Worksheet on present and future values
October 12, 2008
1 Revision on simple and compound interest
In a previous work sheet we encountered the formula:
f = p(1 + r)y (1.1)
Where f is the future value of an amount p invested for y years at an annual interest payment at
the rate of r%.
If the interest is paid n times a year then for a single year we have:
f = p(1 + r/n)n (1.2)
Suppose that you wished to compare two interest rates over the same period but where the interest
was paid at different intervals. For example imagine two banks one of which pays interest semi-
annually and an other which pays interest quarterly. In order to make this comparison it is useful
to convert to an equivalent annual rate. That is the rate which would give the same compound
return at the end of the year as the rate we are comparing. It follows that:
p(1 + r/n)n = p(1 + ra ) (1.3)
Where ra is known as the effective annual rate.
Rearranging this we can eliminate p and have:
ra = (1 + r/n)n − 1 (1.4)
1. 8% is the nominal interest rate quoted for a 1-year deposit with the interest paid in quarterly
installments. What is the equivalent annual rate, that is the rate that quoted when the
interest is paid in a single sum at the end of the year? [Ans: Imade it 8.24%]
2. 5% is the nominal interest rate quoted for a 1-year deposit when the interest rate is paid
all at maturity. What is the quarterly equivalent? Note you can do this by rearranging the
formula, but explore Maple’s ability to solve the problem without doing that. i.e. Use the
solve command. [Ans: I made it 4.91%]
1.2 Continuous compounding
The effect of compounding interest increases with the frequency of the payments because there
is an increasing opportunity to earn interest on those payments. It is interesting to consider what
happens as these payments become more and more often, in fact until they are continuous. In
these circumstances we have:
f = pery (1.5)
1. An amount of 2,340.00 is deposited in a bank paying an annual interest rate of 3.1%,
compounded continuously. Find the balance after 3 years. [Ans: 2568.06].
In practice no bank that I know of offers continuously compounded interest. The nearest practical
equivalent is interest paid on a daily basis which is quite a common practice.
The effective annual rate of interest calculated from a daily rate for m days is given by:
365 rd m/365
ra = (e − 1) (1.6)
2 The present value of future cash ﬂows
For any ﬁnancial instrument, the price an investor ir prepared to pay in the net present value
(NPV) of the future cash ﬂows to which he or she is entitled as a result of owning it. This
depends on the interest rate offered by the instrument (the yield) and the number and timing of
Consider a Certiﬁcate of Deposit (CD) issued by a bank that pays one coupon at maturity in d
days time. If the coupon rate is denoted by r then the future value at that time follows by simple
f = (1 + r)d/365 (2.1)
The more interesting question to ask is, “what is that worth to me today?” That is what is
the present value? The present value p is the is the investment needed at the current yield i
to achieve the same amount on maturity. For i we need to consider the interest payable on a
risk free investment over a similar period and the yield on a Treasury Bond1 would be a good
approximation to that value. In the U.S. a Treasury Bill or T-Bill would be equivalent. Thus we
f (1 + r)d/365
1 + id pm /365
Where d pm is the number of days from purchase to maturity and d is the number of days from
now to maturity. The price would usually be quoted based on a face value of 100.
1. A CD was issued by Barclays Bank on 1st January 2008 and it carried a coupon of 10%
payable at maturity after one year. The face value was £100. What is the value of this CD
3/4 of the way through the year if the current yield of a risk-free bond is 4%?
We will continue with the topic of the present value of future cash ﬂows when we come to
consider the pricing of bonds and equities.
3 Market data
To ﬁnish off today’s exercise we will move away from Maple for a while. Try to ﬁnd the share
price history of Barclays Bank (BARC) on the Yahoo ﬁnance web site (http://uk.ﬁnance.yahoo.com).
1 These used to be called gilt-edged securities
Download the history for the lastt three months as a comma separated (csv) ﬁle and load it into
an Excel spreadsheet. Only the CLOSE price is really of interest to us.
Note that the latest data is at the top of the column in excel and the earliest at the bottom. It
would be much more convenient if it were the other way round. How might you invert the data?
Mostly we are more interested in daily returns that prices. The obvious formula for this is:
rd = p(pt − py )/py (3.1)
That is the daily return is given by the change in price from yesterday’s close divided by yester-
Can you calculate this in the Excel spreadsheet?
Can you calculate the average return and the standard deviation of returns in Excel? A great deal
of ﬁnancial theory is based on calculating those two values.
Now calculate the natural log of the closing prices in another column, and then form a column
which is the difference of the log of today’s price from the log of yesterday’s price. How do these
values compare with the returns you calculated previously? If they are similar, why do you think