Suppose a rich relative, "Uncle Bill" offers to give you one of two, mutual exclusive gifts just for being his niece:
If you choose Gift A, he would give you $100 a year from now, $100 two years from now $100 three years from now and $250 six years from now.
If you choose Gift B, he would give you $325 five years from today and another $325 six years from today.
Which gift should you pick?
To answer this question, you need to know a few things: Is this guy really your uncle? Mom says he's her brother, but are you sure? In other words, is this guy true to his word? Will he keep his promise over the full six years? What's the expected rate of inflation over the next six years? What rate can you earn investing your money? Are you able to save what Uncle Bill gives you or are you likely to go to the mall with each installment?
We will talk more about all of these factors later, but for now let's assume you will save the money and you can earn 10% per year by investing it. i=.10.
Here's the two gifts:
A 100 100 100 250 0 1 2 3 4 5 6 B 325 325 0 1 2 3 4 5 6To solve this dilemma, you can't just say, "Oh, with A, I get only $550 but with B, I get $650, so I'll take B." Remember, you can make 10% on your investments and with Gift A, you get the money a lot sooner and can earn lots more interest on it.
To answer the question, then, you need to take all the dollar amounts to the same, common point. It could be time 0 (present value), it could be time 6 (future value), it could be time 4. It doesn't matter. Whichever gift is better at time 0 will also be better at time 6 or time 4. It just has to be a common point for all cash flows in order to be able to "compare apples with apples."
Since we're making the choice today at time 0, let's start with calculating the present value of both gifts at time 0. For each gift, discount all the payments back to time 0 using an interest rate of 10%.
Let's start with A: The first $100 is a full year away. It's worth only 100/(1.10)1 or 90.91. The second $100 is two years in the future so its value today is only 100/(1.10)2 82.64. The third $100 is worth only 100/(1.10)3 or 75.13 today. The 250 payment can be discounted back to the present by 250/(1.10)6 or 141.12. Add up the four present values: 90.91 + 82.64 + 75.13 + 141.12 = 389.80.
Before we look at Gift B to see how it stacks up, let's think about what that 389.80 means.
389.80 is the amount you could put away today in a savings account that earns 10% interest and be able to withdraw 100 at the end of each of the next three years and 250 at the end of year 6 and have nothing left over.
| Time | Interest | Value | Withdrawal | Remaining |
|---|---|---|---|---|
| 0 | - | 389.80 | - | 389.80 |
| 1 | 38.98 | 428.78 | 100 | 328.78 |
| 2 | 32.88 | 361.66 | 100 | 261.66 |
| 3 | 26.17 | 287.83 | 100 | 187.83 |
| 4 | 18.78 | 206.61 | - | 206.61 |
| 5 | 20.66 | 227.27 | - | 227.27 |
| 6 | 22.73 | 250 | 250 | 0 |
Now let's do the same Present Value calculations for Gift B. The first 325 is discounted back 5 years at 10%. Its value at time 0 is 325/(1.10)5 = 201.80. The other cash payment is worth 325/(1.10)6 = 183.45. Therefore, Gift B is worth 201.80 + 183.45 = 385.25. Your kid brother would be willing to pay you only 385.25. Gift A is worth more and is the better gift given these conditions.
What about time 6? What if compound all the dollar amounts to the end of each gift? Won't the bigger cash flows of Gift B win out? Let's find out.
The future value of Gift B is easy to calculate: 325*(1.10)1 + 325 = 682.50. Bigger cash flows but no time to earn any interest on them. Gift A has smaller payments but lots more time to earn interest by investing them. The future value of A at time 6 is 100*(1.10)5 + 100*(1.10)4 + 100*(1.10)3 + 250 = 690.73. Another narrow victory to A. Whichever gift wins at time ) will also win at time 6 or at time 3 or 4.
Notice that if you take the present value of A (389.90) and compound this lump sum for 6 years at 10% you get: 389.90 * (1.10)6 = 690.73!!! You can go forwards and backwards with these lump sums.
If you have already learned about annuities, you can easily get A's present value using a quicker method than we used above.
A 100 100 100 250 0 1 2 3 4 5 6The three 100 payments is an annuity and its present value at time 0 is equal to 100 * (PVIFa - 10% - 3) = 248.69. Add to that the PV of the 250 (250/(1.10)6) = 141.12 and you get the 389.81.
