Lecture 3: Financial Math & Cash Flow Valuation

Lecture 3: Financial Math& Cash Flow Valuation

C. L. Mattoli

1(C) 2008 Red Hill Capital Corp.,

Delaware, USA

Intro This week we cover topics in chapters 4 &

5 of the textbook. One of the most basic principles in

finance is that a dollar, now, is worth more than a dollar, received later. Thus, money actually has a time value.

If you have money, now, you could put it into a bank account, earn interest, and have more money, later (future value).


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Intro If you get money, later, you will have

lost the opportunity to invest (opportunity cost).

In investment, we invest money, now, to get cash flows, in the future, whether we invest in equipment to make noodles or we buy the securities of companies.


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What is Investment???? Although you may have heard your friends

and the news talk about investment, what really is investment?

Common things that you think of are stocks and real estate, but still what is an investment about?

Take stocks, for example. You buy a share of stock of China Telecom. You pay money now (PV) for a piece of paper from China Telecom.


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What is Investment???? For your present outlay, you are

expecting future money. You might get cash dividend payments, once or twice a year. Plus you will get a cash inflow when you later, in the future, sell the stock.

The difference, by the way, between the price you buy the stock at and the price you sell it for, Plater – Pnow = Capital gain or loss, depending on if the price that you sell it at is higher or lower than when you bought the stock.


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What is Investment???? The same applies to buying real estate, like

a house, an apartment, or an office complex. You will hope to get a capital gain when you sell it, and, in the mean time, you can rent it out and get periodic future cash flows from people paying their rents.

Buy a noodle machine and make noodles to sell: you pay money out for the noodle machine and space to house the business. You make an investment, in the noodle maker and some flour, eggs, water, and salt, and you make noodles to sell at the market, every day.


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What is Investment???? Again, you have a present outlay of

money, you get future cash flows from the sale of your noodles, and you expect to come out, somehow, ahead.

The thing is: you pay out money, now, instead of putting it in bank and earning interest, and you want to be ahead of the future money that you would get for putting your money in bank instead of making some other kind of investment.


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What is Investment???? At this point you should be beginning to

perceive the problem. Money, at different times, will, somehow, not have the same value.

Therefore, finance asks the question: what is money, received later, worth to us, right now (present value of future cash flow).

These themes will be one of the foundations for further study, in the course. So, be sure that you understand them before you go on.


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Future Value


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Future Value: Simple Interest In finance, we talk about (percentage)

rates of return, and usually, annual percentage rates (APR) of return.

Interest (and interest rates) on savings in a bank is an example of a rate of return.

In that regard, if I put $100 in a bank account that earns a 10%/year interest rate of return, then, I will earn 10% of that $100, in a year, or 10%$100 = $10.


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Future Value: Simple Interest Therefore, at the end of a year, I will have

$110, in bank = $100, original principal, plus $10, interest earned on the principal at 10%/year.

We can put this into a simple equation form as FV1 = P + r P = P(1 + r), where r is the annual rate of return = interest rate, in this case.


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Future Value: Simple Interest We have used the notation, FV1, to

indicate that this is the value of your savings account, (1 year) in the future, and we call it the future value.

If you earn interest on that principal for n years, where n can be > 1 or n < 1, then, you will get rP for n years, or nrP.


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Future Value: Simple Interest The simple interest equation becomes

FVn = P + nrP We collect terms inside a parentheses and

get FVn= P(1 +nr).

In n years, you will have P(1+nr) dollars in future value.


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Future Value: Compounded More common, actually, than simple

interest is compound interest. If I put $100 (P) in bank for a year, at the

end of a year, I will have $110 [FV = P(1+1r)], in the bank.

If I leave that money, in the bank, I will earn interest on the whole thing, which, after the first year, is P + rP = P(1+r).


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Future Value: Compounded At the end of 2 years, I will have FV2 =

$110(1+10%) = $121 = $100(1+10%)*(1+10%) = $100(1+10%)2 = P(1+r)2.

What has happened is that we have earned interest, not only on the original P that I put in the bank, but also on the interest that we earned in the previous year.


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Future Value: Compounded We can see also play around a little with

the equation to understand what it is telling us.

