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:

• Define and recognize the definitions of the following terms: interest rate (rate of interest), simple interest, compound interest, accumulation function, future value, current value, present value, net present value, discount factor, discount rate (rate of discount), convertible m-thly, nominal rate, effective rate, inflation and real rate of interest, force of interest, equation of value.
• Given any three of interest rate, period of time, present value, current value, and future value, calculate the remaining item using simple or compound interest. Solve time value of money equations involving variable force of interest.
• Given any one of the effective interest rate, the nominal interest rate convertible m-thly, the effective discount rate, the nominal discount rate convertible m-thly, or the force of interest, calculate any of the other items.
• Write the equation of value given a set of cash flows and an interest rate.

Measurement of interest

Terminologies

In the following, we will introduce some terminologies used in the measurement of interest. Template:Colored definition Template:Colored definition Template:Colored definition Template:Colored remark Template:Colored example Template:Colored definition Template:Colored remark Template:Colored definition Template:Colored remark Template:Colored definition Template:Colored remark We denote the interest earned during the $n$th period ($n$ is a positive integer) from the date of investment (i.e. from the beginning of $n$th period to the end of $n$th period by $I_n$.

By definition, $I_n=A(n)-A(n-1)$, in which

• $A(n)$ is the accumulated value at the end of $n$th period, and
• $A(n-1)$ is the accumulated value at the start of $n$th period.

Simple interest

For Template:Colored em interest, under the Template:Colored em interest rate $i$, the interest earned during each period is calculated according to the principal (and so is constant), i.e. the interest earned is $iA(0)$, i.e. $A(n)-A(n-1)=iA(0)$ for each positive integer $n$. So, $A(n)=A(0)+(A(1)-A(0))+\cdots +(A(n)-A(n-1))=A(0)+inA(0)=A(0)(1+in).$ Since $A(n)=A(0)a(n)$, $a(n)=1+in$ for each nonnegative integer $n$. Intuitively, we may expect that the same form of accumulation function also holds for other nonnegative Template:Colored em. Template:Colored definition Template:Colored remark Template:Colored example Template:Colored example

Compound interest

For Template:Colored em 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 $k$ and compound interest rate $i$, at the end of first year, the interest earned is $ki$, and thus the accumulated value is $k+ki=k(1+i)$.

Thus, at the end of second year, the interest earned is $k(1+i)i$, and so the accumulated value is $k(1+i)+k(1+i)i=k(1+i{)}^2$

Using the same argument, at the end of $n$th year, the interest received is $ki(1+i{)}^{n-1}$, and the accumulated value is $k(1+i{)}^n$. We obtain the accumulation function with nonnegative integer $n$ as input here, namely $a(t)=(1+i{)}^n$.

Intuitively, we may expect that the same form of accumulation function also holds for other nonnegative Template:Colored em. This motivates the definition of compound interest. Template:Colored definition Template:Colored example Template:Colored example Template:Colored remark

Effective discount rate

The effective interest rate was defined as a measure of interest paid at the Template:Colored em of the period. However, there are also Template:Colored em, denoted by $d$, which is a measure of the interest paid at the Template:Colored em of the period. Template:Colored example We can see from this example that the effective Template:Colored em rate is a percentage of the principal, while the effective Template:Colored em rate is a percentage of the balance at the Template:Colored em of the year. Thus, we can define Template:Colored em more precisely as follows: Template:Colored definition Template:Colored remark Template:Colored example Template:Colored example Template:Colored example Template:Colored example

Simple discount rate

For simple discount, the interest paid is calculated according to the accumulated value at the end of $n$th period. That is, the interest paid at the beginning of each period is $dA(n)$ (constant), i.e. $A(m)-A(m-1)=dA(n)$ for each positive integer $m$.

So, $$A(0)=A(n)-(A(n)-A(n-1))-\cdots -(A(1)-A(0))=A(n)-dnA(n)=A(n)(1-dn).$$ Since $A(n)=A(0)a(n)\iff A(0)=A(n)a^{-1}(n)$ , $a^{-1}(n)=1-dn\Rightarrow a(n)=\frac{1}{1-dn}$ for each nonnegative integer $n$ such that $dn<1\iff n<1/d$ (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. Template:Colored definition

Compound discount rate

For Template:Colored em 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 $A(n)=k$ and the compound discount rate is $d$. At the beginning of $n$th year, the interest paid is $kd$, and so the balance at the beginning of $n$th year is $k-kd=k(1-d)$.

Since the balance at the end of $n-1$th year (which is the same as that at the beginning of $n$th year) is $k(1-d)$, the interest paid at the beginning of $n-1$th year is $k(1-d)d$, and thus the balance at the beginning is $k(1-d)-k(1-d)d=k(1-d{)}^2$.

Using the same argument, the balance at the beginning of first year is $k(1-d{)}^n$, i.e. $A(0)=k(1-d{)}^n=A(n)(1-d{)}^n$, and we can see that $a^{-1}(n)=(1-d{)}^n\Rightarrow a(n)=\frac{1}{(1-d{)}^n}$ similarly for each nonnegative integer $n$. This motivates the following generalized definition similarly. Template:Colored definition Template:Colored example

Nominal rates

We have discussed effective interest and discount rates. For those effective rates, the interest is paid exactly Template:Colored em 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 Template:Colored em, 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") $m$thly per measurement period is $i^{(m)}$ ($d^{(m)}$), and its value is a Template:Colored em value only, in the sense that the actual rate used in the calculation for each payment is $i^{(m)}/m$ ($d^{(m)}/m$) Template:Colored em, rather than $i^{(m)}$ ($d^{(m)}$), and this is the $m$thly rate. Template:Colored example Template:Colored example

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 $m$ in "compounded $m$thly" becomes larger and larger, to the infinity. We call this "compounded Template:Colored em".

