Time Value of Money I

Simple and Compound Interest

Interest and Interest Rate

Let

P=\text{principal},\qquad i=\text{interest rate per period}

We want to find the balance after (n) periods.

Banking / Financial Market

In a classical banking system:

flowchart LR D[Depositors] -->|Lend money| B[Banks] B -->|Pay interests| D B -->|Lend money| C[Companies and Individuals] C -->|Pay interests| B

Banks act as financial intermediaries.

\text{Spread} = \text{Loan Rate} - \text{Deposit Rate}

Funds may flow through:

  • Indirect finance: through financial intermediaries;
  • Direct finance: through financial markets, e.g. bond issuance.
flowchart LR L[Lenders / Savers] FI[Financial Intermediaries] FM[Financial Markets] B[Borrowers / Spenders] L -->|Funds| FI FI -->|Funds| B L -->|Funds| FM FM -->|Funds| B

Different financial products have different interest rates, but these rates are related within the financial system.

\text{Interest is charged in borrowing-lending relationships.}

Simple Interest

Only the original principal earns interest.

Period Beginning Balance Interest Ending Balance
1 ​P ​Pi ​P(1+i)
2 ​P(1+i) ​Pi ​P(1+2i)
​\vdots ​\vdots ​Pi ​\vdots
​n ​P[1+(n-1)i] ​Pi ​P(1+ni)

Therefore,

F=P(1+ni)

and total interest is

I_{\text{simple}}=nPi

Compound Interest

Both the principal and accumulated interest earn interest.

Period Beginning Balance Interest Ending Balance
1 ​P ​Pi ​P(1+i)
2 ​P(1+i) ​P(1+i)i ​P(1+i)^2
​\vdots ​\vdots ​\vdots ​\vdots
​n ​P(1+i)^{n-1} ​P(1+i)^{n-1}i ​P(1+i)^n

Therefore,

F=P(1+i)^n

and total interest is

I_{\text{compound}}=P(1+i)^n-P

Simple Interest vs. Compound Interest

Simple Interest Compound Interest
Interest earned on Principal only Principal + accrued interest
Future balance ​P(1+ni) ​P(1+i)^n
Growth Linear Exponential
  • Simple: principal earns interest
  • Compound: balance earns interest

The longer the investment horizon, the larger the difference between simple and compound interest.

Time Value of Money

Time Value

Money has time value because money available today can be invested and earn a return over time. Therefore, receiving the same amount of money today is generally better than receiving it in the future.

Present Value and Future Value

Present Value (PV) is the value of a future amount of money measured at the present time.

Future Value (FV) is the value of a present amount of money measured at a future time.

If ​P is the present value, ​F is the future value, ​i is the interest rate per period, and ​N is the number of periods:

F=P(1+i)^N

Finding ​F from ​P is called compounding.

P=\frac{F}{(1+i)^N}

Finding ​P from ​F is called discounting.

Interpretations of Present Value and Future Value

The meanings of PV and FV depend on the context.

Present Value

For a future cash flow ​F:

P=\frac{F}{(1+i)^N}
  • Deposit: ​P is the amount that must be deposited now to obtain ​F in the future.
  • Loan: ​P is the amount that can be borrowed now if ​F will be repaid in the future.

Future Value

For a present cash flow ​P:

F=P(1+i)^N
  • Deposit: ​F is the future balance of the account.
  • Loan: ​F is the future repayment amount.

Some Notations and Terminology

The compound factor is

(F/P,i,N)=(1+i)^N

and

F=P(F/P,i,N)

where ​(F/P,i,N) means find ​F given ​P at interest rate ​i per period for ​N periods.

The discount factor is

(P/F,i,N)=\frac{1}{(1+i)^N}

and

P=F(P/F,i,N)

where ​(P/F,i,N) means find ​P given ​F at interest rate ​i per period for ​N periods.

Notation rule: the first letter is the value to be found, and the second letter is the given value.

Cash-Flow Diagrams

A cash-flow diagram is a graphical representation of cash flows occurring at different times.

  • The horizontal line represents time.
  • Time ​n means the end of period ​n and the beginning of period ​n+1.
  • An upward arrow represents a cash inflow.
  • A downward arrow represents a cash outflow.
  • The diagram depends on the point of view, so the same perspective must be used consistently.

