lesson 1: the time value of money -...

Lesson 4.1 Page 428 of 915.

Lesson 1: The Time Value of Money

Would you rather have $ 1000 today, or a promise of $ 1000 a year from now? It is onlynatural to prefer the money now, and the justification for this preference will be preciselyquantified in this lesson. Furthermore, we will explain opportunity cost, the time value ofmoney, and treasury notes & treasury strips markets.

It is very customary in B2B (Business-to-Business) transactions to give the buyer 60 daysto pay for a purchase; some companies even allow 90 days. Often when vendors want toreduce this window of time, buyers become upset and threaten (or even consider) changingsuppliers. Let’s examine why!

Suppose your company prints books. Also, suppose you need $ 100,000 worth of paperin the second quarter of 2010 (April 1, 2010 to June 30, 2010). You might place the orderon March 31st and have the money to pay for it. If so, then you pay $ 100,000 on the spot.Alternatively, you might instead hold payment for 90 days if the supplier allows it. Duringthat time, you can earn interest on the $ 100,000 by investing the money in a bank account.If the account pays 6% interest (compounded monthly) at the end of 90 days you wouldhave

A = (100, 000)(1 + 0.06/12)3 = (100, 000)(1.005)3 = 101, 507.51

and your company could keep the $ 1507.51 as profit without e↵ort, just by deferringpayment. This profit through capital investing is a common revenue stream for businesses,and is called cash flow management or capital management .

Do the same problem, but with 60 days to pay, and then again with 180 days. Howmuch does the firm pocket for delaying its payment in each case? [Answer: $ 1002.50 and$ 3037.75.] Can you see now why buyers wish to have long windows during which to pay?

We’re going to see the term prevailing rate a lot over this lesson, and throughout the restof the chapter as well. This term can be thought of as the rate at which you can invest asum of money for a relatively short amount of time, with little risk of loss. There’s a lotof wiggle room in the definition of the “prevailing rate,” and for good reason.

In reality, a manager in a firm has a wide variety of choices available to him when heor she has unallocated capital for a short window of time. Each choice has its own risks,rate of return, and requirements. Usually, through either the commercial paper market(see Page 208) or the treasury-bill market (which we will learn about in this lesson) onefinds a very good balance of extremely low risk and a publicly-known rate of return. Forthat reason, temporary “extra cash” will often be invested into (or be borrowed from) thesemarkets, and the “going rate” there is the prevailing rate for any firm.

Luckily for you, any math problem will either tell you the prevailing rate outright or askyou to solve for it mathematically. You are not required to—nor should you ever attemptto—estimate the prevailing rate from intuition. Like with all interest rates, tiny changeshave drastic consequences, and so one must always endeavor to be precise.

COPYRIGHT NOTICE: This is a work in-progress by Prof. Gregory V. Bard, which is intended to be eventually released under the Creative

Commons License (specifically agreement # 3 “attribution and non-commercial.”) Until such time as the document is completed, however, the

author reserves all rights, to ensure that imperfect copies are not widely circulated.


Suppose I have a payment coming to me with a dollar value of $ 10,000. How much is thepayment really worth to me? The question may sound foolish, but the answer depends onthe date of the payment—whether the payment is coming tomorrow, next week, or 30 yearsin the future.

As a dollar value is promised further into the future, its present value to us actuallydiminishes by a rate known to us—the prevailing rate. We can, however, use the techniquesin this lesson to find a future payment value that is equivalent to the desired value today.The formula we will use is called the “Time Value of Money Formula,” and in fact it is thesame formula mathematically as compound interest.

Up to this point we have used the term “amount” to refer to the end value of a loan, and“principal” to the beginning of a loan. As we drift toward more sophisticated and complexfinancial instruments, it becomes necessary and desirable to follow the terminology of thefinance community.

The value “at this moment” of any financial instrument is called its present value, and isdenoted PV . Naturally, the future value is denoted FV .

This terminology is a generalization of “principal” and “amount” respectively, andapplies to things as simple as a treasury bill (which you’ll learn about very shortly) andas complex as a 30-year fixed-rate mortgage that has (because it is monthly) 360 distinctpayments.

In particular, the Time-Value-of-Money concept is nearly always described in terms ofPV and FV .

