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The Time Value of Money

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The Timeline Suppose you are lending $1,000 today and the loan will be repaid in two annual payments. The timeline looks like this: The first cash flow at date 0 (today) is represented as a negative amount 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? 1 2 +$500 +$550 -$1,000

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**The 1st Rule of Time Travel**

A dollar today and a dollar in one year are not equivalent. ($1,050 > $1,000 is not relevant.) 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 $1,050 at a later date? To answer this, you will have to compare the alternatives to decide which is more valuable. One factor to consider: How many periods is “later”? Another is: What does it cost to move money across a period of time?

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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 is:

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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, and need 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 = $8,053.55 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. Taking $5,000 today is only equivalent to receiving $8, in 5 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**

What is today’s value of your opportunity to receive $10,000 in five years. If the interest rate is 10%, what is today’s value of the future cash? $10,000 received in five years is worth: $10,000 ÷ (1.10)5 = $6, today which can be compared to the alternative of $5,000 today. Alternatively: $6,209×(1.10)5 = $10,000 so if you had $6, today it would become $10,000 in 5 years if the interest rate is 10%. Thus today’s value of $10,000 to be received in 5 years must be just $6,

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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?

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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. Borrow $ today and pay $420 for the claim. The claim will just repay the loan and you keep the $61.59.

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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 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%?

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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

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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

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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. 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

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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

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**Interest Rate Quotes and Adjustments**

The Effective Annual Rate (EAR) Indicates the total amount of interest (as a percent of investment) that will be earned at the end of one year The EAR considers the effect of compounding Also referred to as the effective annual yield (EAY) or annual percentage yield (APY)

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**Interest Rate Quotes and Adjustments**

Adjusting the Discount Rate to Different Time Periods If 4% is the effective annual rate what is the 6-month rate? Earning a 4% return annually is not the same as earning 2% every six months. (1.04)0.5 – 1= – 1 = = 1.98% Note: n = 0.5 since we are solving for the six month (or ½ year) rate Note: this implicitly assumes that the six month rate is the same for both six month periods in the year.

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Example Problem Suppose your bank account pays interest monthly with an effective annual rate of 5%. What is the interest you will earn in one month? What is the interest you will earn in two years? We know from above that a 5% EAR is equivalent to earning (1.05)1/12 -1 = .4074% per month. We can use the same formula to find the two year rate (1.05)2 – 1 = 10.25% (in particular, not 10%).

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**Annual Percentage Rates**

The annual percentage rate (APR), indicates the amount of simple interest earned in one year. Simple interest is the amount of interest earned without the effect of compounding. The APR is typically less than the effective annual rate (EAR, or the interest you will actually earn/pay). APR is simply a communication device, a way to communicate a periodic interest rate where the period is not a year.

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**Annual Percentage Rates**

The APR itself cannot be used as a discount rate (except in one very special case). The APR with k compounding periods each year is simply a standardized way of quoting the actual interest earned each compounding period:

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**Annual Percentage Rates**

Converting an APR to an EAR The EAR increases with the frequency of compounding. Continuous compounding is compounding every instant. A very useful mathematical abstraction we will not dwell upon.

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**Annual Percentage Rates**

If the APR is 6% what are the EARs for different compounding intervals? Annual compounding: ( /1)1 – 1 = 6% Semiannual compounding: ( /2)2 – 1 = 6.09% Monthly compounding: ( /12)12 – 1 = % Daily compounding: ( /365)365 – 1 = % A 6% APR with continuous compounding results in an EAR of approximately %.

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Example You are offered a chance to buy a perpetuity paying a $100 per year (beginning in one year) for $1,650 today. You are able to borrow and lend money at a 6% APR with monthly compounding. Should you buy the perpetuity? First we must find the EAR since the perpetuity makes annual payments. The effective annual rate is (1+0.06/12)12 – 1 = %. The value of the perpetuity is $100/ = $1,

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**Determinants of Interest Rates**

Inflation and Real Versus Nominal Rates Nominal Interest Rate: The rates quoted by financial institutions and used for discounting or compounding cash flows Real Interest Rate: The rate of growth of your purchasing power, after adjusting for inflation

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**Determinants of Interest Rates**

The Real Interest Rate The Nominal Interest Rate

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**Example Problem Solution**

In the year 2006, the average 1-year Treasury rate was about 4.93% and the rate of inflation was about 2.58%. What was the real interest rate in 2006? Solution Using the equation given above, the real interest rate in was: (4.93% − 2.58%) ÷ (1.0258) = 2.29% Which is approximately equal to the difference between the nominal rate and inflation: 4.93% – 2.58% = 2.35%

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**The Yield Curve and Discount Rates**

Term Structure: The relationship between the investment term and the interest rate Yield Curve: A graph of the term structure

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**Term Structure of Risk-Free U. S**

Term Structure of Risk-Free U.S. Interest Rates, January 2004, 2005, and 2006

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Example When the yield curve is not flat we must discount future cash flows at the appropriate rates. Compute the present value of a risk-free, three-year annuity of $500 per year, given the following yield curve:

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Example Solution Each cash flow must be discounted by the corresponding interest rate: It is very important to note that even though this is a 3- year annuity, we cannot use the annuity formula to find its present value.

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