So, take FV = [$100+$10](1+10%) = $100 + $10 + $10 + $1 = P+rP+rP+Pr2 = P+2rP+Pr2 = [P(1+2r)]+[(Pr)r].

The first part of the equation, [P(1+2r)], is the amount that you would have gotten, if you were only getting simple interest.


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Future Value: Compounded The second term, [(Pr)r], represents

interest, r, on the interest $’s, rP, that you earned in the first year.

This is referred to as compounding, and you earn compound interest on your principal.

For any number of years, n, the future value equation with compounding of interest is given by FVn = P(1+r)n.


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Simple vs. Compound You earn more money with compounding. Shown: the value of money versus time into

the future for simple interest and compounding.

$

N = years

Compound

Simple


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Present Value: the Inverse of Future Value


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Intro If you have money, now, you will have a

larger amount, in the future, because you can put it in bank or some other investment.

If you have $100, now, and your opportunity to invest, a rate that you can get from an investment, is 10%/year, a year from now it will be $110.


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Intro Thus, $110, a year from now, has a

value, now, it’s present value, of $100 because if You put $100 in bank now it would be $100, in a year from now.

We wrote, before: FV = (1+r)PV, and we just reverse that equation to get: PV = FV/(1+r), the present value of an amount of money that we would get in the future, a year from now.


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Intro The PV of $110, received in a year, is PV

= FV/(1+r) = $110/(1+10%) =$100. We already knew that should be the

answer because we knew that if we put $100 in bank, now, at 10%/year, we would have had $110, a year from now.

We refer to this as discounting a future amount back to the present.

PV and FV are just inverse relations of each other.


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PV of any future payment

Assuming that interest is compounded, the PV of an amount of money, FVn, that will be received n years from now, is PV = FVn/(1+r)n.

Again, all that we have done is to reverse the future value equation: PV = FVn/(1+r)n


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PV of any future payment We usually refer to r as the discount rate

or the required rate of return (RRR), and the PV is called discounted future cash flow.

It just tells us what a future cash flow is worth to us, today, given that we could invest (opportunity lost; opportunity cost) it, if we had it now, and earn r rate of return compounded to that future time.


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The language of interest rates Interest rates are normally quoted

up to two decimal places, e.g., 4.03% (= in numbers 0.0403).

The decimal places, which represent hundredths of percents, e.g., the above rate is 4 and 3 one-hundredths percent, are called basis points (bp).

Thus 0.03% is commonly called 3 basis points.


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The language of interest rates Then, you might hear an interest rate

quote, like 30 bp over the BAB rate. So, if the BAB rate is 4.50%, the quote would mean 4.50% + 30bp = 4.50% + 0.30% = 4.80%

Bank accepted bills (BAB’s) = a type of short-term corporate debt paper, called a bill, like a dollar bill, and payment is guaranteed by a bank.


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Time value One thing that you should notice from this

simple equation is that the value, now, of a future payment of money decreases, the further into the future that we receive it, i.e., as n gets larger.

The other important thing that you should notice is that, for given n, the PV decreases as r increases. Thus, we say that the PV and the discount rate are inversely related. As r increase, PV decreases, and vice versa.


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Time value Often, in using the PV concept to value future

money, we will know FV and n, and we will have to decide on an appropriate discount rate, r, to value the future payment, in the present.

As we shall learn, that will be based on opportunity to invest.

After all, the PV concept is based on the fact that, if we have money now instead of getting it in the future, we could invest it.


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Time value Different people may, in fact, have

different required rates of return for the same situation.

We shall also see that for other situations, we can read the market’s required rate of return for certain types of future cash flows.

In the next slide we show plots of PV vs. n for various r’s.

Various symbols are used for r, like k or j.


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Time value vs. n for varied r

0%

5%

10%

15%


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In Truth: Consumption vs. Saving People give up buying things, now, to save or

invest money, and get money in the future. There is a reason that people should not just

put the money under their mattress and hide it.

There is always inflation, in the world. Inflation means that the $10 that you have

now will buy you less in the future because the prices of everything are going up.

(C) 2008 Red Hill Capital Corp., Delaware, USA 31

In Truth: Consumption vs. Saving If inflation is, for example, 10% per year, that

means that $10 will only buy $10/(1 + 10%) of the same goods and services, next year.

Thus, if someone wants to not consume, now, they will want to consume as much or more, in the future.

