Warm-Up

1.         If interest rates are 6% annually, what is the future value of $83.96:

Remember that the relationship between money today (present value) and money in the future (future value) is given by FV = PV*(1 + r)t where FV is future value, PV is present value, r is the per period interest rate, and t is the number of periods. If we know any three of the variables, we can solve for the fourth.

a.   One year from today?

FV = PV*(1 + r)t = 83.96*(1 + 0.06)1 = 89.00

b.   Two years from today?

FV = PV*(1 + r)t = 83.96*(1 + 0.06)2 = 94.34

c.   Three years from today?

FV = PV*(1 + r)t = 83.96*(1 + 0.06)3 = 100.00

2.         If interest rates are 6% annually, what is the present value of $100 received:

a.   One year from today?

PV = FV/(1 + r)t = 100/(1 + 0.06)1 = 94.34

b.   Two years from today?

PV = FV/(1 + r)t = 100/(1+ 0.06)2 = 89.00

c.   Three years from today?

PV = FV/(1 + r)t = 100/(1+ 0.06)3 = 83.96

3.         If rates are 6% annually, what is the present value of three equal annual payments of $100, with the first payment occurring one year from today?

There are two ways to do this. The first is to use the fact that this is an annuity with three payments. Remember that an annuity is a stream of cash flows with one equal payment per period. We are looking for the present value of an annuity at time 0.

0               1             2              3

???           100          100          100

C「 1    ]      100「 1        ]

PV=  r |L1 -  (1+r)t 」| =  0.06L(|)1 - (1+ 0.06)3 = 267.30

The other way to do this is to realize that when cash flows are at the same point in time, we can add them up. In problem 1, you were asked to get the present value of $100 three different times: one, two, and three years into the future. Question 2 essentially asks you for the present value of all three of those. The present value of each that was calculated in Question 1 is AT time zero. Since they are all at the same point in time, we can add them up. 94.34 + 89.00 + 83.96 = 267.30. This is precisely what we got when we used the annuity formula. Remember that when cash flows are at the same point in time, we can add or subtract them.

A Little Harder

4.   During the height of the internet craze, startup dotcoms were looking for any way to get people to go to their site.   One such site, Grab.com (http://www.grab.com) offered a free lottery in which you were asked to pick 7 numbers between 1 and 77.  If anyone matched all

7 numbers that grab.com randomly selected, they would win the grand prize of $1 billion (YES – ONE BILLION, it is not a typo).  However, as in all lotteries, you would not get all the money right away.  Rather, the winner would have the following options to collect their prize:

Option 1:  An immediate payment of $170 million (paid on the day the winner claims their prize)

Option 2:  First 20 years - annual payments of $5 million

Next 10 years - annual payments of $10 million

Next 9 years – annual payments of $20 million

A final payment of $620 million in the fortieth year

(NOTE: For  Option  2,  all payments  are made  at  the end of the year, thus  the winner wouldn’t get the first payment of $5 million until 1 year from the date they claimed their prize.  This also means that they would get the $620 million at the end of the 40th year.)

Assume the interest rate is 10 % per year, compounded annually and that this rate will remain unchanged for the next 40 years.

(Incidentally, no one won the grand prize. Not so shocking when you realize that the chances of winning were 2.4 billion to one).

If you had won the lottery, which option would you take and why?

You will want to take the option that has the greatest present value. The present value of Option 1 is simply $170 million because you get it right away. The present value of Option 2 needs to be calculated.

Option 2 basically consists of 3 annuities and a final lump sum payment. The three annuities are:

Annuity 1: $5 million per year in years 1-20

Annuity 2: $10 million per year in years 21-30

Annuity 3: $20 million per year in years 30-39

The key to this problem is making sure that all the cash flows get back to year zero. Recall the formula for present value of an annuity is:

c「 1 ]

PV(annuity) = r |L1 - (1 + r)t 」|

Where c is the per period payment, r is the per period rate, and t is the number of periods. In this case, r is always 0.10 since the rate is compounded annually.

We need to get the present value of all the cash flows and add them up. A time line is very helpful for this problem. Notice that we can separate the big time line into separate time lines for each of the annuities:

We can use the annuity formula to get the present value of each annuity

Notice though that these values while all “present values” are NOT all at the same point in time so we cannot simply add them up.

The present value of the first annuity IS at time zero because it starts payments in year 1. However, notice that “present value” for the second annuity is really 20 years from now. Similarly, for the third annuity the “present value” is actually at time 30. We must bring everything back to the same point in time.