For Gift B, it's a little more complicated to get its present value at time 0. The two 325s are a deferred annuity. First take the value of the two 325s one period prior to the first payment--time 4. PV4 = 325 * (PVIFa - 10% - 2) = 564.05. Then discount this lump sum back 4 more periods to time 0. PV0 = 564.05/(1.10)4 = 385.25.
Annuities? Now that I've brought them up, let's review how they work. Here's the no nonsense, quick and dirty summary: Use R * (PVIFa - i% - n) to find the present value of an annuity of n payments of R dollars each discounted at i% to ONE PERIOD PRIOR TO THE FIRST PAYMENT. Got that? Good! Your tables and your calculator do it that way. So should you. Calculator buttons: PMT = R, I = i per period, n = # payments, hit PV last.
What if one period prior to the first payment isn't what you want. Too bad. The calculator does it that way. Therefore, you need to adjust the calculator's answer. Take the lump sum answer you get with the above annuity formula and discount it as a lump sum back however many more periods you want it. But remember, the annuity formual already has taken the payments back that one period prior to the first payment. Don't double count it.
Perpetual Annuities
Today you put $2,000 into an investment that pays 10% interest a year. How much will be in the investment a year from now? Easy! 2000*(1.10)1 = 2200. Obviously, you earned 200 in interest over the past year. You can withdraw the 200 in interest that you earned and leave the 2000 untouched. Leave the 2000 in for another year and you'll have 2200 a year later. Again, you can take the 200 in earned interest out of the investment and not touch the 2000 original principal. As long as the interest rate is 10%, you can theoretically take out 200 a year for ever (in perpetuity). Suppose someone offers you an infinite stream of $200 a year. What is that offer worth to you today? It's the present value of the infinite stream discounted at the going rate of interest. In class we showed that the (PVIFa - i% - n periods) is equal to 1/i for n = infinity. What looks complicated in the beginning is the easiest to work with in the end. The perpetual annuity formula works like any other: PV = R * (1/i) or PV = R/i. The promise of an infinite stream of 200 a year is worth 200/.10 = 2000 if the interest rate is 10%. The promise is worth only 1000 if the interest rate is 20% [1000=(200/.20)]. Why? Well, you could invest 1000 today at 20% and produce those same 200 annual withdrawals for ever. So, you wouldn't pay someone more than 1000 for the promise if you could earn 20% on your investment.
Suppose the interest rate is back to the original 10% but first 200 payment won't be until the end of year 4. Now what's the promise of 200 a year for ever worth today? PV of the stream is 200/.10 = 2000. But that's the present value ONE PERIOD PRIOR TO THE FIRST PAYMENT (just like any other annuity). So it's the present value at the end of year 3. You need the 2000 one year ahead of time so you can earn the 200 interest during the fourth year so you can then take it out without touching the 2000 principal. So you need to discount it back to time 0 by using the lump-sum 2000/(1+.10)3 = 1502.63. A very common and costly error on exams is to bring it back 4 periods rather than 3. This is incorrect because the annuity formula (1/i) already brings it back to the end of the third year.
Effective Rates
If the frequency of compounding is other than annual, it becomes necessary to distinguish between the effective annual rate and the stated nominal rate (APR--annual percentage rate). If a bank offers to pay 12% interest a year but to compounded it quarterly, what they are really saying is that they will pay (12% / 4 quarters) = 3% a quarter or period for four quarters or periods a year. Got that? If they compound the interest monthly, then they give (12%/12 periods per year) or 1% a month. Both of these savings plans beat simply paying 12% a year with annual compounding. Why? Because you can earn interest on the interest and emerge at the end of the year with more money. Let's use the 12% annual interest with quarterly compounding. Invest 100 in the account. One quarter later, the bank pays you 100*.03 = 3. You leave the 103 in for another quarter. At the end of this 3 month period, they'll pay you 103 * .03 = 3.09 in interest. Leave the full 106.09 in for another quarter and you'll earn 106.09 * .03 = 3.183 in interest. Now you're up to 106.09 + 3.183 = 109.273. The fourth quarter brings in 3.278 and your account is worth 112.551. This is better than the 112 you would have gotten had the bank paid 12% interest, compounded annually. OK, so 55 cents is not that big a deal. But it adds up when investing millions, and the compounding could be more frequent than just quarterly (like daily--waht MasterCard and Visa charge you on your unpaid balance).
To get this 112.551 a lot faster than how we did it above, just take (100* effective rate). The effective rate is equal to the (stated annual rate/frequency of compounding