The “Time Value of Money Equation” is merely a manipulation of the compound interestformula. We start with the original, rewriting A as FV and P as PV :

FV = PV (1 + i)n () FV

(1 + i)n= PV

Sometimes it is easier to substitute n = mt and i = r/m to obtain

FV = PV (1 +r

m)mt () FV

(1 + rm )mt

= PV

which is equivalent.

Many students do not treat the Time Value of Money Equations as a new equation, but just see it as compound interestslightly relabeled.





Suppose that your firm has sold some equipment to a factory and that payment was duefrom them today, for a total of $ 500,000. They call on the phone, apologize, and tell youthat exactly sixth months from now they will be able to pay you, but no sooner. How muchmoney has the six-month delay cost your company? To figure this out we need to know thePV associated with the FV of $ 500,000 six months from now. Suppose the prevailing rateis 6% compounded monthly. The formula tells us

PV =FV

(1 + i)n=

500, 000

(1 + 0.06/12)6=

500, 000

1.03037 · · · = 485, 259.03

So it comes to pass that your company was expecting $ 500,000 and instead has receiveda promise worth $ 485,259.03 today. Thus, the loss comes to

500, 000� 485, 259.03 = 14, 740.97

The value of money decays over time, thus you really aren’t getting paid in full, eventhough it is the same dollar amount agreed upon at the beginning of the transaction. Onehopes that the corporate lawyers have written some sort of “late fee” into the contract—thisis sometimes forgotten.

In the previous problem, what would the loss be if the $ 500,000 payment were 3 monthslate? 9 months late? 12 months late? [Answer: For 3 months, $ 7425.62; for 9 months,$ 21,947.65; and for 12 months, $ 29,047.33.]

Minor note: The prevailing rate was compounded monthly, so all values of time shouldbe calculated monthly as well. Doing so will ensure that your calculation is as exact aspossible.

Let’s suppose that you are a financial trader at the capital management desk of a majorcorporation. You are an expert in commercial paper, which is discussed on Page 208. Threecompanies are selling IOUs, all written by the same highly prestigious and trusted majorcorporation, General Hydraulic. Coincidently, each IOU has a face value of one milliondollars. One IOU, call it “Note A,” is due in 60 days and is going for $ 994,000. AnotherIOU is due in 90 days (call it “Note B”) and is going for $ 985,800. Finally, the third IOU,call it “Note C,” is due in 30 days and is going for $ 995,100. Which one should you buy?The prevailing rate for you is 5.5% compounded monthly. You only have enough cash tobuy one of them.

Given your prevailing rate, the correct way to approach these notes is to calculate whattheir present value is to you, given that their future value is one million. Then you comparethe present value with the actual asking price. If the asking price is higher, then it is not agood deal at all. If the asking price is lower, then it is attractive, with a larger gap beingmore attractive. More plainly, the lower the asking price, the more attractive the deal. Let’sdo that calculation now.




Lesson 4.1 31 of 915.

Which Note? t n Computation of the Present Value Asking Price Di↵erence

Note A 60 days 2 FV(1+i)n = 1,000,000

(1+0.055/12)2 = 1,000,0001.00918··· = 990, 895.97 $ 994,000 $ -3104.03

Note B 90 days 3 FV(1+i)n = 1,000,000

(1+0.055/12)3 = 1,000,0001.01381··· = 986, 375.08 $ 985,800 $ 575.08

Note C 30 days 1 FV(1+i)n = 1,000,000

(1+0.055/12)1 = 1,000,000

1.0045833= 995, 437.57 $ 995,100 $ 337.57

The column marked “di↵erence” is the present value minus the asking price. From our calculations, we learn that“Note A” is a really bad note and would result in overpayment. “Note B” is a di↵erent story—it will deliver the 5.5%compounded monthly plus an additional $ 575.08, so this is by far the best option. Therefore, you should buy “Note B.”

It is interesting to see that “Note C” is also good, but not as good as “Note B.” They both yield a profit, and so ifyou had enough available cash then you would take both. However, “Note B” has a higher profit, and so because youcan only a↵ord one note, you should take “Note B.”