This is an illustration that the annuity formula assumes that payments start at the end of the year. Thus, the formula for annuity 1 brings it back to time 0, but the formula only brings annuity 2 back to year 20 and annuity 3 back to year 30. We need to discount those values back to time zero by using the present value of a lump sum.

PV =

So the present value of annuity 2 (61.4457) needs to be discounted an additional 20 years and the present value of annuity 3 (115.1805) needs to be discounted an additional 30 years.

PV = = 9. 1335

PV = = 6.6008

Additionally, we need to discount the final payment of $620 million back 40 years.

PV = = 13.6989

0    1                   19     20     21                          29        30     31                                39        40

5                   5       5       10                       10           10     20                                20        620

42.5678

61.4457

9.1335

115.1805

6.6008

13.6989

Now, we have all the cash flows back at time zero and we can simply add them up. 42.5678 + 9.1335 + 6.6008 + 13.6989 = 72.001

Thus, the present value of Option 2 is about $72 million. Since the present value of Option 1 is $170 million, you would select Option 1.

Incidentally, you could have easily answered this question using your financial calculator and the cash flow worksheet. On the BA-II+ enter the cash flow worksheet by pressing <CF>. To clear a worksheet you need to press <2nd><CE/C> which is listed as CLR Work.

You should now see CF0 which stands for the cash flow at time 0. Since you get no money immediately, leave it at 0 and press the down arrow <↓>

C01 is the 1st cash flow in the stream. Here it corresponds to 5, so press 5<↓>

Luckily, you do not need to individually enter all 20 cash flows of $5 million. F01 stands for the frequency with which you get C01. Since you get 20 consecutive payments of 5, you can enter 20<Enter><↓> (Note: by default frequency will be set to 1).

C02 stands for the 2nd cash flow in the stream. Despite the fact that the $10 million comes at time 21, it is the 2nd cash flow because you already told the calculator to include 20 payments of 5 million.

10<Enter><↓>

You get 10 payments of 10 million, so F02 is also 10

10<Enter><↓>

Similarly:

C03 =20<Enter><↓>

F03 = 9<Enter><↓>

C04 = 620<Enter>

There is no need to include F04 since by default it is 1.

You have now let the calculator know the entire time line of the payments. To get the present value of it, press the net present value key <NPV>. It will ask you for an interest rate. The rate we are using is 10%, but on the BA-II+, interest rates are not entered in decimal for so: 10<↓>

You should now see NPV=0. To have it calculate the present value of that cash flow stream at 10%, simply press the compute button <CPT>. You should get an answer of 72.001 million which matches what we calculated doing everything with the formulas.

NOTE: Your financial calculator is a very helpful tool if you understand how to use it. It would be wise to invest some time learning the functions of your calculator.

5.         Grab.com did not put up the billion dollars on their own; rather, they paid several million dollars to an insurance company who stood by to take the risk of paying a winner.  Being a  major  player  in  the  insurance  industry,  one  of Berkshire  Hathaway’s  subsidiaries (National Indemnity Company) insured the risk.   Suppose that someone had won the billion  and  elected to  take the  deferred  stream  of payments.   Assume  that National Indemnity Company decided it would rather just make a single deposit into an account earning 10% compounded annually such that:

The deposit is made on the day the winner claims their prize Any interest earned in the account will be reinvested into the account.

At the time each payment is due to the winner, the money will be withdrawn from

the account.

They make the smallest deposit needed to fulfill their obligations.  In other words,

when the final payment is made to the winner, the account will have no money left over.

How much needs to be deposited into the account to meet all their criteria?

While it may seem that this is a complicated problem, the fact is that no calculations need to be done because you have already answered this exact question in #5. The present value of Option 2 was $72.001 million. Recall that the concept underlying present value is that it is the amount which makes you indifferent (at the rate of 10%) between the lump sum and the proposed stream of cash flows. In other words, $72.001 million today is equivalent to the following stream of payments:

First 20 years - annual payments of $5 million

Next 10 years - annual payments of $10 million

Next 9 years – annual payments of $20 million

A final payment of $620 million in the fortieth year

Conceptually, it also means that depositing the $72.001 million into an account today would be sufficient to allow the account to pay out the entire stream of cash flows, accounting for reinvested interest. If this seems difficult to believe, take a look at the Grab.com excel worksheet on Blackboard. You will see this problem worked out year by year and notice that the starting value of $72.001 (subject to rounding) allows the firm to make each payment and leaves the account with a zero balance at the end of the 40 years.

Time value of money is a very useful concept and the annuity formula is incredibly powerful because it takes into account all of the concerns dealing with reinvested interest. Pretty cool, isn’t it?