Present Value Calculations

In pricing the value of a security, we consider that humans discount future returns.

• Reason we need to discount value: Value of Money
• This is the reverse of Future Value Calculations.

def. Discounted Cash Flows (DCF) are used to price assets.

• Each cash flow is discounted by the discount rate. They are summed to get the present value of all the cash flows
• We always want asset price↑, risk↓
• rate of return $\propto$ volatility ← knowing how to trade off these two is important

To calculate the DCF of some future cash flow $FV$’s current present value $PV$:

$$\begin{array}{rl}PV& =\frac{FV}{{\left(1+\frac{r}{k}\right)}^{t\times k}}\\ & =\frac{FV}{(1+q{)}^n}\end{array}$$

• $r$ is the Annual Percentage Rate [= quoted rate]
• $k$ is how many times to compound every year
  • → thus $r/k$ is the interest rate per compound period (=$q$)
• $t$ is the number of years
  • → thus $t\times m$ is the number of **total compounding periods (=$n$)

You can calculate the present value of multiple identical cash flows of amount $C$:

$$\begin{array}{rl}PV& =\frac{C}{(1+r{)}^1}+\frac{C}{(1+r{)}^2}+\cdots +\frac{C}{(1+r{)}^n}\\ & =\frac{C}{r}\left(1-(1+r{)}^{-n}\right)\end{array}$$

What is the value of Interest Rate $r$?

• In individual investments/loans, $r$ determined usually by looking at similar assets in the market
• $r\propto$ Inflation; if inflation is high, money isn’t worth much in the future, so lender demands more interest rate

Net Present Value §

$$NPV(r)=PV(r)-\text{Initial Investment}$$