Time Value of Money

Time Value of Money Dr. Suha Alawi

Chapter Outline 4.2 Perpetuity 4.3 Annuities 4.5 Solving for Variables Other Than Present Value or Future Value

Learning Objectives Value a serious of many cash flows Value a perpetual series of regular cash flows Value a annuity series of regular cash flows Compute the cash flow of a loan

4.2 Perpetuities Perpetuities A perpetuity is a stream of equal cash flows that occur at regular intervals and last forever. Here is the timeline for a perpetuity: The first cash flow does not occur immediately; it arrives at the end of the st period

4.2 Perpetuities Using the formula for present value, the present value of a perpetuity with payment C and interest rate r is given by: Notice that all the cash flows are the same. Also, the first cash flow starts at time 1.

4.2 Perpetuities Let’s derive a shortcut by creating our own perpetuity. Suppose you can invest $100 in a bank account paying 5% interest per year forever. At the end of the year you’ll have $105 in the bank – your original $100 plus $5 in interest.

4.2 Perpetuities Suppose you withdraw the $5 and reinvest the $100 for another year. By doing this year after year, you can withdraw $5 every year in perpetuity:

4.2 Perpetuities Present Value of a Perpetuity (Eq. 4.4)

Example 4.3 Endowing a Perpetuity Problem: You want to endow an annual graduation party at your alma mater. You want the event to be a memorable one, so you budget $30,000 per year forever for the party. If the university earns 8% per year on its investments, and if the first party is in one year’s time, how much will you need to donate to endow the party?

Solution: Plan: The timeline of the cash flows you want to provide is: This is a standard perpetuity of $30,000 per year. The funding you would need to give the university in perpetuity is the present value of this cash flow stream Example 4.3 Endowing a Perpetuity (cont’d)

Example 4.3 Endowing a Perpetuity (cont’d) Execute: From the formula for a perpetuity,

Example 4.3 Endowing a Perpetuity (cont’d) Evaluate: If you donate $375,000 today, and if the university invests it at 8% per year forever, then the graduates will have $30,000 every year for their graduation party.

Example 4.3a Endowing a Perpetuity Problem: You just won the lottery, and you want to endow a professorship at your university. You are willing to donate $4 million of your winnings for this purpose. If the university earns 5% per year on its investments, and the professor will be receiving her first payment in one year, how much will the endowment pay her each year?

Solution: Plan: The timeline of the cash flows you want to provide is: This is a standard perpetuity. The amount she can withdraw each year and keep the principal intact is the cash flow when solving equation 4.4. Example 4.3aEndowing a Perpetuity (cont’d)

Example 4.3aEndowing a Perpetuity (cont’d) Execute: From the formula for a perpetuity,

Example 4.3a Endowing a Perpetuity (cont’d) Evaluate: If you donate $4,000,000 today, and if the university invests it at 5% per year forever, then the chosen professor will receive $200,000 every year.

4.3 Annuities Annuities An annuity is a stream of N equal cash flows paid at regular intervals. The difference between an annuity and a perpetuity is that an annuity ends after some fixed number of payments

4.3 Annuities Present Value of An Annuity Note that, just as with the perpetuity, we assume the first payment takes place one period from today. To find a simpler formula, use the same approach as we did with a perpetuity: create your own annuity.

4.3 Annuities With an initial $100 investment at 5% interest, you can create a 20-year annuity of $5 per year, plus you will receive an extra $100 when you close the account at the end of 20 years:

4.3 Annuities The Law of One Price tells us that because it only took an initial investment of $100 to create the cash flows on the timeline, the present value of these cash flows is $100:

4.3 Annuities (The Present Value) Rearranging:

4.3 Annuities (The Present Value) We usually want to know the PV as a function of C, r, and N. Since C can be written as $100(0.05)=$5, we can further re-arrange:

4.3 Annuities (The Present Value) In general:

Example 4.4 Present Value of a Lottery Prize Annuity Problem: You are the lucky winner of the $30 million state lottery. You can take your prize money either as (a) 30 payments of $1 million per year (starting today), or (b) $15 million paid today. If the interest rate is 8%, which option should you take?