Suppose there are three notes each with face value of $ 2,000,000. “Note D” is goingfor $ 1,982,815.63 and is due in 45 days. “Note E” is due in 60 days and is going for$ 1,976,308.19. Finally, “Note F” is going for $ 1,988,935.85 and is due in 30 days. Supposethat the prevailing rate is 7%, compounded daily. What is the present value of each noteto your company? Which should you buy? Please use a 360-day year.

[Answers:

• “Note D” is worth $ 1,982,578.02 which is $ 237.60 below the asking price;

• “Note E” is worth $ 1,976,804.49, actually $ 496.30 more than the asking price;

• “Note F” is worth $ 1,988,368.42, which is $ 567.42 less than the asking price.

• Obviously, “Note E” is the note to buy because the other two are worth less than theasking price! ]

Suppose there is a class action lawsuit against a chemical corporation. The chemical com-pany knows it has broken the law, so it wishes to settle out of court. They estimate thatthe trial, if it would occur, would take 4 years once all the preliminaries were done, the trialcame up on the docket, ran its course, and the appeals were finished. So, to save itself theaggravation of litigation, the chemical company agrees to pay the plainti↵s $ 72 million ex-actly 4 years from now. Their o↵er is accepted. What amount must the chemical companydeposit now, in an account earning 8% compounded monthly, to have that amount on thatdate?

You can either see this is as an ordinary compound interest problem, or as asking “whatis the present value of a note for $ 72 million dollars due in 4 years?” Let’s approach it thelatter way:

PV =FV

(1 + i)n=

72, 000, 000

(1 + 0.08/12)48=

72, 000, 000

1.37566 · · · = 52, 338, 281.78

Therefore the company must place about 52.3382 · · · million dollars in the account at thistime, to have the $ 72 million ready to be paid 4 years from now.





It is likely that in the previous box, your calculator might disagree with mine, particularly at the digits representingthe pennies. A hand-held calculator has limited precision, but you should always be able to get your answer to matchmine to at least five, if not six, significant figures.

Now we will re-examine the previous problem with more realism, and from the point of viewof the plainti↵s. Because the plainti↵s are individual people, they do not have access to thetreasury note/commercial paper markets to earn that type of rate. Alternatively, some ofthe plainti↵s might have credit card debt, perhaps at 19.95%, compounded daily. (This toois a prevailing rate, because they’d like to pay that o↵.) The earlier the plainti↵s get theirlarge payment, the sooner they can pay o↵ their debts as individuals.

You will perform the calculation yourself in the next box.

Using 19.95% as the prevailing rate, what is the present value for the plainti↵s of the 72million dollar promise? Should they counter-o↵er the chemical company with a settlementof 50 million, but delivered immediately? For a change, use a 365-day year.

[Answer: The present value for them is $ 32,423,521.22, and so $ 50 million immediatelywould be much better.]

We talked about Treasury Bonds on Page 210 and US Savings Bonds on Page 222. Thereis another type of financial instrument that the government uses to raise money for its day-to-day fluctuations in wealth. (After all, taxes usually arrive very irregularly; think aboutwhat April 15th alone does to the US government!) This day-to-day financing is done withTreasury Bills , which are remarkably simple.

A treasury bill could be thought of as a promise from the U.S. Government for a specifiedpayment at a later date. This date is called the maturity date. If I buy a $ 10,000 USTreasury Bill, then I know that on the maturity date, I will receive the face value of thebill as a payment from the government. These bills are sold at auction, just like at an artauction (or your favorite internet-auction website). The maturity date is usually 13 weeksor 26 weeks after the date of the auction, or sometimes 4 weeks after.

Frequently, some bills trade hands several times before they mature, being bought andsold on secondary markets. As you can imagine, these transactions can be modeled perfectlyas simple Time Value of Money Equation problems.





Let’s suppose there is an auction for a $ 250,000 treasury bill, and the winning bid is$ 247,000.00. The maturity date is 26 weeks into the future. (A banker would say thatthe duration is 26 weeks.) What does that suggest about the prevailing rate, compoundedannually? In other words, what rate of interest would make this price the fair PV for theFV of $ 250,000?

