>>Hello. Today we’re going to talk about chapter six, discount cash flow evaluation. We will first address how to calculate a PV, FV for multiple cash flows. Then we move on and talk about special multiple cash flows. Annuity, perpetuity. Lastly, different ways of quote interest rate and three type of loans. We see multiple cash flow example every day. Investment proposals, saving for retirement, mortgage, auto loan payment, or scholarship fund. The first task today is how to find out the PV of multiple cash flows. We can use formula to discount each cash flow, find the PV of a single cash flow, and [inaudible]. We can also use financial calculators, use cash flow keys to find out the PV of them. Let’s first take a look at cash flow keys for PV questions. So, all these keys are located in second row of BA II Plus, cash flow keys, and you will see the initial cash flows. So I’m going to first clear everything in the cash flow keys. So I’m going to press second and clear work. So cash flow zero is initial, could be outflow, negative, or zero, or positive. Then you can use arrow key, go down. This ask you what is the first cash flow. Then where F frequency ask how many times we have the same amount of cash flow, how many times, how many repeats. After you end all these cash flows, you will press the net [inaudible] will ask you to enter the interest rate. And you can use arrow key, move down, once you see that [inaudible], you are able to compute. So let’s take a look of this following example. There is an investment proposal. The initial requirement is 5000. Then this will give you three positive cash flows in year one, two, three. So the discount 10 percent. If we use formula, so we can use one [inaudible], two [inaudible], three [inaudible]. This count at 10 percent. One year discount, two discount, and three year discount. Now let’s practice using cash flow keys. The first let’s press cash flow button, so we enter the five 5000 negative, press plus and negative to change sign, click enter. Remember, you have to enter the number first and then click enter to register. Then move down. First year cash flow 1000, enter, down. Second cash flow, 2000, enter, down. So we keep the default frequency at 1 because only repeat one time. Third year cash flow 3000, enter, down. So that’s it. We go to the next button [inaudible]. This count as 10. Enter 10, ten, enter. Go down. Once we see MPV, we going to click on compute. So we got 184.07. I want you now pause, take a moment, practice this PV question. I want you to try both formula way and financial calculator approach. So that is the first task today, calculating PV for multiple cash flow. There’s another type of question. We made a few deposits saving for future, and we want to know how much money we have in the end. So that type is locating for future value of multiple cash flows. Again, we can use a formula. However, for the financial calculators, we don’t have a cash flow key shortcut. What we’re going to do is for each single cash flow we count how many years to compounding and find out future value for single of them, each of them. Then add them up. I would suggest to you to use [inaudible] calculator keys to store result from each future value. For example, there are three deposit, 1000, 2000, and 3000. We want to know if account offer you 10 percent how much total money you will have by the end of year three. So let’s first look at how many years compounding, how many years in the payment we are receiving. For the first cash flow, which you deposit by the end of year one, there are two more year interest payment before year three. For the second year cash flow, then one more year compounding, one more year interest payment, while the last cash flow we deposit at the end of year three, so we don’t earn any interest payment. So what we’re going to do is we’re going to use formula, compounding 10 percent, for two more year, one more year, and solve. Or we can use financial calculator keys. So let’s try that. So the first cash flow we save for two years. Let’s first clear everything, clear [inaudible] money. The first cash flow, two years, two years, we have 10 percent. And the initial cash flow is 1000. I put a negative for sign convention. Then compute. FV. Right. So I’m going to use the recall function, put result into the [inaudible]. So I’m put [inaudible] store into key one. Then I move on to the second flow, which is 2000, say for one more year, so N is 1. [inaudible] against 10, PV is 2000 negative. Then I’m going to compute FV. Right. Okay, again, I’m going to store into another key, store into 2. So we’re going to add now three result, 3000 itself, the third year cash flow, result from first calculation and second calculation. So I’m going to add 3000 plus the call result from one. And add, the call result from two. The total is $6410. So, I would like you to practice, find out the future value for this multiple cash flows. So let’s conclude how to look for PV and FV for multiple cash flows. As you see, these cash flow from year to year, they are not necessarily the same amount. They could have very different amount. Now, next, we’re going to talk about special cash flows where every period the amount is the same. So the first type, special cash flow is called perpetuity, where the same amount is paid or received every period, and this last forever. The common example is preferred stock or university endowment, they set a fund, use the annual interest payment to offer scholarship and so on and so forth. To solve this type of perpetuity, we actually only use formula. So the value of this type of a cash flow is that cash flow, periodic cash flow amount payment over discount rate. Let’s look at one example here. Perpetuity for preferred stock. The preferred stock pays a quarterly dividend of seven percent based on the par value of 100. Required rate of return is 20 percent. What’s the value of this preferred stock? Use formula. Every quarter the company pays $7, 100 times $7, preferred stock, required rate of