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The Time Value of Money

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The Timeline Suppose you are lending $10,000 today and the loan will be repaid in two annual $6,000 payments. The timeline looks like this: The first cash flow at date 0 (today) is represented as a negative sum because it is a cash outflow. The others are positive as they represent cash inflows. Timelines can represent cash flows that take place at the end of any time period. We want to ask, is this loan a good idea?

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**The 1st Rule of Time Travel**

A dollar today and a dollar in one year are not equivalent. ($12,000 > $10,000 is not how its done.) It is therefore only possible to compare or combine values at the same point in time. Which would you prefer: A gift of $1,000 today or $1,210 at a later date? To answer this, you will have to compare the alternatives to decide which is worth more. One factor to consider: How long is “later?” Note, there are “three rules of time travel”

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**The 2nd Rule of Time Travel**

To move a cash flow forward in time, you must compound it. The future value of a cash flow in n periods:

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**The 2nd Rule of Time Travel**

Suppose you have a choice between receiving $5,000 today or $10,000 in five years. You can earn 10% on the $5,000 today, but want to know what the $5,000 will be worth in five years. Clearly the current interest rate, here the 10%, is an important consideration as is the five year horizon. The time line looks like this:

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**The 2nd Rule of Time Travel**

In five years, the $5,000 would grow to: $5,000 × (1.10)5 = $8,053 The future value of $5,000 at 10% for five years is $8,053. You would be better off forgoing the gift of $5,000 today and taking the $10,000 in five years. What if you need money today, does that change the decision?

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**The 3rd Rule of Time Travel**

To move a cash flow back in time you must discount it. The present value of a future cash flow n periods in the future:

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**The 3rd Rule of Time Travel**

Suppose you are offered an investment that pays $10,000 in five years. If you expect to earn a 10% return, what is the value of this investment? The $10,000 is worth: $10,000 ÷ (1.10)5 = $6,209 Alternatively: $6,209×(1.10)5 = $10,000 so if you had $6,209 today it would become $10,000 in 5 years at 10%. Thus today’s value of $10,000 to be received in 5 years must be $6,209.

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**Using the Rules of Time Travel**

Assume that an investment will pay you $5,000 now and $10,000 in five years. The time line would like this:

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**Using the Rules of Time Travel**

You can calculate the total present value of the combined cash flows by adding their values today. The present value of both cash flows is $11,209. Note that adding these values is now appropriate since we have put them at the same date.

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**Using the Rules of Time Travel**

You can calculate the future value of the combined cash flows by adding their values in Year 5. The future value of both cash flows is $18,053.

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**Using the Rules of Time Travel**

Note it doesn’t matter when you combine them, the answer remains the same. In other words, these values are equivalent 11,209 today and 18,053 in 5 years (if the interest rate is 10%). Or no value is created or lost moving money across time at the current interest rate. We commonly use present value as today is a fixed time and usually costs are incurred today.

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**Valuing a Stream of Cash Flows**

Based on the first rule of time travel we can derive a general formula for valuing a stream of cash flows: if we want to find the present value of a stream of cash flows, we simply add up the present values of each. The Present Value of a Cash Flow Stream

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Example Suppose you have the opportunity to purchase a claim to a series of cash flows such that you would receive $100 in one year, $200 in two years and $300 in three years. The current interest is 10%. What is the present value of these payments? How much would you be willing to pay to purchase this claim? If you are able to purchase the claim for $420 are you better off than you were without this opportunity? By how much? $ is the present value of the payments and you should be willing to pay any thing up to this amount. Borrow and pay 420 for the claim. You have today and can use the claim to repay the loan.

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Solution The solution of course is to simply find the total present value of all the future cash flows (the sum of the present values of the individual cash flows. Now we compare to the proposed cost: The NPV tells us how much better off we are in terms of dollars today from the purchase.

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Example You have the opportunity to purchase a security that promises a series of cash flows such that you will receive $100 in one year, and $100 each year for after that for the following six years (a total of seven payments). What is the present value of these payments if the current interest rate is 4%? There must be an easier way. $ The easier way to do highly structured cash flows is with the formulas we will now turn to.