We start with the basic Time Value of Money Equation:

FV = PV (1 + i)n

250, 000 = (247, 000)(1 + i)1/2

250, 000

247, 000= (1 + i)1/2

1.012145 · · · = (1 + i)1/2

(1.012145 · · · )2 = (1 + i)

1.024439 · · · = 1 + i

i = 0.0244390 · · ·

Thus, i = 2.44390 · · ·% is the rate per compounding period (per half-year), so doublethat (r = 4.88780 · · ·%) is the correct nominal rate per year. Ordinarily, one would write4.88%.

What if the winning bid

• were $ 240,000? [Answer: 17.0138%.] (By the way, that is extraordinarily high.)

• Or if it were $ 248,000? [Answer: 3.23881 · · ·%.]

• How about $ 247,500? [Answer: 4.06081 · · ·%.]

In these auctions, the bids are reported in the form 98.000, 98.500, 97.500, et cetera. Whatthis means is that the bidder is willing to pay 98%, 98.5%, 97.5%, et cetera, of the face valueof the treasury bill. If many di↵erent bills of many di↵erent face values are being sold, it ismuch easier to look at a list of these reports than at the sales prices, if you wanted to havean idea of how the market is moving.

Note that the bid report should never be over 100.000. A bid report such as 101.000would imply that someone is willing to pay $ 101,000 today in return for receiving $ 100,000after 26 weeks. Clearly, no one would actually do that. Likewise, a bid of exactly 100.000would represent someone giving the government an interest-free loan for 26 weeks.

A Pause for Reflection. . .Suppose you are taking this course with a friend, and an exam is looming on the horizon.Your friend emails you about the previous box, asking “Why is it that the bid is never over100.000? Why are 98.000, 97.500, and 97.000 possible but not 101.000?” Write a response,trying to make the explanation as intuitive as possible. That means you should use veryfew numbers.





For interest compounded semiannually, what represents a higher interest rate: a bid reportof 97.000 or a bid report of 98.000? If the note is for 26-weeks, what interest rates areprevailing? [Answer: Because a bid report of 97.000 indicates r = 6.18% and a bid reportof 98.000 indicates r = 4.08%, the bid report being 97.000 is the situation with a higherinterest rate.]

It is an unfortunate choice of terminology, but in the bond market, if a treasury bill with facevalue $ 250,000.00 sells for $ 246,250.00, then the word “price” does not refer to $ 246,250.00.Rather, the term “price” refers to

246, 250.00

250, 000.00· 100 = 98.500

which we called the “bid report.” Instead, the $ 246,250.00 is referred to as “the cost.” Itis okay if you find this irritating—I find it irritating too. This terminology, so far as I amaware, is only used in the treasuries market.

If the prevailing rate on treasury bills is 4% (compounded annually), what should I expectthe cost of a 13-week treasury bill with face value $ 500,000 to be? And of a 26-weektreasury bill?

The formula tells us

PV =FV

(1 + i)n=

500, 000

(1 + 0.04)1/4=

500, 0004p1.04

= 495, 121.36

as well as

PV =FV

(1 + i)n=

500, 000

(1 + 0.04)1/2=

500, 000p1.04

= 490, 290.33

Let’s try some variations on the previous box. What if the rate were

• Only 3%? [Answer: For 26-weeks, $ 492,664.63, and for 13-weeks, $ 496,318.76.]

• Instead 5%? [Answer: For 26-weeks, $ 487,950.03, and for 13-weeks, $ 493,938.27.]

• As high as 6%? [Answer: For 26-weeks, $ 485,642.93, and for 13-weeks, $ 492,769.18.]

As you can see, these prices are highly sensitive to the interest rate. That is why whenthe Federal Reserve Board debates changing the interest rate, however necessary that mightbe, it is like attempting to fine tune an analog radio’s dial with a sledge hammer.

A Pause for Reflection. . .Suppose you show these last two boxes to someone who has not read this lesson. They arepuzzled as to why the bonds get less expensive as the interest rate increases. How wouldyou explain to them why a rise in the interest rate causes the price of the bonds to fall,while a decrease in the interest rate causes the price of the bonds to rise?





As we described on Page 210, the standard US Treasury bond that lasts t years pays interestsemiannually. This means there will be 2t payments, which are calculated using simpleinterest. The face value is then returned at the end. So, for example, if you buy a $ 10,000treasury note that is 20 years in duration, then you will receive 40 interest payments plusthe final return of the principal. If the interest rate is 5% per year, then that is $ 500 peryear or $ 250 per payment. Those payments are called coupons.