That means that people will set a base rate for lending or investing their money equal to at least inflation.

As a result, we have a beginning gauge to see how much people will require for returns.


Real life example: commercial bills A commercial bill (CB) is a ST debt security

that is issued by corporations. A CB is called a discount security because it

pays no formal interest payment, but, instead, it pays only its face value (FV) at maturity.

In order for someone to make money on their investment, therefore, they buy it at a discount (less than) face value, and their profit will be the difference between what they paid for it (PV) and what they get at the end: FV.


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Real life example: commercial bills We can use this example security to study

the PV/FV concept. What follows is an example of how we can

use thoughts and words to make up an equation.

The return from investment is equal to the money that you earn, E, from the investment.

The rate of return, r, from investment is the earnings, E, divided by the original investment, PV, at time = 0.


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Real life example: commercial bills In general return can have two parts, actual

intermediate time cash flow income, INCi, and the final profit or loss, the capital gain/loss, from selling, at a future date.

CGL, the capital gain or loss is just FV – PV, the difference between what you bought it at, PV, and what you sold it for, PV.

Thus the general rate of return equation will be ROI = [INC + CGL]/PV.


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Real life example: commercial bills Then, for the CB, which has no INC term,

but only a gain on sale at the end maturity, the rate of return on investment will be (FV – PV)/PV, the profit divided by the initial investment.

In finance, we usually like to look at rates of return for a one-year period. That is just, again, so that we can compare one return to another (remember, finance is a comparative science).


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Real life example: commercial bills What we have found, above, in the CB

equation, so far, is a return over the period from the purchase date to expiration.

A return earned over the [period of time that you hold an investment is referred to as a holding period return (HPR).

So, assume that our CB matures d days from now. Then, we make our rate of return in d days.


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Real life example: commercial bills To convert our HPR into an annual rate of

return (APR = annual percentage return or rate), first we divided the HPR by d and get a daily rate of return. Then, we multiply the daily rate of return by 365 days to get an APR return, r.

Thus, converting the words, in the last point, into equation, we find r = annual rate = [(HPR for d days)/days = daily rate] x 365 days/year = annual amount of daily return = {[(FV – PV)/PV]/d}365 = [FV/PV – 1](365/d).


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Real life example: commercial bills Thus, we have started from first principles,

just knowing what we want, in words, and making that into a solvable equation.

That is how the problems in this course will be. You will be given, mostly, word problems, and you will have to find which equation will help you solve the problem.

We know FV, so we will usually be solving for either PV, given an r, or r, given a price, PV, that you paid for it.


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Real life example: commercial bills Rearranging the equation, we get PV =

FV/[1+r(d/365)]. That is just the simple interest PV equation,

PV = FV/[1 + nr), with n = d/365. Commonly, the interest rate on a CB (and

other debt securities) is called an annual yield or yield-to-maturity (YTM).

For example, assume that you purchase a $1,000 FV CB maturing (coming due for pay off) in 90 days, and your annual RRR is 8%. What is the price that you should pay for it?


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Real life example: commercial bills PV = $1000/(1+(90/365)8%) = $980.66. That is the amount that you should pay if you

want to earn (RRR) 8% annual return. In fact, in this simple case of CB, we can

actually solve the equation for the YTM, by rearranging, again.

Solving, r = [FV/PV – 1]x365/d gives us the YTM, if we know PV and d.

Assume, for example, that a 180-day $1000 FV CB is priced at $900. What is its annual yield?


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Real life example: commercial bills We have $900 = $1000/(1+(180/365)r). Solving for r, we get r = [($1000/$900) – 1]

(365/180) = 0.2253 = 22.53%. Thus, the annual yield is 22.5%. Note: CB’s and other debt instruments

are usually quoted on yield, not the actual dollar price.

Thus, if you call a broker to get the price of a CB, he will not say $950, he will say 8%.


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YTM more general concept. In the CB example, we said that the rate earned

is called the YTM. For more general investments, we can talk of

the YTM isa an “average” rate of return over the life of an investment.

Thus, if an investment earns r1, one year, and r2, another year, the average YTM rate would be: FV = PV(1 + r1)(1 + r2) = PV(1 + rYTM)2.

A similar definition will apply when we have more than one cash flow.