Future and Present Value of Cash-Flow Streams

Value of a Single Cash Flow

A single cash flow can be moved to another time point by compounding forward or discounting backward.

FV of a Single Cash Flow

For a cash flow ​C at time ​t, its value ​n periods later is

FV=C(1+i)^n

PV of a Single Cash Flow

For a cash flow ​C at time ​t, its value ​n periods earlier is

PV=C(1+i)^{-n} =\frac{C}{(1+i)^n}

Moving forward in time uses compounding, while moving backward in time uses discounting.

General Cash-Flow Stream

For a cash-flow stream ​C_0,C_1,\ldots,C_n, the value of the stream at a given time is the sum of the values of all individual cash flows at that time.

Future Value of a Cash-Flow Stream

FV=\sum_{t=0}^{n} C_t(1+i)^{n-t}

Present Value of a Cash-Flow Stream

PV=\sum_{t=0}^{n}\frac{C_t}{(1+i)^t}

At any general time ​k, cash flows before ​k are compounded forward and cash flows after ​k are discounted backward.

Annuities

An annuity is a series of equal cash flows ​A occurring at the end of each period from time 1 to time ​N.

Future Value of an Annuity

F=A\frac{(1+i)^N-1}{i}

Define

(F/A,i,N)=\frac{(1+i)^N-1}{i}

then

F=A(F/A,i,N)

Sinking Fund

If ​F is given,

A=F\frac{i}{(1+i)^N-1}

Define

(A/F,i,N)=\frac{i}{(1+i)^N-1}

then

A=F(A/F,i,N)

Present Value of an Annuity

P=A\frac{1-(1+i)^{-N}}{i}

Define

(P/A,i,N)=\frac{1-(1+i)^{-N}}{i}

then

P=A(P/A,i,N)

Capital Recovery

If ​P is given,

A=P\frac{i}{1-(1+i)^{-N}}

Define

(A/P,i,N)=\frac{i}{1-(1+i)^{-N}}

then

A=P(A/P,i,N)

The four annuity factors describe the conversions ​A \leftrightarrow F and ​A \leftrightarrow P.

Deferred Annuities

A deferred annuity is an annuity whose first payment is postponed for several periods.

If the first payment occurs at time ​J+1 and the last payment occurs at time ​N, there are ​N-J payments.

Present Value

First calculate the annuity value at time ​J, then discount it back to time 0.

P=A(P/A,i,N-J)(P/F,i,J)

Future Value

The future value at time ​N is

F=A(F/A,i,N-J)

For a deferred annuity, first identify the time immediately before the first payment and treat the remaining cash flows as an ordinary annuity.

Arithmetic Gradient Series

An arithmetic gradient series is a cash-flow series that increases or decreases by a constant amount ​G from one period to the next.

The standard pattern is ​0, G, 2G, \ldots, (N-1)G, so the gradient component starts at time 2.

Future Value

F=G(F/G,i,N)

where

(F/G,i,N) = \frac{1}{i}(F/A,i,N)-\frac{N}{i}

Present Value

P=G(P/G,i,N)

where

(P/G,i,N) = \frac{1}{i} \frac{1-(1+i)^{-N}}{i} - \frac{N}{i(1+i)^N}

Equivalent Annuity

An arithmetic gradient series can be converted into an equivalent annuity with the same PV or FV.

A=G(A/G,i,N)

where

(A/G,i,N) = \frac{1}{i} - \frac{N}{(1+i)^N-1}

Geometric Gradient Series

A geometric gradient series is a cash-flow series that increases or decreases by a constant percentage ​f from one period to the next.

If the first cash flow is ​A_1, the cash-flow pattern is ​A_1, A_1(1+f), A_1(1+f)^2, \ldots, A_1(1+f)^{N-1}.

Present Value

For ​f\neq i,

P= A_1 \frac{1-(1+f)^N/(1+i)^N}{i-f}

For ​f=i,

P=\frac{A_1N}{1+i}

Future Value

For ​f\neq i,

F= A_1 \frac{(1+i)^N-(1+f)^N}{i-f}

For ​f=i,

F=A_1N(1+i)^{N-1}

A geometric gradient changes by a constant percentage, while an arithmetic gradient changes by a constant amount.