To be more precise, we would like to know the value of $\underset{m\to \mathrm{\infty }}{\lim }{i}^{(m)}$ during the "infinitesimal" time interval $[t,t+1/m]$ which tends to be simply the time point $t$, and we call this Template:Colored em at time $t$, denoted by ${\delta }_t$. Now, we would like to develop a formula for ${\delta }_t$.

For nominal interest rate $i^{(m)}$, we have the following relationship between $A(t)$ and $A(t+1/m)$ by definition (treating $1/m$ of year as measurement period, then the effective interest rate during the period $[t,t+1/m]$ is $i^{(m)}/m$ by definition): $$A(t+1/m)=A(t)(1+i^{(m)}/m)\Rightarrow i^{(m)}=\frac{m(A(t+1/m)-A(t))}{A(t)}=\frac{A(t+1/m)-A(t)}{1/m}\cdot \frac{1}{A(t)}$$ So, taking limit, $$\underset{={\delta }_t}{\underbrace{\underset{m\to \mathrm{\infty }}{\lim }{i}^{(m)}}}=\frac{1}{A(t)}\cdot \underset{m\to \mathrm{\infty }}{\lim }\frac{A(t+1/m)-A(t)}{1/m}=\frac{1}{A(t)}\cdot \underset{\stackrel{\text{ def }}{=}{A}^{\prime }(t)}{\underbrace{\underset{1/m\to 0}{\lim }\frac{A(t+1/m)-A(t)}{1/m}}}=\frac{A^{\prime }(t)}{A(t)}.$$ This motivates the definition of Template:Colored em. Template:Colored definition Template:Colored remark Template:Colored proposition

Proof. $${\delta }_s=\frac{a^{\prime }(s)}{a(s)}\iff \underset{0}{\overset{t}{\int }}{\delta }_{s}ds=\underset{0}{\overset{t}{\int }}\frac{a^{\prime }(s)}{a(s)}ds\iff \underset{0}{\overset{t}{\int }}{\delta }_{s}ds=\underset{0}{\overset{t}{\int }}\frac{1}{a(s)}d(a(s))\iff \underset{0}{\overset{t}{\int }}{\delta }_{s}ds=\ln a(t)-\underset{=\ln 1=0}{\underbrace{\ln a(0)}}\iff a(t)=\exp \left(\underset{0}{\overset{t}{\int }}{\delta }_{s}ds\right).$$

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Proof.

• Without loss of generality, suppose the measurement period is $[t,t+1]$. Then,

$$\frac{a(t+1)}{a(t)}=\exp \left(\underset{t}{\overset{t+1}{\int }}\delta ds\right)\iff \frac{(1+i{)}^{t+1}}{(1+i{)}^t}=\exp \left(\delta \right)\iff \delta =\ln (1+i).$$

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Present, current and future values

From previous sections, we have seen that money has a Template:Colored em because of the interest, in the sense that $1 today will worth Template:Colored em $1 after a period of time (assuming positive interest rate).

To be more precise, an investment of $k$ will accumulate to $k(1+i)$ at the end of one period, in which $i$ is the effective interest rate during the period. In particular, the term $1+i$ is called Template:Colored em, since it Template:Colored em the investment value at the beginning Template:Colored em its value at the end. Graphically, it looks like the following Template:Colored em (a graph represents statuses at different time).

   *----------*
   |          |
   |          v
 k |            k(1+i)
---*----------*----
  beg        end

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 Template:Colored em (or Template:Colored em), the "reverse" calculation is essentially calculating the Template:Colored em value of the investment, given its accumulated (or Template:Colored em) value at the Template:Colored em.

To be more precise, we would like to calculate the principal (or Template:Colored em, denoted by $PV$) such that it accumulates to $k$ (which is Template:Colored em, denoted by $FV$) at the end of one period. Using equation to describe this situation, we have $$PV(1+i)=FV\iff PV=FV\cdot \frac{1}{1+i}$$ in which $i$ is the effective interest rate during this period. The term $\frac{1}{1+i}$, which is denoted by $v$ , is the Template:Colored em, since it "discounts" the future value to the present value.

   *----------*
   |          |
   v          |
 k/1+i        | k     
---*----------*----
  beg        end

The term Template:Colored em (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 Template:Colored em that date, while some payments are made Template:Colored em that date.

We have discussed how to calculate the present value for one period, but we can Template:Colored em the result to Template:Colored em period. To be more precise, we would like to also calculate the present value given the future value at the end of $t$ periods. We can use the accumulation function $a(t)$ to describe this situation in general . $$PVa(t)=FV\iff PV=FV\cdot \frac{1}{a(t)}=FV{a}^{-1}(t)$$ in which $a^{-1}(t)$ is the Template:Colored em function of $a(t)$ .

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 Template:Colored em.

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 Template:Colored em interest rate and Template:Colored em interest rate.

For Template:Colored em interest rate, it is the same as the "normal" interest rate discussed previously, and thus is denoted by $i$.