BAB’s A major form of short-term money market

debt paper issued by companies is called commercial bills.

Commercial bills can be guaranteed by a bank.

Then, they are called bank accepted bills (BAB’s).

Since the bank is liable for payment, BAB’s trade on the credit rating of the bank, not the underlying company issuer.


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BAB’s Because they are guaranteed by a bank, BAB’s

are the only security that is considered to be intermediated (financing which involves an intermediary, like a bank) rather than direct financing, getting money directly from investors by selling paper, which is what a BAB really is.

However, even though the company issues securities to investors, since the bank is involved, it is strictly considered intermediated finance.

That is often a trick question in exams.


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How many days in a year? Again, this might sound like a stupid

question, but actually it is a trick question.

Australia uses 365 days in a year to calculate yields, which seems logical, and you might ask, what else would it be.


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How many days in a year? The answer is that the U.S. and many

other countries use 360 days (30 days per month x 12 months) in their quotes and calculations of yields.

Thus, if you get a quote on a CB of 8%, in Australia, it will not have the same meaning, or value, as an 8% quote in the U.S.


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Need 3 to get the 4th There are 4 variables in FV/PV

equation: PV, FV, r, and n. So, we need to know 3 to get the 4th. Thus, in problems, you will be given

enough information to figure out 3 variables, and you will be required to calculate the one unknown.


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Need 3 to get the 4th We might know PV, n, and r, then, we

can get FV. If we know FV, n and r, we can find PV. Etc.

So, if we are told that we will receive $1,000, a year from now, and that, if we had money, we could earn 10%/year over the next year, then we know FV, r, and n, and we can solve for PV = $1,000/(1.1) = $909.09.


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Need 3 to get the 4th The meaning of that is that $1,000, a year from

now, is worth only 909.09 to us now, given our lost opportunity to invest.

In this simple situation of only 1 Cash Flow, in present and future, we can also solve for r, if we know, PV, FV, and n.

Then, if FV = PV(1+r)n , solve for r as:1. (1+r)n = FV/PV2. [(1 + r)n]1/n = [(1+ r)n/n] = 1 + r = [FV/PV]1/n 3. r = [FV/PV]1/n – 1


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Need 3 to get the 4th Such an easy solution for r might not

be possible with multiple CF’s, as we shall discuss next lecture.

An example, in the book, looks at a bequest of about $1,000 that Ben Franklin left to my home state of Pennsylvania, 200 years ago.

200 years later, it had grown to $2 million.


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Need 3 to get the 4th So, what was the annual interest rate

over the period? Solving r = [FV/PV]1/n – 1 = r =

[2,000,000/1,000]1/200 – 1 = 3.87%, which is not a very high rate of annual return, but it really adds up over 200 years.

What this really demonstrates is the power of compounding.


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Need 3 to get the 4th The authors also list a heuristic (back-of-

envelope, rule of thumb, approximate way to solve): the rule of 72, which says that for reasonable rates of return, it takes approximately 72/r (in %) years to double your money.

For example, say you want to double your money in 8 years.

Then, $2 = $1(1 + r)8.


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Need 3 to get the 4th You could solve the equation, formally, as

was done on the previous page. Using the rule of 72, we get 72/r% = 8, so r =

72/8 = 9%. Similarly, if we know r, FV, and PV, we can

find n. One way to do that is to use FV/PV factor

tables (in the back of the book) to find the right n that corresponds to our numbers of PV, r, and FV.


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Need 3 to get the 4th Again, in the simple case of one cash flow,

we can solve the PV/FV equation to find n, in terms of the other three variables.

The technical way to solve is this. First, take the natural logarithm of both sides of the equation, FV/PV = (1+r)n ; so, ln(FV/PV) = n ln(1+r), where ln is the natural logarithm function.

Then, solving for n = ln(FV/PV)/ln(1+r).


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The meanings and uses of PV & FV Investments offer money, in the future, in

return for money given, now, in the present. The calculation of PV adjusts money that will

be received in the future to a value, now = PV, by accounting for our lost opportunity to invest.

Then, if we have a certain RRR, and we know how much the future cash flows will be, we can find a PV = FV/(1 + RRR)n.