return 20 percent. So this gives us the value of this preferred stock should be $35 per share. Perpetuity can be also used to evaluate endowment scholarship fund. Suppose there’s a potential donor, want to find out. The money set up as scholarship fund. Every year, this fund uses the annual interest payment to pay out 10 scholarship, which was 1500. So how much initial money should you put into this fund. Again, this is the perpetuity example. I will ask you, use calculator, find out. The answer is 300,000. So perpetuity is the same payment and it continue forever. There’s another type of special multiple cash flow, which annuity where the same amount is paid or received every period but for a couple periods. We this all the time. We see the mortgage payment, auto loan payment. We see the deposit for retirement. Saving for college education. So there’s two type. For the loan question, you borrow first, then pay back later. For the retirement question, you save first, you want reach a goal in the end. So we call this annuity PV question and annuity future value question. The formula for annuities for PV it is the same amount. Annuity payment times the PVIFA, the annuity present value interest factor. And for the future value, you can also use factor. Annuity, future value [inaudible]. Or you can use original formula. For the calculator, we will still use the third row in the BA II Plus where among five keys we only actually use four of them to solve for PV of FV question. So just pause a moment every time, ask yourself, are we solving for a long question or [inaudible] question. Some conventions still hold, which means you cannot have all the value positive. One of the values should be negative. Let me talk about annuity factor. This is used very, very popular in practice. So, some of the commonly used annuity factors are listed in the appendix A3, A4 table. But you can calculate by your own for any special kind. For example, annuity present value interest factor at 10 percent, three years. According to appendix 8.3, this is 2.4869. But you can do this by yourself with the calculator. Let’s find out the annuity factor using the calculator. First, clear time value money. I entered 3 into N. Ten into IY. Then I put 1, negative 1 into payment. Then compute PV. So I get 2.49 because I have two decimals. I can change decimal into 4 in the [inaudible]. But you get this idea. Let’s do one practice. Suppose you’re considering buy a new car. You know how much you can afford every month. You can afford up to 632 a month. The bank will quote you a loan 12 percent interest rate compounding monthly. You’re going to pay off in the next four years. So how much you can borrow today? So let’s a long question, borrow first, pay back later. A note of caution is regarding interest rate. The bank always quotes you annual rate APR 12 percent. But you are paying monthly cash flow. So to do this, you have to convert annual rate to the month rate for your calculation. So in other words, here we use monthly one percent. We use 48 months. Use the financial calculator, we will put 48 into N, 1 percent 1 into IY. So let’s try second [inaudible]. So I’m going to this time, okay enter 48 into N. The rate is 12, divided by 12, which is 1 percent. I put into IY. Then monthly payment is 632 negative into payment. This is done conventionally. Then I will compute how much I borrow today, that’s PV question, PV. Okay. So I got it. Another example, suppose you are able to borrow 20,000 for a new car. Bank offer you 8 percent annual rate. If you take a four-year loan, how much are your monthly payment? Very similar. Right. So, again, to do some preparations. Make sure we use monthly interest rate. Okay. Make sure we count the number of months, number of payment, 48 instead of 40. So let’s try again, let’s see. Clear everything. So I’m going to put four years times 12, which is 48, into N. I use 8 divided by 12 to get monthly rate IY. Then I borrow 20,000, put into PV, then compute payment. Right, 488.26. Okay. So I want you to pause a moment and practice these two questions. In practice, we sometimes want to find out how much money you have if we [inaudible] the time [inaudible]. So that’s saving for the time that requires [inaudible]. It is a type of looking for future value for annuity. For example, you are saving for retirement. Every year you deposit 2000. They [inaudible] you 7.5 percent compounding rate. How much will you have in 40 years? So with calculate the formula or calculator. Okay. So, again, I’m going to clear all my numbers. Okay. So I say for 40 years the rate is 7.5. Then I, every year I put a negative, I put 2000 into retirement fund, which is repeating payment. Then I compute for how much money in the end is the future value question. So I have 450, 4513. Another example, suppose you are able to borrow 20,000 at 9 percent compound monthly rate. You pay $497.71 a month. So how long do you have to wait until you paid off the loan? So just looking for the number of payments, question ask yourself, is this annuity PV question or future value question? Remember, if we borrow first for long question, it’s PV question. If we are saving for retirement, that’s a future value question. But in this case we borrow first, pay back later. Long question. PV question. Let’s again clear time value money. So, the N, we are looking for N, so start with the rate, 9 percent annual rate, divide by 12. So that’s interest rate. Then we borrow 20,000. I put negative in parentheses. Then monthly you pay for 97.71. So that’s our monthly payment. This question we are solving for the period. So it’s a 48 months, right, remember. So if we ask how many years, we divide by 12 months. It’s four year. So I’m going to ask you now, pause a moment, to do more annuity practice. I’m going to talk about annuity. Do in real life when you sign up at mortgage payment or sign a lease contract, the first payment normally is due today, the time you sign up the contract. So this is the cash flow paying at the gain of the period