# Financial Math FM/General Cash Flows and Portfolios

## Cashflow

A cashflow is a series of payments and withdrawals made at different times.

 Investments ${\displaystyle C_{0}}$ ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle \cdots }$ ${\displaystyle C_{n}}$ Time ${\displaystyle t_{0}}$ ${\displaystyle t_{1}}$ ${\displaystyle t_{2}}$ ${\displaystyle \cdots }$ ${\displaystyle t_{n}}$

Unless otherwise stated, compound interest is assumed so that the present value of the cash flow at time ${\displaystyle t=0}$, also called the current value, is ${\displaystyle PV=\sum _{j=0}^{n}{C_{j}v^{j}}}$

Example
Ivan offers to cut you in to a project of his. His project requires an investment from you of $10,000 now at time ${\displaystyle t=0}$ and it promises to pay$3,000 at time 2, $4,000 at time 3 and$5,000 at time 10. At an annual effective interest rate of 10%, should you take him up on his offer?
Solution
${\displaystyle PV=10000v^{0}-3000v^{2}-4000v^{3}-5000v^{10}}$
${\displaystyle v=1.1^{-1}}$
${\displaystyle PV=-2587}$

You should not take him up on his offer. The present value of the cash flow is negative.

As we can see from this example, the way to analyze profitability of an investment is to find the present value of the inflows minus the present value of the outflows. This is called the net present value of an investment.

## BAII Calculator usage

The TI BAII plus has a cash flow function to deal with uneven cash flow streams. Here's an example cash flow.

Period Cash Flow
0 -500
1 100
2 0
3 300
4 0
5 500

To solve this cash flow you must enter this series of commands.
CF 2ND CE|C 500 +/- ENTER ↓ 100 ENTER ↓ ↓ 0 ENTER ↓ ↓ 300 ENTER ↓ ↓ 0 ENTER ↓ ↓ 500 ENTER CPT NPV 10 ENTER ↓ CPT CF Enters into cash flow mode. 2ND CE|C will clear any previous work saved in the calculator. Next you'll see CF0= where you'll enter the first payment of -500.
TVM Calcs

## Duration

1. The candidate will be able to define and recognize the definitions of the following terms: a. Yield rate/rate of return b. Dollar-weighted rate of return/Time-weighted rate of return Current value Duration (Macaulay and modified) Convexity Portfolio Spot rate Forward rate Yield curve Stock price, stock dividend 2. The candidate will be able to: a. Calculate the current value of a set of cash flows. b. Calculate the portfolio yield rate. c. Calculate the dollar-weighted and time-weighted rate of return. d. Calculate the duration and convexity of a set of cash flows. e. Calculate either Macaulay or modified duration given the other. f. Use duration and convexity to approximate the change in present value due to a change in interest rate. g. Calculate the price of a stock using the dividend discount model.