The meanings and uses of PV & FV That PV is actually the price that we

should pay, now, for that investment. We use the FV concept, naturally, when we

want to know how much something will be worth, sometime, in the future.

Thus, if we invest money, now, buy buying a security or other investment for PV-dollars, and we earn r annual return on this investment, for n years, we will have FV = PV(1+r)n total dollars at the end of that n-year time period.


The Market’s Apparent RRR. If we know what we will get in future payments

from investment, then, we can decide on what RRR (required rate of return … by us) we need for giving up our money, now = PV = price we initially pay for the investment.

After we have decided what we think is the proper price to pay, given what we think is a reasonable RRR, we can call a broker, for example, if it is a stock or a bond that we have valued, and find out the price that is being asked in the actual market.


The Market’s Apparent RRR. At that point, if the price in the market is equal to

or below our calculated fair price, we would buy it. If the market price is above our calculated PV based on our own personal RRR, then, either we will not buy it, or we will settle for making less of a return (PV and RRR, inversely related).

In that regard, from the actual price that an investment is trading for in a market, we can find out the market’s apparent aggregated, at the moment, RRR.

For example, taking our only example security, a BAB, assume that a FV = $1,000 BAB


Equations don’t know what time it is. Do not give too much power to the PV/FV

equation. Remember, you are the boss, and the

equation is supposed to work for you. So, you have to know what it is exactly

that the equation does, then, you can make it work for you in many different situations.


Equations don’t know what time it is. The equation FV = PV(1 + r)n means that,

if you put money, PV, into an investment and leave it there for n years, you will have FV dollars at the end of the n-year period.

For example, we could ask: what will the value of $1,000 be in 4 years from now, if we put it into an account that earns 10% compound annual interest, one year from now?


Equations don’t know what time it is. So, we make the equation work for us. First,

if we put it in bank, one year from now, and want to know how much we will have 4 years from now, then, it will be in the account for 4 – 1 = 3 years.

So, the answer is FV4 = PV1(1 + r)3 = $1,000 (1.1)3 = $1,331.

We could also ask what a $500 payment that will be come five years from now will be worth 2 years from now, if our opportunity cost interest rate is 6%.


Equations don’t know what time it is. In this case, we want to look at a value, 2 years

into the future discounted to that point in time from five years from now.

Again, we use the PV/FV equation, and we call the answer PV, even though it is not in the present, it is more like a prior value than a present value.

Using the equation the way that it is meant to be used, we want to discount back a value from 5 years from now to 2 years from now, or 5 – 2 = 3 years, and the value will be PV2 = $500/(1.06)3 = $419.80, two years from now.


Equations don’t know what time it is. Moreover, we could go from there to ask what is

that money worth now? Then, it is money two years from now, so we discount bask that value 2 years to find PV0 = FV2/(1 + r)2 = $419.80/(1.06)2 = #373.62, which is the same answer we would have gotten by taking PV0 = FV5/(1 + r)5 = $500/(1.06)5.

Often, as we value more complicated cash flow situations, we will use the PV/FV equations, to bring different cash flows to different times and apply them again to bring them back to the present.

Just remember, you know what time it is but the equations don’t.


Selling before maturity One of the most important things that secondary

markets provide is maturity transformation or conversion.

That is just fancy words for a simple fact: the existence of a secondary market where already issued securities can be sold by one owner to a new owner, opened 5 days a week during the day, means that no security has to be held to maturity.

That, again, is liquidity. In this case, liquidity means are offered to investors, and they feel better about investing, in the first place.


Selling before maturity Indeed, since a corporation and its shares of

stock have no fixed maturity, you could never get your money out of a corporation unless there was a means to sell, like a market.

As we just discussed, in the preceding slides, the PV/FV equation does not know what time it is, it only knows how to relate two values, whose times are separated by a number of years, n.

To go along with the means of making our own decisions of holding periods (making our own maturity), we can use the PV/FV equation also to find values in other years.


Selling before maturity For example, we buy a bond security that will

pay $1,000, two years from now, and we pay $900.

From that information we can find out the annual YTM = RRR that the investment will pay from: r = [FV/PV]1/n – 1 = 5.4%.

Assume that we want to sell the security, one year later because we suddenly need money to buy a new car, and we get $940, for it, one year from now.


