Finance for Actuarial

A summary of Finance for Actuarial course:

Cash Flows; Time & Rounding Conventions; Glossary of (some) Financial Products
Simple & compound interest; Rate of Discount; Nominal Interest; Accumulation Factors; Force of Interest
Discounted, Accumulate & Present Value; Continuous Cash flows.
Annuties Immediate & Due; present and future values; increasing & perpetuities…
Loan schedules; Level Installments; APR and Flat rate
Equations of Value and Yield.

(This covers about half of the Institute and Faculty of Actuaries CT1 exam — though you should probably work on context if you want to pass the exam.)

Cash Flow

Def [Cash flow] A cash flow is a vector of times $t_j$ and amounts $c_j$ , $j=1,...,n$

$\Big( (t_1,c_1), (t_2,c_2), ... (t_n, c_n) \Big)$

If $c_j>0$ then it is an inflow and if $c_j<0$ it is an outflow.

We will generally consider the values of $c_j$ (and times $t_j$ ) to be fixed an deterministic. However, later it might be better to weight the probabilities of different outcomes. These are then called generalized cash flows.

Time

Def [Exact number of days] The exact number of days between two dates is the number days from the first given date to the second (excluding the first or last date)

E.g. There are $9$ days betweens 1st and 10th April. There are $144$ days between 29th April and 20th September.

Def [The Simplified Convention] The simplified convention assumes every month has 30 days and every year has 360 days. So under the simplified convention the number of days between two dates $d_1 / m_1 / y_1$ and $d_2 / m_2 / y_2$ is

Q. How many days are there between 29th April and 20th September under the simplified convention?

A. 141.

Rounding

Numbers are often rounded so that they fit with natural units, or so they are simpler (and thus hopefully reduce human error). There are no fixed rules for rounding and you need to be mindful of “round off error”, as many small errors can add up!

Here are some rules of thumb

Number in currency are rounded to the nearest hundredth.
Number in currency larger than 100,000 are rounded ot the nearest one.
General numbers of modulus less that 1 are rounded to the nearest tenthousandth.
General numbers with modulus larger that 100,000 are rounded to the nearest hundredth.

A Glossary of financial products

Here is an imcomplete and perhaps simplified list of finacial products. Some are risky and some are risk-free. All of these are examples of cash flows.

Savings account: A deposit in a band that recieves interest. Often recieves fixed interest, at least for an initial period.

Bond: After purchase, you receive regular payments called coupon payments and on a specific date of maturity you receive an amount of money called the par value or face value of the bond.

Zero-coupon bond: A bond without coupons. (face value is typically higher than purchase price).

Share: A unit of ownership in a public company (aka a coperation).

Annuity: A contract sold by an insurance company. After making one or more payments to a purchase the annuity. The buyer recieves periodic payments unit death (basically this is a pension).

Interest only Loan: The borrower is paid an amount of money. The borrower pays interest on regular invervals at maturity the full amount is due back. (Similar to a bond but the rights are different).

Repayment Loan: The borrower is paid an amount of money. He repays the full amount plus interest in a series of periodic payments.

Interest Rates

When you deposit money in a savings account, in reture over time you recieve payments per unit of money saved. This is called interest. In general interstes rates can be fixed (for a time-period) or variable. Rates depend heavily on the economic states at the time the savings account is initiated.

Def [Effective rate of interest / AER] if $1$ unit of capital invested at time $t$ is returned as $1+ i(t)$ units of capital at time $t+1$ , then $i(t)$ is called the effective rate of interest for period $t$ to $t+1$ .

Def [Simple Interest] Simple interest is added over time. In particular, under simple interest, an initial capital $c$ is returned as

after $n$ units of time.

Suppose that interest is fixed over time $i(t)=i$ for all $t$ . Then show that over $n$ time units and initial investment $c$ is worth

Simple interest is rarely used in practice. Really you earn interest on your interest, so called compound interest. Unless stated otherwise when we say interest we mean compound interest.

Def [Compound Interest] Compound interest is multiplied over time. In particular, under compound interest, an initial capital is returned as

Ex. Suppose that interest is fixed over time $i(t)=i$ for all $t$ . Then show that over $n$ time units and initial investment $c$ is worth

Rate of Discount

Def [Rate of Discount] You are loaned $\pounds 1$ that you must repay after $1$ year with a rate of interest $i$ . You pay interest in advance and effective rate of interest is $i$ . You are thus given $\pounds 1-d$ where

Here $d$ is called the (effective) rate of discount.

Ex. Show that

$d= \frac{i}{1+i}\ .$

Nominal Rates

So far (fixed) interest is compounded as

where $t$ is an integer number of years. The same expression applied for fractions of time (e.g. weeks, months, quarters).

Def [Nominal Rate of Interest] Over a time interval of length $h$ if an amount of $\pounds 1$ increases to

then $i_h$ is called the nominal rate of interest. Note we apply the notation that $i^{(p)} = i_{1/p}$ , e.g. $p=12$ for monthly interest.

