Financial Math FM/Time Value of Money

Learning objectives.
The Candidate will understand and be able to perform calculations relating to present value, current value, and accumulated value.
Learning outcomes.
The Candidate will be able to:
Measurement of interest.
Terminologies.
In the following, we will introduce some terminologies used in the measurement of interest.
=\frac{A(t)}{k}&lt;/math&gt;. 
We denote the interest earned during the formula_1th period (formula_1 is a positive integer) from the date of investment 
(i.e. from the beginning of formula_1th period to the end of formula_1th period by formula_5. 
By definition, formula_6, in which 
Simple interest.
For interest, under the interest rate formula_11, the interest earned during each period is calculated according to the principal (and so is constant), i.e. the interest earned is formula_12, i.e. formula_13 for each positive integer formula_1.
So, formula_15 
Since formula_16, formula_17 for each nonnegative integer formula_1. 
Intuitively, we may expect that the same form of accumulation function also holds for other nonnegative . 
Compound interest.
For interest, the interest earned for each period is calculated according to the accumulated value at the beginning of that period.
To be more precise, with principal of formula_19 and compound interest rate formula_11, at the end of first year, the interest earned is formula_21, and thus the accumulated value is formula_22. 
Thus, at the end of second year, the interest earned is formula_23, and so the accumulated value is formula_24 
Using the same argument, at the end of formula_1th year, the interest received is formula_26, and the accumulated value is formula_27. 
We obtain the accumulation function with nonnegative integer formula_1 as input here, namely formula_29. 
Intuitively, we may expect that the same form of accumulation function also holds for other nonnegative . 
This motivates the definition of compound interest.
=\frac{(1+i)^{n-1}\big((1+i)-1\big)}{(1+i)^{n-1}}=i.&lt;/math&gt;
Effective discount rate.
The effective interest rate was defined as a measure of interest paid at the of the period. 
However, there are also , denoted by formula_30, which is a measure of the interest paid at the of the period. 
We can see from this example that the effective rate is a percentage of the principal, 
while the effective rate is a percentage of the balance at the of the year.
Thus, we can define more precisely as follows:
Simple discount rate.
For simple discount, the interest paid is calculated according to the accumulated value at the end of formula_1th period. That is, the interest paid at the beginning of each period is formula_32 (constant), 
i.e. formula_33 for each positive integer formula_34.
So,
formula_35
Since formula_36 , 
formula_37 for each nonnegative integer formula_1 such that formula_39 (so that the accumulation function is defined).
Similarly, we may intuitively expect that the same form of accumulation function holds for other nonnegative numbers, which motivates the following definition.
Compound discount rate.
For discount, the interest paid at the beginning of each period is calculated according to the balance at the end of that period.
To be more precise, suppose formula_40 and the compound discount rate is formula_30. 
At the beginning of formula_1th year, the interest paid is formula_43, and so the balance at the beginning of formula_1th year is formula_45. 
Since the balance at the end of formula_46th year (which is the same as that at the beginning of formula_1th year) is formula_48, 
the interest paid at the beginning of formula_46th year is formula_50, and thus 
the balance at the beginning is formula_51.
Using the same argument, the balance at the beginning of first year is formula_52, i.e. formula_53, and we can see that formula_54 similarly for each nonnegative integer formula_1. 
This motivates the following generalized definition similarly.
Nominal rates.
We have discussed effective interest and discount rates. For those effective rates, the interest is paid exactly per measurement period (either at the beginning (for discount rates) or at the end (for interest rates)).
However, the interest can be paid more than once per measurement period, and the interest and discount rates, 
for which the interest is paid more than once per measurement period, 
are called , rather than effective, rates.
The reason for calling those rates as "nominal" is that the notation for the nominal interest (discount) rate payable (or "convertible" or "compounded") formula_34thly per measurement period is formula_57 (formula_58), 
and its value is a value only, in the sense that the actual rate used in the calculation for each payment is formula_59 (formula_60) , rather than formula_57 (formula_58), and this is the formula_34thly rate. 
&amp;\Rightarrow&amp; 1+\frac{i^{(12)}}{12}&amp;=\big((1-0.03)^{-6}\big)^{1/12}\text{ or }-\big((1-0.03)^{-6}\big)^{1/12}(\text{rejected})\\
&amp;\Rightarrow&amp; i^{(12)}&amp;=12(0.97^{-6/12}-1)\approx 0.184.
&lt;/math&gt;
Force of interest.
We have discussed nominal interest rates, and in this subsection, we will discuss what will happen if the compounding frequency gets higher and higher, i.e. the formula_34 in "compounded formula_34thly" becomes larger and larger, to the infinity. 
We call this "compounded ".
To be more precise, we would like to know the value of formula_68 during the "infinitesimal" time interval formula_69 which tends to be simply the time point formula_70, and we call this at time formula_70, denoted by formula_72.
Now, we would like to develop a formula for formula_72.
For nominal interest rate formula_57, we have the following relationship between formula_75 and formula_76 by definition (treating formula_77 of year as measurement period, then the effective interest rate during the period formula_69 is formula_59 by definition): 
formula_80 
So, taking limit,
formula_81
This motivates the definition of . 
Present, current and future values.
From previous sections, we have seen that money has a because of the interest, in the sense that $1 today will worth $1 after a period of time (assuming positive interest rate).
To be more precise, an investment of formula_19 will accumulate to formula_83 at the end of one period, in which formula_11 is the effective interest rate during the period.
In particular, the term formula_85 is called , since it the investment value at the beginning its value at the end. 
Graphically, it looks like the following (a graph represents statuses at different time). 
We would often like to do something "reverse" to calculating the accumulated value given the principal. That is, calculating the principal given the accumulated value.
Since the principal is the investment value at initial time, which is often (or ), 
the "reverse" calculation is essentially calculating the value of the investment, given its accumulated (or ) value at the .
To be more precise, we would like to calculate the principal (or , denoted by formula_86) such that it accumulates to formula_19 (which is , denoted by formula_88) at the end of one period.
Using equation to describe this situation, we have
formula_89
in which formula_11 is the effective interest rate during this period.
The term formula_91, which is denoted by formula_92 , is the , 
since it "discounts" the future value to the present value. 
The term (which is "at the middle" of present and future values) is sometimes used. It means the value of the payments at a specified date, and some payments are made that date, while some payments are made that date.
We have discussed how to calculate the present value for one period, but we can the result to period.
To be more precise, we would like to also calculate the present value given the future value at the end of formula_70 periods.
We can use the accumulation function formula_94 to describe this situation in general .
formula_95
in which formula_96 is the function of formula_94 .
Also, given multiple future values, we can calculate the total present value of these future values by summing up all present values corresponding to these future values.
Equations of value.
For two or more payments at different time points, to compare them fairly, we need to accumulate or discount them to a common time point, so that the effects on payments from the time value of money are eliminated.
The equation which accumulates or discounts each payment as in above is called the .
Indeed, we have encountered equations of value in previous sections, since an example of equations of value is calculating present values of multiple payments ("present" is the common time point).
The concepts involved in equations of value have been discussed previously.
Inflation and real interest rate.
In previous sections, we have not consider the effect from inflation, and we will introduce how interest rates changes under inflation.
Because of the inflation, there are two types of interest rates, namely interest rate and interest rate.
For interest rate, it is the same as the "normal" interest rate discussed previously, and thus is denoted by formula_11.
- formula_107
- formula_108 
- formula_109 
- formula_110 
&lt;/quiz&gt;