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Formula For Calculating Interest Rates

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					Formula Sheet

Interest rate calculations

Simple interest

When money is put on deposit for a relatively short period of time, simple interest may be
applied. The relevant parameters are i the annual rate of interest, d the number of days the cash
will be on deposit, and a the amount of cash on deposit. We denote the interest received by r.
This can we worked out as follows:
                                                   aid
                                              r=                                            (1.1)
                                                   365
Here we express the interest rate i as 0.05 for say 5%. Often it is more useful to know how much
cash will be returned at the end of the deposit period and this is obviously:
                                                       d
                                       FV = a(1 + i       )                                   (1.2)
                                                      365
Where FV stands for Future Value.


Compound Interest

Interest is paid on deposits at set intervals of time, perhaps monthly, quarterly or annually. The
interest is added to the amount on deposit and so increases the interest payable in future periods.
This is known as compound interest. If interest is paid on an annual basis at rate i, and an amount
a is on deposit for y years then the final value is given by:

                                         FV = a(1 + i)y                                       (1.3)

If interest is paid monthly on a savings account and the interest rate applied is i% per annum, it
is important to be able to compare this arrangement to what is known as the effective rate. That
is the rate which would give the equivalent interest if paid once only at the end on the year. The
general formula for this is:
                                                  i n
                                       ie = 1 +         −1                                    (1.4)
                                                  n


                                                                                                 1
Where ie is the effective rate and n is the number of payments per year. For example my savings
account applies an interest rate of 0.045 per annum every month. The effective rate is just below
4.6% per annum rather than the 4.5% applied. Clearly a daily accrual of interest would be more
favourable still and this comes out as a effective rate just over 4.6%.

This formula can of course be rearranged to calculate the nominal rate given the effective rate.

                                                      1
                                       i = n((1 + ie ) n − 1)                                  (1.5)



Continuous Interest

There is no reason to make a day the shortest period over which interest is applied. Why not an
hour, a minute, or a microsecond? The limit is the continuous rate where interest is applied at
vanishingly small increments of time so as to be in effect continuous. The growth is of course
exponential and given by the following equation:


                                            FV = aeiy                                          (1.6)

where y is again the number of years which might be expressed with decimal places.



Bond Pricing

Bonds issued by stable governments are regarded as risk-free and the current coupon offered is
a good guide to a value that can be used in calculating the risk-free discount rate in finding the
present value of future cash flows.

For a given expected rate of return i, a cash flow C years in the future has a present value (PV )of:


                                                    C
                                          PV =                                                 (1.7)
                                                 (1 + i)N

where N is the number of years ahead the cash flow occurs. The expected yield i is can be taken
as the current risk-free rate for government bonds. A more accurate valuation might use a future
profile of i rather than a single value. The value of a bond is simply the sum of all the future cash



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flows expressed in present value terms. The price is expressed per 100 units of the bond. This
gives:


                                                         Ck
                                      PV = ∑                     ndk                         (1.8)
                                                 k           i   365
                                                      1+     n


where Ck is the k’th expected cash flow, dk is the number of days until Ck , and n is the number of
payments per year.

If inflation increases the yield will grow and the bond will become less valuable. Conversely if
inflation falls then bond prices rise. There is a thriving market in bonds and the prices can be
used to calculate an accurate value of future inflation as anticipated by the market.

Bond coupons are paid at equal intervals, say every 0.5 years. That is to say a bond with a coupon
of 10% will pay exactly 5% half yearly. At some point arbitrary point in time it is convenient
to express equation 1.8 in terms of the coupon rate R and W which is the fraction of the coupon
period between the date of purchase and the next coupon being received.

                                                                                 
                                             1 − (1+1i )N
                                100  R                  n                 1       
                        P=        i
                                                                  +        i
                                                                                            (1.9)
                             (1 + n )W  n              1              (1 + n )N−1 
                                                 1−       i
                                                      (1+ n )




Returns

In finance we are often more concerned with returns than prices. A return r at time t is defined
as:


                                                 (pt − pt−1 )
                                          rt =                                             (1.10)
                                                     pt−1

Provided rt    1.0 this can be expressed in the more convenient form:


                                       rt = ln pt − ln pt−1                                (1.11)



                                                                                                3
Risk

The standard deviation (σ) of a time series is equated with risk and is given by the square root of
the variance.

