In this set of lectures, we will talk about time value
of money. After a brief introduction, we will go
over the different possible interpretations of
interest rates, talk about the concept of future
value both for a single cash flow and then a series
of cash flows. Then we will talk about present value
again for a single cash flow and a series of cash flows.
And finally we will work on some problems which involves
solving for rates, number of periods, size of annuity
payments and so on. A brief introduction:
If you have $100 today vs an option to receive $100
after 3 years, so this is time 0, this is at the end of 3
years. What would you prefer? Obviously, you would
prefer $100 today, even though we have $100 in both
cases, you prefer $100 today. This means that there
has to be some value associated with time because
you are putting more value on the $100 that you are
getting today relative to the hundred dollars at
a later point in time. The money today or the value
today is called the present value. At times this
could be an investment which you make at time zero.
The value at a future point in time is called the future
value. Lets say that you are indifferent between
$100 today vs a $110 after 1 year. This $110 is then
referred to as a future value at the end of year 1.
The relationship of the link between present value
and future value is established through an interest
rate. And what we are going to get. What we are going
to do in this reading is essentially talk about these
concepts. Present value, future value, and the way
we link these two concepts using interest rates.
The central theme is interest rates. So let's talk
about the different interpretations of interest
rates. Interest rates can be interpreted as; number
one, required rate of return or a discount rate or an
opportunity cost. Say you lend $900 today and receive
$990 after 1 year. So if you put this on a timeline
time 0, you lend 900, so let's put a minus sign to indicate
money out and then after one year you get 990 back.
The fact that you are willing to do this you are willing
to give $900 today on the condition that you get $990
after 1 year means that to engage in this transaction
you are requiring a return of 10% I'm sure you can
calculate this given that you are lending 900 today
or investing 900 today and getting 990 after 1 year
that interest rate is 10% and the required rate of
return interpretation is simply saying that you
require a return of 10% to engage in this overall
transaction. So that is the required rate of return
interpretation. Discount rate is also straightforward
and we'll cover it briefly here and in a lot more detail
later, but if you think of the money that you are getting
after one year, which is 990 you can discount 990
at 10% to get the present value of 900 and therefore
the 10% can also be thought of as a discount rate.
And finally opportunity cost. Let's say that you
had taken the $900 and spent it on something else.
What you have done done is you have forgone the 10%
return that you might have received. And if you remember
the concept of opportunity cost is the cost of what
you forgo. In this particular case, what you have
forgone is the 10% return therefore 10% can also be
thought of as a opportunity cost. We shift gears
a little bit now and look at interest rates from
a investor perspective. As an investor we can think
of interest rate as a sum of multiple components.
The first component is the real risk-free interest
rate. This is the rate that you get on a security that
has no risk and is extremely liquid and we make an
assumption here that there is no inflation. We can then
add on an inflation premium. Inflation premium is the expected
annual inflation in the upcoming period. We can also then
add the default risk premium. This is the additional
premium that an investor requires because of the risk
of default and you can understand this through a
simple example. Let's say that you lend some money
to person A and also lend some money to person
B to both these people you are lending $100 initially
B has a high risk of default so you are worried that B might
not pay, then you might demand a higher return from
B, because of that risk of default. That additional
return that you will demand from B because of
the risk of default is called the default risk premium.
Next we have liquidity premium. This is the premium
and investor demand because of the lack of liquidity
of a investment. Here again, think of two Investments
C&D, which are similar in all regards. The only difference
is that investment C is liquid, extremely liquid,
whereas investment B is not that liquid, clearly.
as a investor we will demand a higher return turn on
D because it is not liquid. That additional return
that we demand is called the liquidity premium.
And finally we have maturity premium. The idea being that
if you have two securities again, let's say, E and F, and
E has a maturity of one year and F has a maturity
of let's say four years, because F has a longer maturity,
it has a higher risk in terms of the price of F being
more sensitive to changes in interest rate. And
this is the concept that you will understand better
when we do fixed income securities, but for now you
can just take it as a given that F has a higher risk
because of the longer maturity and obviously an
investor will demand some compensation for that
higher level of risk, so the sum of the real risk-free
rate and the inflation premium is often referred
to as the risk-free rate or the nominal risk-free
rate. So if our real risk-free rate is 3% and the inflation
premium is 2% then the nominal risk-free rate is
5% If you hear the term risk-free rate then the assumption
is that we are talking about the nominal risk-free
rate. Let's look at a practice question, Jill Smith
wishes to compute the required rate of return which
of the following premiums is she least likely to
include. The correct answer is C, nominal premium
and we say that because while this sounds like a nice
term, but it is clearly not included in the list of
premiums that you saw on the previous slide. It is
possible that you'll see a question like this on
the exam. So make sure that you know, all the premiums
really well. Next question; the correct answer is
C. The required rate of return is the minimum rate
of return. So A and B are correct options, C is not correct
because of this, this is a mistake, and since we are
looking for least likely true, C is the choice. Another
practice question, say you have five different Investments
and we have the maturities for each investment, the
liquidity, default risk, and interest rate for each
investment. Why is there a difference between the
interest rate on investment A and investment B?
They have the same maturity they have the same default
risk, but interest rates are different and the answer
is, because of the difference in liquidity. Does
it make sense that the return on B is higher. The answer
is yes because B has lower liquidity. And therefore
investors will demand a higher return for investing
in B relative to investing in A. That higher return happens
to be 0.5% so in our simplistic example here, 0.5% is
the liquidity premium. Next question is estimate
the default risk premium. And before you go on, try
to solve this on your own. If you look at Investments
D and E, notice that they are both 3-year investments.
They have a different liquidity in the sense that
D has high liquidity E has low liquidity, default risk
is different and interest rates are different.
What we need to do is make the liquidity the same.
Let's say, D for some reason goes from high
liquidity to low liquidity. Will it change the interest
rate? The answer is yes, if the liquidity becomes
low, then investors will require a higher return
and we've already determined that the liquidity
premium is .5% So we add .5% and we have 3.5% as a return.
Then what we can do is the following, if we have this newer version
of D. Let's call it D bar or D compliment so
this has a low liquidity? Versus E which also has
low liquidity. Default Risk hear is high, the difference
between the new version of D and E is 0.5%. This means
that the default risk premium is equal to 0.5%. And
finally, can you calculate the upper and lower limits
for the interest rate on investment C? Notice that between
B and C. The only difference is that C has a longer
maturity which means that the interest rate on C
must be greater than B so we can say that r has to
be greater than the return on B which is 2.5%. So
r has to be greater than 2.5 and if you look at the low
liquidity version of D, that has a return of 3.5%. Other
than that it is similar to C because liquidity
is the same, default risk is the same. So maturity
is less. So maturity for investment C is less than
the maturity of investment D. That means that r
has to be less than 3.5 so the range for r has to be
between 2.5 and 3.5.