Example 4.4 Present Value of a Lottery Prize Annuity (cont’d) Solution: Plan: Option (a) provides $30 million in prize money but paid over time. To evaluate it correctly, we must convert it to a present value. Here is the timeline:

Example 4.4 Present Value of a Lottery Prize Annuity (cont’d) Plan (cont’d): Because the first payment starts today, the last payment will occur in 29 years (for a total of 30 payments). The $1 million at date 0 is already stated in present value terms, but we need to compute the present value of the remaining payments. Fortunately, this case looks like a 29-year annuity of $1 million per year, so we can use the annuity formula.

Example 4.4 Present Value of a Lottery Prize Annuity (cont’d) Execute: From the formula for an annuity,

Example 4.4 Present Value of a Lottery Prize Annuity (cont’d) Execute (cont’d): Thus, the total present value of the cash flows is $1 million + $11.16 million = $12.16 million. In timeline form:

Example 4.4 Present Value of a Lottery Prize Annuity (cont’d) Execute (cont’d): Financial calculators or Excel can handle annuities easily—just enter the cash flow in the annuity as the PMT:

Example 4.4 Present Value of a Lottery Prize Annuity (cont’d) 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 $14 million  8% = $1.12 million 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.

4.3 Annuities Future Value of an Annuity (Eq. 4.6)

Example 4.5 Retirement Savings Plan Annuity Problem: • Ellen is 35 years old, and she has decided it is time to plan seriously for her retirement. • At the end of each year until she is 65, she will save $10,000 in a retirement account. • If the account earns 10% per year, how much will Ellen have saved at age 65?

Example 4.5 Retirement Savings Plan Annuity (cont’d) Solution Plan: As always, we begin with a timeline. In this case, it is helpful to keep track of both the dates and Ellen’s age:

Example 4.5 Retirement Savings Plan Annuity (cont’d) Plan (cont’d): Ellen’s savings plan looks like an annuity of $10,000 per year for 30 years. (Hint: It is easy to become confused when you just look at age, rather than at both dates and age. A common error is to think there are only 65-36= 29 payments. Writing down both dates and age avoids this problem.) To determine the amount Ellen will have in the bank at age 65, we’ll need to compute the future value of this annuity.

Example 4.5 Retirement Savings Plan Annuity Execute: Using Financial calculators or Excel:

Example 4.5 Retirement Savings Plan Annuity Evaluate: By investing $10,000 per year for 30 years (a total of $300,000) and earning interest on those investments, the compounding will allow her to retire with $1.645 million.

Solving for Variables Other Than Present Value or Future Value: Computing a Loan Payment Problem: • Suppose you accept your parents’ offer of a 2007 BMW M6 convertible, but that’s not the kind of car you want. • Instead, you sell the car for $61,000 and use that money for a down payment on an Aston Martin V8 Vantage Roadster. You got a real bargain at $110,000! • The bank offers you a 5-year loan with equal monthly payments and an interest rate of 4% per year. • What will be the payment on the loan?

Example 4.8bComputing a Loan Payment (cont’d) Solution: Plan: • The loan amount is $110,000 – $61,000 = $49,000 • Note, we need to use the monthly interest rate. Since the quoted rate is an APR, we can just divide the annual rate by 12: r = .04/12 = .0033

Example 4.8bComputing a Loan Payment (cont’d)

Example 4.8bComputing a Loan Payment (cont’d) Execute (cont’d): • Using a financial calculator or Excel:

Example 4.8Computing a Loan Payment Problem: • Your firm plans to buy a warehouse for $100,000. • The bank offers you a 30-year loan with equal annual payments and an interest rate of 8% per year. • The bank requires that your firm pay 20% of the purchase price as a down payment, so you can borrow only $80,000. • What is the annual loan payment?

Example 4.8Computing a Loan Payment (cont’d) Solution: Plan: • We start with the timeline (from the bank’s perspective): • Using Eq. 4.8, we can solve for the loan payment, C, given N=30, r = 8% (0.08) and P=$80,000

Example 4.8Computing a Loan Payment (cont’d) Execute: • Eq. 4.9 gives the payment (cash flow) as follows:

Example 4.8Computing a Loan Payment (cont’d) Execute (cont’d): • Using a financial calculator or Excel:

Time Value of Money