Selling before maturity Then, we can calculate the YTM when we sell it,

and we can find out what our actual HPR was over the year.

First, don’t take the designations, PV and FV too literally. We can find the YTM on a one-year-left to maturity bond that costs $940 and pays $1,000, one year later as: PV = $940 = $1000/(1 + r), or r = FV/PV – 1 = (FV – PV)/PV = $60/$1000 = 6%.

What we actually earned on the investment during the year that we held it was (FV – PV)/PV = $40/$900 = 4.44%.


Selling before maturity What we earned and what the next guy earns by

holding it for the second year are below and above, respectively, the original 2-year annual YTM that we would have gotten, if we held it for the whole two years, which we calculated, above, to be 5.4%.

This shows that we can use the PV/FV equation to find values in any year, not just the present and one year, in the future.

When you understand that, it will be much easier for you to solve real problems that you might get on an exam.


Only one time Usually, we want a present value, now at

time = 0, or at a future year, n years from now.

The real point of time value is that the same cash flow will have different values at any different time.

Next week, we will consider multiple cash flows that come in different years.

To apply time value to those situations, we must focus on the point of time value, which is that any one cash flow has different values at any different time.


Only one time Then, when we consider more than one cash

flow, we must take them all to one particular point in time to be able to add them up.

Unusually, that will be now (present value) or at one particular year in the future.

However, after we take them all to one time and are able to add them, then, we can take that whole summed value to any other time just by using the simple one-cash-flow PV/FV equation and moving the summed up one value to a different time.


Trailer: put and call provisions Since there is a problem in this weeks

problem set that discusses put and call provisions on bonds, lets take a closer look at these concepts.

Normally, a bond has a definite maturity when the loan represented by the bond is due to be fully paid off (liquidated).

If a discount bond, for example, promises to pay face value = FV = $1,000 in 2020, then, we decide on a proper RRR, figure out how long it is from when we buy it until the maturity date.


Trailer: put and call provisions For example, if our RRR = 10%, and since it

is, now 2009, there are 11 years before payment, so the proper price to pay is $1000/(1.1)9.

Bonds and other securities can have so-called put and call provisions.

A call provision will give the company the right, for example, to call in the bond for retirement, early, say, beginning in 2015 and thereafter.


Trailer: put and call provisions Then, valuation of the bond becomes more

complicated because we no longer have a definite value for M = years to maturity since it could be called away from the holder in 2015, 2016, 2017, 2018, 2019, or 2020.

The reason that a company would do that is that it might want to give itself options, in case interest rates come down dramatically, in future years, and the company does not want to be stuck paying a high interest rate after general interest rates have declined.


Trailer: put and call provisions Bonds can also have put provisions. In

that case the bondholder may be given the option to sell back the bond to the company at a specified price before final maturity.

In that case, bond holders are given a way out of the bond investment, in case interest rates go up, in the future, which would make their bond values go down.


The creativity of finance One of the basic reasons for the development

of modern finance is valuation of investments.

From the simple concepts of stock and bond securities, the securities industry has expanded to include many different kinds of paper, which are much more complicated than stocks or bonds with specific cash flows and one maturity.


The creativity of finance Today, new securities are packaged by

combining old securities and new concepts, just like the callable and putable bonds that we discussed in the preceding slides.

The ability to value complicated packages of complicated cash flows means that there will always be need of people who can understand them and can develop means of valuing them.

Even as accountants, you will be called upon to do those kinds of things, and this course can give you a firm beginning basis to do so.


Tutorial ProblemsLearning activity● Work through the chapter review

and self-test problems on page 110 and the critical thinking and concept review on page 111. Attempt questions and problems 1, 3, 4, 5, 8, 13, 15, 16, 19 and 24 on pages 111 to 114.


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Exam caliber question You can buy a car and pay $2000, now or

2200, two years from now. 1. If your opportunity cost of funds is 7%/year,

which is the better option?2. What is your reason for your answer in part

1?


Use USQ on line

Use the resources that USQ has on-line. You can get daily emails about questions other students are having problems with, and you can read about writing and referencing assignments.

Use me … email, text message, or private talk.


END


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Lecture 3: Financial Math & Cash Flow Valuation

Documents

red hill capital

future money

capital gain

kind of investment

present outlay of money

later future value

periodic future cash

cash inflow