Time Value of Money
Annuity Factor
Present value of a $1 annuity.
Discount factor
Present value of a $1 received in year t.
Equations for PV
Pv=(future value after t periods)/(1+r)^t
Perpetuity
Stream of level cash payments that never ends.
How to fine the value of future dates?
To find the value at some future date of a stream of cash flows, calculate what each cash flow will be worth at that future date and then add up these future values.
Delayed perpetuity
the amount x (1/(interest rate))^year). This will show the today's value of the total amount.
Value of investment formula
(amount) X ( 1 + (interest rate)) ^ (# of years)
How are present values calculated?
using compound interest.
Present value of t-year annuity
= C [ (1/r) - (1 / r(1+r)^t) ] = payment X annuity factor
Future value of annuity of $1 a year
= present value of annuity x (1 + r)^t ((1 + r)^t-1 / r) ; C x FV of Annuity = desired amount
Present Value of Annuity Due
=present value of ordinary annuity x (1 + r)
The equation of PV of perpetuity
C (cash payment) / r (interest rate)
Annuity
Equally spaced level stream of cash flows, with a finite maturity
Compound Interest
Interest earned on interest.
Simple Interest
Interest earned only on the original investment; no interest is earned on interest.
Present Value (PV)
Value today of a future cash flow. We are trying to find out how much we need to invest now to product the given amount of the number of year to be receive from now.
How do we calculate the present value of a future cash flow?
When we are asking how much that cash flow would be worth today. If there is more than one future flow, we simply need to work out what each flow would be worth today and then add these present values.
Future Value
Amount of which an investment will grow after earning interest.
Discounted cash-flow
Another term for the present value of a future cash flow.
Discount rate
Interest rate used to compute present value of future cash flows
Annuity Due
Level stream of cash flows starting immediately
Compound Growth
The value increase each period by the factor (1+growth rate). The value after t period will equal the initial value times (1+ growth rate)^t. When money is invested at compound interest, the growth rate is the interest rate.
How important is it to use present values when comparing alternative patterns of cash payment?
You should never compare cash flows occurring at different times without first discounting them to a common date. By calculating present values, we see how much cash must be set aside today to pay future bills.