_{q}} \right)-\dfrac {1}{r}$ 3 Pool Average Life with CPR Prepayments (3.1)$W=D_{n} \left( \dfrac {1-C^{n}} {c}+d\dfrac {d^{n}-C^{n}}{C-d} \right)$ The Spreadsheet, The Derivation 4 Geometric Series (One of the Tricks) (4.4) $Q=\dfrac {q-q^{n+1}}{1-q}$ 5 Quasi-Geometric Series (The Other Trick) (5.1)$S=\dfrac {q-(n+1)q^{n+1}+nq^{n+2}} {(q-1)^{2} }$

Symbols

 FF# Symbol Description 1 $L$ Monthly payment 1 $P$ Principal amount 1,2,3 $n$ Term of fully amortizing mortgage, in months 1,2,3 $a$ Annual interest rate 1,2,3 $r$ Monthly interest rate $r=\dfrac {a}{12}$ 1,2,3 $d$ Monthly discount factor $d=\dfrac {1}{1+r}$ 1,2,3 $D_{n}$ $D_{n}=\dfrac {1}{1-d^{n}}$ 2 $D_{q}$ $D_{q}=\dfrac {1}{1-d^{q}}$ 1,2,3 $W$ Weighted average life, in months 2 $m$ Months until balloon payment 2 $q$ Months from balloon to amortization end $q=n-m$ 3 $c$ Monthly prepayment rate 3 $C$ Monthly survivorship factor $C=1-c$ 4,5 $q$ Any real number other than 1 4 $Q$ Sum of finite geometric series $Q=q+q^{2}+...+q^n$ 4,5 $n$ Number of terms in series 5 $S$ Sum of quasi-geometric series $S=q+2q^{2}+...+nq^n$