Week 4: Chapter 4: The Time Value of Money- Stream of Cash flows
What is an annuity? What is the difference between an annuity and a perpetuity?
Annuity is a stream of equal cash flows arriving at a regular interval, and ending after a specified time period. -Difference: annuity ends and perpetuity is forever
How do you calculate the Future Value of an annuity? Example attached
By either using the calculator, or Using the formula: FV (Annuity)= C x 1/r [(1+r)^N- 1]
How do you calculate the value of an annuity? How do we calculate the PV for annuity? *in attached picture
Initial Investment amount MINUS the Present Value of the principal that will be left in the account in 20 years.
Using the same general formula, how do we calculate the Present Value of a Perpetuity? Example: Endowing a perpetuity and grad party attached
PV (C in perpetuity) = C/r The PV is basically saying, if you invest that amount at the rate per year forever, you will be able to fund that perpetuity every year.
What is Growing Annuity? How do we calculate the PV of a growing annuity? Formula
A Stream of N growing cash flows, paid at regular intervals -It is a growing perpetuity that eventually comes to an end Formula attached
4.3 Solving for Variables Other Than Present Value or Future Value Finding Annual payment: For spme situations, we use the present and/or future values as inputs and solve for the variable we are interested in. Example shown in attachment
Efficient if you use the Financial Calculator.
How do we calculate perpetuity? Example: Withdrawing $5 each year in perpetuity (reading attached) What is the general formula for it?
We calculate the value of a perpetuity by creating our own perpetuity. The Valuation Principle tells us that the value of a perpetuity must be the same as the cost we incurred to create our own identical perpetuity. C = r x P We invest an amount(P) in a bank account with the interest rate (r), Every year we can withdraw interest we;ve earned, C = r x P, leaving the principal (P) in the bank. Our cost for creating perpetuity is only the Principal, the value of receiving C in perpetuity is therefore the upfront cost P.
By the end of this chapter I should be able to..
•Value a series of many cash flows •Apply shortcuts to value special sets of regular cash flows called perpetuities and annuities •Compute the number of periods, cash flow, or rate of return in a loan or investment
4.1 Valuing a Stream of Cash Flows What is a stream of cash flows?
-A series of cash flows lasting several periods -Can be represented as a timeline similarly to single cash flow
What is the Rate of Return? How do you find the ideal interest rate required by a firm to get a present value of what you expect equal to the present value of what you give up? Finding interest rate:
-The rate at which the value of an amount invested, grows to exactly fund the future cash flows received by the investment -Use the calculator! Example attached
CHAPTER QUIZ
1. Steps for PV: -Compute present value of each individual cash flow -Combine the PV 2. Time value of money: the value of that money today isn't the same in the future. Value of (c) will be smaller PV= c/r 3. Regular deposits into savings account, monthly home mortgage payments, pension payments. Monthly, weekly, yearly.. 4. Annuity has a time period, perpetuity is forever 5. Commercial real estate, rental cash flows could be considered indefinite and will grow over time. 6. Calculate present value of the annuity, or the future value of the annuity. The value of the cash flow would be initial Investment amount MINUS the Present Value of the principal that will be left in the account in the end of the time period 7. Use the calculator!!! The rate at which the present value of the benefits exactly offset the cost.
4.4 Growing Cash Flows What is a growing perpetuity? How do we calculate Present Value of a growing perpetuity? Example attached
Growing Perpetuity is a stream of cash flows that occur at regular intervals and grow at a constant rate forever. PV (growing perpetuity)= C / r-g
4.2 Perpetuities and Annuities What is a perpetuity? Example? How does the formula for PV differ from the formula PV for other cashflows?
Perpetuity is a stream of equal cash flows that occurs at regular intervals, and lasts forever Eg. British government gives out consol, which promises the owner fixed cash every year forever. -First cash flow does not occur immediately -Arrives at end of the period (Payment in arrears) Formula: The C (cash flow) value remains constant and are all the same. Also, c0 = 0, first cash flow starts at time 1
How do we calculate the Present Value for a stream of cash flows? (2 steps) How do we then calculate the future value? *EXAMPLE Included in image attached
Steps for PV: 1. Compute present value of each individual cash flow 2. Combine the PV This gives you the total PV for the steam of cash flows. FV: refer to example -Compute the bank balance each year accounting for interest and new deposits
Example of Calculating Present Value of Annuity
◗ Evaluate The reason for the difference is the time value of money. If you have the $15 million today, you can use $1 million immediately and invest the remaining $14 million at an 8% interest rate. This strategy will give you 14mil x 8% per year in perpetuity! Alternatively, you can spend 15 million - 11.16 million = 3.84 Million today and invest the remaining $11.16 million, which will still allow you to withdraw $1 million each year for the next 29 years before your account is depleted.