In order to convert the standard deviation to annualised volatility it is necessary to divide by one
                                                                                     √
over the square root of the sampling period. This is the same as multiplying by 252 for daily
      √                         √
data, 52 for weekly data and 12 for monthly. Thus the annualised volatility for a daily time
series is given by:

                                                 √
                                            υ=    252 σ                                      (1.12)




Time Series Analysis

The mean of a time series pi for i = 1 . . . M is given by:


                                                 1 M
                                            p=     ∑ pi                                      (1.13)
                                                 M i=1


and the variance by:


                                          1 M
                                 σ2 =         ∑ (pi − pi)(pi − pi)                           (1.14)
                                        M − 1 i=1



Portfolio Theory

Let r to denote the actual return on a asset and r denote the average return. In finance theory the
expected return is almost always a synonym for the average return. Where we have a portfolio
of N assets we will denote their returns by ri where i = 1 . . . N. The return of the portfolio as a
whole will be denoted using the subscript p that is r p . The standard deviation of return i will be
denoted by σi and the fraction by value of a portfolio represented by asset i will be denoted by
wi which we describe as the weighting.



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The expected rate of return of a portfolio is the weighted sum of the expected rate of return of its
components.

                                                    N
                                             r p = ∑ wi ∗ ri                                 (1.15)
                                                    i=1


The variance of the portfolio is given by:

                                                N    N
                                       σ2 = ∑
                                        p           ∑ wiw j σi, j                            (1.16)
                                               i=1 j=1




The CAPM

There is a relationship between return and risk, which may be assumed to be linear. We know
two points on this line, the risk and return of the market, for example the FTSE 100 index and
the risk-free rate of return. This allows us to define expected return as a function of risk:


                                                (rM − r f )
                                     E(σ) =                 σ+rf                             (1.17)
                                                   σM

Where rM is the market rate of return and r f is the risk-free rate.

The slope of this line is clearly:


                                               (rM − r f )
                                                                                             (1.18)
                                                  σM

It is useful to be able to see whether an equity is more or less risky that the market as a whole
and this is termed ins β which can be calculated as follows:

                                                        σi,M
                                               βi =                                          (1.19)
                                                        σ2M


In other words it is the ratio of the cross-variance of the share and the market to the variance of
the market. These terms can be found in the covariance matrix found from the two time series of
returns.



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Option pricing

The Black-Scholes formula for calculating the value of European call and put options is given
below. We first define:

    • K is the exercise or strike price in pence

    • S is the current or spot price in pence

    • T is the time to maturity in years

    • σ is the annualised volatility.

    • r is the risk-free annual interest rate, say on one year gilts

Then calculate d1 and d2 and then use the cdf function in a formula for call or put option value.


                                                S           2
                                            ln( K ) + (r + σ )T
                                                           2
                                   d1 =              √
                                                    σ T
                                                    √
                                  d2 =      d1 − σ T
                             C(S, T ) =     SΦ(d1 ) − Ke−rT Φ(d2 )
                             P(S, T ) =     Ke−rT Φ(−d2 ) − SΦ(−d1 )


Φ() is the standard normal cumulative distribution function.



Company Valuation

Valuing a company is a difficult and contentious process. It could be argued that this is the job
of a stock market, but it is useful to be able to estimate a share value using simple rules of thumb
to give an indication of the assumptions the market is making.

The first part of such a valuation is the “liquidation value” of the company which is essentially
the value of its assets minus its liabilities and should be available from a balance sheet showing
recent accounts. The book value usually excludes intangible assets such as goodwill and often
intellectual property, but this is a debatable point. If there are S shares issued and the book value
is B, then the value per share due to this component is B/S.

Companies often pay a dividend to shareholders. This is usually rather less that the amount of
surplus cash generated after tax and interest payments and the ration between that amount and



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the dividend is known as the “coverage ratio.” It is the total cash generated that adds shareholder
value and so if the company gives a dividend of D units for each share and the coverage ration
is R, then this is considered to be a cash flow per share of RD. This can be brought back to
present value in exactly the same way as a bond coupon. If the future free cash flows per share
are summed at their present value and the book value per share added to that then a valuation
of the company has been made. However this neglects two issues. One is the expected rate of
return i which will be higher than that for government bonds because the risk is greater. However
a precise figure is rather difficult to estimate. One could use the CAPM to estimate this or make
a comparison with similar firms. The other factor that was neglected is growth. If the free
cash flows are expected to increase each year as the company grows then this will have a large
effect on the valuation of the company. Anything in the future is problematic to predict and so
a company valuation can never be a highly precise calculation, it will always involve the use of
“reasonable assumptions.”




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