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Perpetuity A stream of equal payments, starting in one period, and made each period, forever. Forever?? Please, please remember, this gives the value of this stream of cash flows as of time 0, one period before the first payment arrives. … 1 2 3 C Consol bonds once issued by the British government and preferred stock are examples of such instruments. Also, just as before there is an equivalence between the value of the stream and the PV.

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Growing Perpetuity A growing perpetuity is a stream of periodic payments that grow at a constant rate and continue forever. The present value of a perpetuity that pays the amount C1 next period, grows at the rate g indefinitely when the discount rate is r is: … 1 2 3 C1 C1(1+g) C1(1+g)2 Again note the delay in the first payment. Note also that since the payments change it is important to identify what C1 is.

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Perpetuity Example Place the present value in a bank account, and recreate the payments. Let’s stop at 4 years since “forever” would take a while. Arbitrage argument then implies that the initial balance must be the current value of the perpetuity. Why are these the formulas? This is the formula because if you put the PV in the bank at rate r, you could generate and withdraw C each period and never alter your principal. For the growing case you have PVr in interest but you need to set aside PVg to let principal grow, making the first payout equal to PV(r-g) this must equal C1 to make it match the growing perpetuity. The formula follows. Note the account balance is growing. At what rate? Why must this happen?

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Annuities An annuity is a series of equal payments, starting next period, and made each period for a specified number (3) of periods. If payments occur at the end of each period (the first is one period from now) it is an ordinary annuity or an annuity in arrears. If the payments occur at the beginning of each period (the first occurs now) it is an annuity in advance or an annuity due. 1 2 3 C 1 2 3 C C

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Valuing Annuities We can do a lot of grunt work or we can notice that a T period annuity is just the difference between a standard perpetuity and one whose first payment comes at date T+1. The present value of a T period annuity paying a periodic cash flow of C, when the discount rate is r, is: If we have an annuity due instead, the net effect is that every payment occurs one period sooner, so the value of each payment (and the sum) is higher by a factor of (1+r). Or we can add C to the value of a T-1 period annuity.

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Annuity Example Compute the present value of a 3 year ordinary annuity with payments of $100 at r = 10%. or,

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Annuity Due Example What if the last example had the payments at the beginning of each period not the end? Or,

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**Example: A five year annuity paying $2000 per year, with r = 5%.**

Valuing the payments individually we get: Using the annuity formula we get:

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**Alternatively, suppose you were given $8,658**

Alternatively, suppose you were given $8, today instead of the annuity Notice that you can exactly replicate the annuity cash flows by investing the present value to earn 5%. This again demonstrates that present value calculations provide a literal equality, in that the future cash flows can be converted into the present value and vice versa, if (and only if) the selected discount rate is representative of actual capital market conditions.

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Growing Annuities A stream of payments each period for a fixed number of periods where the payment grows each period at a constant rate. … 1 2 T-1 T C1 C1(1+g) C1(1+g)T-2 C1(1+g)T-1 We can again talk about the difference between two growing perpetuities but we have to make sure the first payment in the delayed perpetuity cancels out the T+1st payment of the initial perpetuity.

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Example What is the present value of a 20 year annuity with the first payment equal to $500, where the payments grow by 2% each year, when the interest rate is 10%? … 1 2 19 T=20 500 500(1.02) 500(1.02)18 500(1.02)19 Last payment amount is $728.41

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**Application: Retirement Planning**

You have determined that you will require $2.5 million when you retire 25 years from now. Assuming an interest rate of r = 7%, how much should you set aside each year from now till retirement? Step 1: Determine the present equivalent of the targeted $2.5 million. PV = $2,500,000/(1.07)25 PV = $2,500,000/ = $460,623 Step 2: Determine the annuity that has an equivalent present value: The alternative is to solve for the future value of a constant stream of payments but that gets tedious. Note, this assumes that you make deposits beginning one year from today.

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**Retirement Planning cont…**

Now suppose that you expect your income to grow at 4% and you want to let your retirement contributions grow with your earnings. How large will the first contribution be? How about the last?

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