What some traders do is buy an instrument that behaves just like a US TreasuryBond. Then, the investor sells each payment separately. There are 41 payments (40 interestpayments plus the face value at the end). Each of these becomes what is called a strip,or a promise of some specific payment on some specific date. After all, it is the federalgovernment making the payments, so payment is very reliable. Thus people may wish tobuy strips, as a safer version of commercial paper (see Page 196).

The popularity of strips as forms of payment creates the treasury strips market. Anothername for a strip is a zero-coupon bond .

A trader acquires a 2-year treasury note with face value $ 20,000. Payment—like most USTreasury Bonds—is semiannual, so there will be 4 interest payments plus the return of theprincipal upon maturity. It turns out this bond was issued at 5%, so each semi-annualpayment will be for $ 500. If the trader wishes to dismember the note into “strips,” therewould be 5 new zero-coupon bonds:

The first will have a face value of $ 500, due in six months; three more at $ 500 just likethe first, will be due in one year, 18 months, and 2 years; finally, the face-value paymentwill be due in 2 years for $ 20,000. How much will each of these strips be selling for, if theprevailing rate remains at 5% semi-annually?

PV =FV

(1 + i)n=

500

(1 + 0.025)1=

500

1.025= 487.80

PV =FV

(1 + i)n=

500

(1 + 0.025)2=

500

1.05062 · · · = 475.90

PV =FV

(1 + i)n=

500

(1 + 0.025)3=

500

1.07689 · · · = 464.29

PV =FV

(1 + i)n=

500

(1 + 0.025)4=

500

1.10381 · · · = 452.97

PV =FV

(1 + i)n=

20, 000

(1 + 0.025)4=

20, 000

1.10381 · · · = 18, 119.01

Repeat the above problem with 4% interest.

[Answer: The coupons would go for $ 490.19, $ 480.58, $ 471.16, and $ 461.92 (inorder from nearest due date to furtherest due date), and the final payment would go for$ 18,476.90.]





The Time Value of Money Equation has uses outside of single payments. Let’s say that youhave a $ 10,000 loan with a di�cult younger brother who has been slow in repaying you.You finally settle that he will make 4 equal payments, paid 1 year from now, 2 years, 3years, and 4 years from now. Your prevailing rate is 7% compounded annually. How muchshould each payment be?

Each payment has the same face value of k dollars. The PV of the first payment will bek/(1+ 0.07)1 and the second will be k/(1+ 0.07)2. Likewise, the third will be k/(1+ 0.07)3

and the fourth k/(1 + 0.07)4. The total present value of these four payments to you is

k(1 + 0.07)�1 + k(1 + 0.07)�2 + k(1 + 0.07)�3 + k(1 + 0.07)�4 = 10, 000

You will observe that we have used one of the laws of exponents, specifically a�n = 1/an

(see Page 241). We can use our calculator to find the values of 1.07 raised to various powers,and obtain

k(0.934579 · · · ) + k(0.873438 · · · ) + k(0.816297 · · · ) + k(0.762895 · · · ) = k(3.38720 · · · )

Finally, we can set the above equal to $ 10,000 and obtain

k = 10, 000/3.38720 · · · = 2952.29

Thus, each of the four payments should be for $ 2952.29.

What if the previous problem were with 6% interest, compounded semi-annually? Howabout 5% interest, compounded monthly? [Answer: $ 2891.86 at 6% and $ 2827.73 at 5%.]

Suppose you have a debt with someone of $ 5000, and they promise to pay you three equalpayments. One will be 2 months from now, and one will be 8 months from now, but thelast will be 12 months from now. How much should each payment be? Use a prevailing rateof 12% compounded monthly. [Answer: $ 1791.32.]

If you noticed that the previous problem looks like a (very short) geometric series, then youare correct—it is! We will use exactly this concept to build up the formulas for mortgagesand other complex financial instruments on Page 469. So if the two boxes above this oneseem complicated, relax—we’ll have a shorter method of doing these sorts of problems beforethe end of the chapter, but not before the end of this lesson.