because this type annuity called annuity due. So in the formula, all you need to do is adjust by 1 plus interest rate. For the calculator, we have to use second [inaudible] second set to change into annuity due calculation. The rest is the same step. For example, you’re saving for a house. You deposit 10,000 every year. [inaudible] give you 8 percent. The first payment is made today. So you want to know how much would you have at end of year three. So a three-year annuity question for start of annuity, beginning of a period. For the regular formula, all you need to do is adjust by 1.08. For the calculator, you want to start at the beginning. Okay. So how do we display the beginning of the calculator? So you go to the calculator, press second. On top of the payment key you see BGN, begin. Place second. Place payment. You see [inaudible] this set into end, so I’m going to use the second set, second, okay I’ll cover [inaudible]. You see now I’m changing to beginning. So you see the BGN is displayed on top of your screen. Even you turn off the calculator, it’s still in this beginning mode. Let’s doe now this question again clear everything. Clear time value money. Three into N, IY is 8 percent. Then we have 10,000 into payment, then compute FV. Okay. Here is a practice for you to try your second begin second set. [Inaudible] this calculation you need to change your begin back to end. So we’re going to again, repeat same procedure second. Payment. Now then second set. Okay. So now, begin is no longer displayed. We’re going to now take a moment, talk about interest rate. As you notice, we mentioned APR a lot time. So the APR, annual percent rate, is a quoted rate. By law all these institutions, banks, they need to give you a quoted rate APR. Even though your act payment could be monthly. For example, you have a 30-year, 6 percent mortgage. The APR is six percent, but the monthly rate is maybe 0.5 percent. So, if you know monthly rate, you can convert APR by a multiple of the number of compounding frequency. Now we’re going to talk about effect annual rate year of. So this is not displayed or communicated to the investors. So this actuary rate will help us to compare different type of products, loans, line of credit, in different investment choices. So, we can think of a bank wants something to compare different instruments. How do we convert APR to EAR or another, another way around? You can use a formula where M means number of compound frequency. In the BA II Plus we can use the calculator I convert. So which is on top of the key number two? Okay, so let’s see. Press second. Second I convert. So you will see num EFF or CV. So here num is equality to APR, the annual rate, APR, while EFF is the equivalent effect annual rate while the CY means compound frequency M. Let’s see by [inaudible] 2 APR. One is 5.25 daily compounding. The other is 5.3 semi-annual compounding. So we want to compare which one actually is better, which one is higher. So we need to convert the EAR for us to compare on the same ground. Okay. So this is formula approach. For the calculator, let’s try the first one. I press here second and I convert in this case 5.25 the num APR so I enter 5.25 enter. Then go down. I have to enter, which is daily frequency 365 enter. Go down. You see EFF ran compute. So we got 5.39. Same thing. You can do this for the second offering. Sometimes it’s best to assume in compound [inaudible] compounding. So in this case, we will use natural E or log to find out APR or EAR. A short cut if you are using financial calculator [inaudible] going to [inaudible] compound frequency to a very big number, like 99999. That is will do the trick. Example, if bank wants compare two loans, they want to both of them have the same EAR of 12 percent. So what APR should we set? One is semiannual, one is monthly. So I’m just display one calculator for the first one. So here the 12 percent EAR, we’re looking for the APR loan. So again, I press second. I convert. And I’m going to clear the work. Okay. So here EFF is set at 12 enter. The frequency is, for the first one is semi-annual, so I enter 2. Then we compute. We got 11.66. Right. Repeat same procedure for the second account. Here I want you to practice. I have a few different loans. Some are quoted at APR. Some are EAR with different frequency. So here’s the solutions, answers. I want you to use the financial calculator to find out the missing ones. You guys will remember when we solve multiple discount, multiple cash flow questions, when we have annual frequency, you have to use annual rate. When the cash flow paid monthly you have to use month in the rate. For example, here is example. Deposit $50 amount [inaudible] amount. This account has APR of 9 percent, compounded monthly. So how much money would you have in account in 35 years? What’s the EAR if that’s the rate. So remember, everything is monthly. Payment a month $50, repeating. APR 9 percent. Have to divide by 12 to get monthly rate. Okay, 35 year, have to multiply by 12 to get the number of months. Right. So in this case, monthly rate is 0.075. Number of months 420. And then we [inaudible]. For the effect of annual rate, we convert 9 percent APR to EAR. Lastly, I’m going to talk about type of loans. We have pure discount, pure discount loans, interest rate loans, amortized loans. Pure discount loans where no interest payment or principal is ever paid until maturity date. While interest rate, interest-only loans pay every period a payment and that principal was paid off in end just like the bond, corporate bond. Amortized loans every period both interest payment and principal is paid, but sometimes principal is paid on a constant same amount, like in the case of mortgage or loan payment or total loan payment. Total payment is the same while the interest component and principal payment are changing every single year. So chapter six we talk about a lot of things. PV, FV for multiple cash flows, annuity, perpetuity, interest rate quotes, and three type of loans. So that all for today.