Ex. Show that, under fixed interest, that $(1+ h i_h) = (1+i)^h\quad\text{and}\quad i_h = \frac{(1+i)^h - 1}{h} \ .$

Ex. [Nominal Rate of Discount] Argue that if there is a notion of nominal discount, $d_h$ , then it satisfies

$(1 -h d_h) (1 + h i_h) =1 \quad \text{and} \quad d_h = \frac{i_h}{1+ h i_h}.$

Accumulation Factor

Thus far interest has been applied in units or fractions of time. With Accumulation factors, we have a working definition of compounding any time length.

Def [Accumulation Factor] If $\pounds 1$ at time $s$ is, as time $t$ worth

then $A(s,t)$ is called the Accumulation Factor. It is the amount that your money has multiplied by from time $s$ to $t$ .

Lemma [Principle of Consistency] Any self respecting Accumulation factor must satisfy

for $t\leq t'\leq t''$ .

Force of Interest

The Accumultion factor is great; it considers nice and general time, but it is a multaplicative factor not a rate of change. For this we consider the Force of Interest.

Def [Force of interest] Assuming the following limit exists

$\delta(t) = \lim_{h\rightarrow 0} \frac{1}{h} \left[A(t,t+h) -1 \right]$

then we call $\delta(t)$ the force of interest at time $t$ .

Note that the above defintion assumes that our Accumulation factor has nice differentiability properties. (This is a limitation.)

Ex. If $i_h(t)$ is the nominal rate of interest applied from time $t$ to time $t+h$ , then argue that $\delta = \lim_{h\rightarrow 0} {i_h}$

Ex. If there is a fixed rate of interest $i$ , argue that $\delta = \log (1+i)$ (Hint: Differentiate $(1+i)^t$ .)

Thrm. The Accumulation factor relates to the force of interest as follows $A(s,t) = \exp \left\{ \int_s^t \delta (u) du \right\}$

This result is really just the Fundamental Theorem of Calculus applied to the $\log A(s,t)$ . You may wish to verify this result for the case of fixed interest.

Discounting and Accumulating

We consider how to evaluate several payments made at different times at a single moment in time.

Def [Discounted Value and Accumulated Value] A cash flow Consider a cash flow

$\bar{c} = ((t_1,c_1),...,(t_n,c_n))$

where for each $i$ you are paid $\pounds c_i$ at time $t_i$ ,

For $s\leq t_i$ $\forall i$ , the Discounted Value of the cash flow is $\text{DVal}(\bar{c}, s) = \sum_{i=1}^n \frac{c_i}{A(s,t_i)}$

where $A(s,t)$ is the accumulation factor.

For $s=0$ , $\text{DVal}(\bar{c},0)$ is called the Present Value of the payment.
For $s\geq t_i$ $\forall i$ , the Accumulated Value of the cash flow is $\text{AVal}(\bar{c}, s) = \sum_{i=1}^n c_i A(t_i,s)$

Note in each case above we essentially relativize money to a single time. From this we can move to different times:

Ex. Argue that for $s\leq r <t_1$

$\text{DVal}(\bar{c},s) A(s,r) = \text{DVal} ( \bar{c},r)$

and find a similar expression between $\text{AVal}(\bar{c},s)$ and $\text{AVal}(\bar{c},r)$ .

Continuous Cash Flow

Def [Continuous Cash Flow] A continuous cash flow over time interval $[a,b]$ is an integrable function

$c_c: [a,b] \rightarrow \mathbb R$

For a continuous cash flow and $s<a<b<t$ we have that

$DVal(c_c,s) = \int_a^b c_c(u) \frac{1}{A(s,u)} du \quad\text{and}\quad AVal(c_c,t) = \int_a^b c_c(u) A(s,u) du$

I’ve called this a theorem as is really just a consequence of the definition for discrete cash flows.

Def [Continuous payment on fixed capital] If you have a fixed amount $c$ and a force of interest $\delta(t)$ then interest is paid according to the continuous cash flow

$c_c(t) = c \delta(t)$

Ex. Convince yourself that the continuous payment of interest above agrees with the interest accrued under fixed interest (and fixed force of interest).

Ex. Convince yourself that if cash flow $\bar{c}$ is the sum of two cash flows

$\bar{c} = \bar{c}_1 + \bar{c}_2$

(Nb. these cash flows could be discrete or continuous, or indeed, a mixture of the two) then argue that

$\begin{aligned} \text{DVal}(\bar{c}, s) &= \text{DVal}(\bar{c}_1, s) + \text{DVal}(\bar{c}_2, s) \\ \text{AVal}(\bar{c}, t) &= \text{AVal}(\bar{c}_1, t) + \text{AVal}(\bar{c}_2, t)\end{aligned}$

where is assumed that $s$ is before the first item and $t$ is after the last item in cash flow $\bar{c}$ .

Annuities

Annuities are cash-flows consisting of regular payments.

