The Time Value of MoneyThe TimelineThe 1st Rule of Time TravelThe 2nd Rule of Time TravelThe 2nd Rule of Time TravelThe 2nd Rule of Time TravelThe 3rd Rule of Time TravelThe 3rd Rule of Time TravelValuing a Stream of Cash FlowsExampleSolutionExamplePerpetuityGrowing PerpetuitySlide 15AnnuitiesValuing AnnuitiesAnnuity ExampleAnnuity Due ExampleSlide 20Slide 21Growing AnnuitiesExampleInterest Rate Quotes and AdjustmentsInterest Rate Quotes and AdjustmentsExampleAnnual Percentage RatesAnnual Percentage RatesAnnual Percentage RatesAnnual Percentage RatesExampleDeterminants of Interest RatesDeterminants of Interest RatesExampleThe Yield Curve and Discount RatesSlide 36ExampleExampleThe Time Value of MoneyThe 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?021-$1,000+$500 +$550The 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?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:(1 )nFV C r= � +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: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 $8,053.55 in 5 yearsWhat if you need money today, does that change the decision?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:(1 )(1 )nnCPV C rr= � + =+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 = $6,209.21 today which can be compared to the alternative of $5,000 today.Alternatively:$6,209×(1.10)5 = $10,000 so if you had $6,209.21 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 $6,209.21.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 0 0 ( ) (1 )= == =+� �N Nnnnn nCPV PV CrExample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?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 $481.59 today and pay $420 for the claim. The claim will just repay the loan and you keep the $61.59.2 3$100 $200 $300$481.59(1.10) (1.10) (1.10)PV = + + =$481.59 $420 $61.59NPV PV Cost= - = - =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%?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.…0 1 2 3C C CrCPV 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:…0 1 2 3C1C1(1+g) C1(1+g)2grCPV1Perpetuity ExampleLevel PerpetuityYear BofY Bal INT@10% Payment EofY Bal1 1000 100 100 10002 1000 100 100 10003 1000 100 100 10004 1000 100 100 1000Growing PerpetuityYear BOY Bal. INT@10% Payment EOY Bal.1 1250 125 100 12752 1275 127.5 102 1300.53 1300.5 130.05 104.04 1326.514 1326.51 132.651 106.1208 1353.04Place 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?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.0123 C C C0123 C C CValuing AnnuitiesWe can do a lot of grunt work or we can notice that a T period annuity is