A grandfather has three grandchildren, aged 5, 8 and 12 years old. He wants them to receivefrom him exactly the same dollar amount on their respective 21st birthdays, and he has$ 1.5 million to divide between the three of them. He knows of a mutual fund that reliablyhas returned 8% per year for a long time. How much should the grandfather deposit in themutual fund under the name of each child, and how much will they each get on their 21st

birthday?First, let f denote what each grandchild will get. (Since all three grandchildren get the

same amount at the end, we only need one variable, not three.) Then we must think, is fan FV or a PV ? Surely it is an FV , because it is in the future.

Next, what is the PV ? Here we will use the symbols PV1

, PV2

, and PV3

to representthe PV to the 5 year old, the 8 year old, and the 12 year old, respectively. Note that the5 year old will turn 21 in 16 years; the 8 year old will turn 21 in 13 years; and the 12 yearold will turn 21 in 9 years.

Now we write

PV1

=FV

(1 + i)n=

f

(1 + 0.08)16= f(1.08)�16

where we have quietly used the law of exponents a�n = 1/an (see Page 241). Likewise,the only thing di↵erent about the grandchildren in this problem is their age, so we can justwrite PV

2

= f(1.08)�13 and PV3

= f(1.08)�9. The calculation will be completed in thenext box.

This is a continuation of the previous box. Now we add up the PV s.

1, 500, 000 = PV1

+ PV2

+ PV3

1, 500, 000 = f(1.08)�16 + f(1.08)�13 + f(1.08)�9

1, 500, 000 = f(0.291890 · · · ) + f(0.367697 · · · ) + f(0.500248 · · · )1, 500, 000 = f(1.15983 · · · )

1, 500, 000

1.15983 · · · = 1, 293, 284.77 = f

Well, that wasn’t so bad! Now we know that each grandchild will get $ 1,293,284.77, and then we can quickly calculate

PV1

= f(1.08)�16 = (1, 293, 284.77)(0.291890 · · · ) = $ 377, 497.49

PV2

= f(1.08)�13 = (1, 293, 284.77)(0.367697 · · · ) = $ 475, 538.12

PV3

= f(1.08)�9 = (1, 293, 284.77)(0.500248 · · · ) = $ 646, 964.37

The previous box had a lot of di�cult mathematics, and it is easy to imagine a stray keypresson the calculator ruining the computation. However, we can easily check our work. If youtotal up those three deposits, you obtain

377, 497.49 + 475, 538.12 + 646, 964.37 = 1, 499, 999.98

Our calculation is very, very close to the desired amount of $ 1.5 million. In fact, hadI used rounding instead of truncation, then it would be exact to the penny.





A Pause for Reflection. . .A person might look at those three deposits and see that the one for the eldest grandchildis double that of the youngest grandchild; this might appear to be the very opposite offairness. Suppose the grandfather is your neighbor, and that he dies eight years after thedivision of wealth. His grandchildren, now 13, 16, and 20 years old, have learned of thedivision of wealth and are upset. You, of course, know about the Time Value of MoneyEquation. Write a letter explaining to them why this is fair, and why an equal divisionwould have been extremely unfair.

Repeat the above problem if there were two grandkids, aged 17 and 12. What does eachchild get on his 21st birthday? How much is deposited in each account?

[Answer: The 17-year-old has a deposit of $ 892,547.30 and the 12-year-old has a deposit

of $ 607,452.69, so that they each will receive $ 1,214,300.74 on their 21st birthday.]

There are several ways to check your work. First, note that

607, 452.69 + 892, 547.30 = 1, 499, 999.99

which is good, because that’s what the grandfather has available. Next

607, 452.69⇥ (1 + 0.08)21�12 = 607, 452.69⇥ 1.99900 · · · = 1, 214, 300.73

892, 547.30⇥ (1 + 0.08)21�17 = 892, 547.30⇥ 1.36048 · · · = 1, 214, 300.74

We have learned the following skills in this lesson:

• To find the present value of future payments.

• To perform calculations involving Treasury notes.

• To calculate the present values of the coupon of a bond.

• To divide a sum of money among irregularly occurring payments.

• As well as the vocabulary terms: capital management, cash flow management,coupons, duration, maturity date, prevailing rate, strips, time-to-maturity, treasurybills, treasury strips, zero-coupon bond.

Coming Soon!




lesson 1: the time value of money -...

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