Def [Annuity Immediate] An Annuity Immediate is the cash flow

$\bar{c} = ((1,1),(2,1),...,(n,1))$

and its present value and future values respectively satisfy

$a_{ \actuarialangle{n} } = DVal(\bar{c} , 0 ),\qquad s_{ \actuarialangle{n} } = AVal(\bar{c} , n )$

Show that, under fixed interest,

$a_{ \actuarialangle{n} } = \frac{1}{i} \left( 1 - \frac{1}{(1+i)^n} \right)$

We could continue writing a bunch of definitions on annuities. Instead here is a big table

Screen Shot 2017-12-12 at 17.01.45.png

Ex. Convince yourself in the table above that for each present value we get the future value by multiplying by $(1+i)^n$ . e.g.

$(1+i)^n a^{(p)}_{\actuarialangle{n}} = s^{(p)}_{\actuarialangle{n}}$

Ex. Convince yourself in the table above to go from an annuity due to and annuity immediate we multiply by $(1+i)^t$ where $t$ is the time between the 1st payment and the second. E.g.

Screen Shot 2017-12-11 at 19.03.49.png

Loan Schedules

In taking a loan, the amount that you take our should equal the amount that you put in, after adjusting for interest. Thus if you take a loan of $L_0$ and pay it back with cash flow/repayment schedule $\bar{c}$ then

$L_0 = \text{DVal} ( \bar{c}, 0)$

Def [Level Installments] You pay a loan back with level installments if all repayments on your loans are equal. I.e.

$\bar{c} = ( (1,x) , ..., (n,x))$

This is an annuity immediate i.e. $\text{DVal}(\bar{c},0) = x a_{{n}\rceil}$ . Thus

$L_0 = a_{\actuarialangle{n}}$

Ex. Argue by the same logic as above, that if $L_k$ is the outstanding loan after $k$ years then

$L_k = x a_{\actuarialangle{n-k}} = L_0 \frac{a_{\actuarialangle{n-k}} }{a_{\actuarialangle{n}}}$

Ex. Convince yourself that it is common sense that the interest and capital paid on the loan at year $k$ are given by

$I_k = i L_{k-1} \qquad C_k = x- I_k.$

Ex. Convince yourself that if the Loan is repaid by a general repayment schedule $\bar{c}$ (cash flow) then same calculations as above apply but we replace $a_{{n}\rceil}$ and $a_{{n-k}\rceil}$ with the discounted value of the appropriate number of remaining payments in the repayment schedule $\bar{c}$ .

APR

Def [Annual Percentage Rate of Charge] Suppose that you arre given a loan $L_0$ and a repayment schedule $\bar{c}$ then the interest in implicitly defined. Specifically, The Annual Percentage Rate of Charge (APR) is the solution $i_0$ that solves

$L_0 = \text{DVal}_{i_0} ( \bar{c},0)$

where the subscript $i_0$ is used to indicate that $i_0$ is the effective interest applied in the above calculation.

Note: that after substitution $x = (1+ i)^{-1}$ we have

$\begin{aligned} \text{DVal}_{i} ( \bar{c},0) &= \frac{c_1}{1+i} + \frac{c_2}{(1+i)^2} + ... + \frac{c_n}{(1+i)^n} \\ & = c_1 x + c_2 x^2 + ... + c_n x^n.\end{aligned}$

In other words we have a polynomial. So finding the APR resorts to us finding roots of a polynomial, which is analytically non-trivial but, in practice, numerically straight-forward – just ask wolfram-alpha!

Def [Flat rate] The flat rate of a loan is

$FR = \frac{1}{n L_0} \left[ \sum_{k=1}^n x_k - L_0\right]$

The flat rate is a bit like APR, except it judges the absolute change in capital per unit of loan. However, capital changes relatively (not absolutely) so people tend not to use the flat rate.

Equations of Value

We apply the same logic that we used to get the APR of a loan to general cash flows.

Def [Equation of Value / Yield] For a cash flow the equation

$0 = \text{DVal}_y( \bar{c},0)$

is called the Equation of Value. If there is a unique solution $y$ to this equation then the solution is called the Yield.

Note: The yield belongs to the interval $(-1,\infty)$ . Again finding the yield involves finding the root of a polynomial, since substituting $x= (1+y)^{-1}$ gives

$\begin{aligned} \text{DVal}_{y} ( \bar{c},0) &= c_0 + \frac{c_1}{1+y} + \frac{c_2}{(1+y)^2} + ... + \frac{c_n}{(1+y)^n} \\ & = c_0+c_1 x + c_2 x^2 + ... + c_n x^n= 0\end{aligned}$

A yield of zero, suggests that payments in the cash flow are neither growing of shrinking. A positive yield suggests that payments are growing (at rate $y$ ). If the yield $y$ is greater than the effective interest, i.e. $y >i$ then this suggests the cash flow is a good one to have, while if $y < i$ then this suggest that that the money is shrinking relative to inflation from interest (not so good). Finally, note that if there is only one change in sign in the coefficients $c_0, c_1,..., c_n$ then there is one solution to the equation of value, so the yield exists.