Time Value of Money - Annuities, Perpetuities, and Compounding

Annuities Introduction

Annuity

An annuity is defined as a series of equal dollar payments made for a specified number of years at regular intervals. They are a fundamental concept in finance, especially in areas like retirement planning, loan amortization, and investing.

Ordinary Annuity

If the payments occur at the end of each period, the annuity is classified as an ordinary annuity. Unless explicitly stated otherwise, when the term "annuity" is used, it typically refers to an ordinary annuity.

Annuity Due

If the payments occur at the beginning of each period, the annuity is an annuity due. An annuity due can be conceptualized as an ordinary annuity where all the payments have been shifted forward by one year.

Ordinary Annuity vs. Annuity Due Visual Representation

graph TD
    subgraph Ordinary Annuity
        A[PMT] --> B[PMT]
        B --> C[PMT]
        C --> D[...]
        D --> E[PMT]
    end
    subgraph Annuity Due
        F[PMT] --> G[PMT]
        G --> H[PMT]
        H --> I[...]
        I --> J[PMT]
    end

    A -.-> 0(Time 0)
    A -- Payment at end of Period 1 --> 1(Time 1)
    B -- Payment at end of Period 2 --> 2(Time 2)
    C -- Payment at end of Period 3 --> 3(Time 3)

    F -- Payment at beginning of Period 1 --> 0(Time 0)
    G -- Payment at beginning of Period 2 --> 1(Time 1)
    H -- Payment at beginning of Period 3 --> 2(Time 2)

    style 0 fill:#fff,stroke:#fff,stroke-width:0px
    style 1 fill:#fff,stroke:#fff,stroke-width:0px
    style 2 fill:#fff,stroke:#fff,stroke-width:0px
    style 3 fill:#fff,stroke:#fff,stroke-width:0px

Ordinary Annuity Timeline: Payments ( $PMT$ ) occur at times $t=1, 2, 3, ext{etc.}$

graph LR
    0(0) --> 1[PMT]
    1 --> 2[PMT]
    2 --> 3[PMT]

Annuity Due Timeline: Payments ( $PMT$ ) occur at times $t=0, 1, 2, ext{etc.}$

graph LR
    0[PMT] --> 1[PMT]
    1 --> 2[PMT]
    2 --> 3[PMT]

Future Value of an Ordinary Annuity (FVAn)

Example: Calculating Future Value of a 3-year Ordinary Annuity

Consider a three-year ordinary annuity with $400 payments at an annual interest rate of 5%.

Cash Flow Breakdown and Accumulation at End of Year 3:

Payment 1 (at end of Year 1): Accrues interest for 2 years. Value at Year 3 = $400 imes (1.05)^2 = 441.00$
Payment 2 (at end of Year 2): Accrues interest for 1 year. Value at Year 3 = $400 imes (1.05)^1 = 420.00$
Payment 3 (at end of Year 3): Accrues interest for 0 years. Value at Year 3 = $400 imes (1.05)^0 = 400.00$

Total Future Value ( $FVA_3$ ) at the end of Year 3 = $441.00 + 420.00 + 400.00 = 1,261.00$

FVAn Equation Solution

The future value of an ordinary annuity can be calculated using the formula:
$FVA_n = PMT imes rac{(1+r)^n - 1}{r}$

Applying this to the example:
$FVA3 = 400 imes rac{(1+0.05)^3 - 1}{0.05}$ $FVA3 = 400 imes rac{(1.157625) - 1}{0.05}$
$FVA3 = 400 imes rac{0.157625}{0.05}$ $FVA3 = 400 imes (3.1525) = 1,261.00$

FVAn Financial Calculator Solution

For a three-year ordinary annuity of $400 at 5%:

N = $3$ (Number of periods)
I/Y = $5$ (Interest rate per year)
PMT = $-400$ (Payment amount, entered as negative for cash outflow)
PV = $0$ (Present value, initially zero for an annuity FV calculation)
FV = $1,261.00$ (Computed Future Value)

Future Value of an Annuity Due (FVA(Due)n)

Example: Calculating Future Value of a 3-year Annuity Due

Consider a three-year annuity due with $400 payments at an annual interest rate of 5%.

Cash Flow Breakdown and Accumulation at End of Year 3:

Payment 1 (at beginning of Year 1 / Time 0): Accrues interest for 3 years. Value at Year 3 = $400 imes (1.05)^3 = 463.05$
Payment 2 (at beginning of Year 2 / Time 1): Accrues interest for 2 years. Value at Year 3 = $400 imes (1.05)^2 = 441.00$
Payment 3 (at beginning of Year 3 / Time 2): Accrues interest for 1 year. Value at Year 3 = $400 imes (1.05)^1 = 420.00$

Total Future Value ( $FVA(Due)_3$ ) at the end of Year 3 = $463.05 + 441.00 + 420.00 = 1,324.05$

Relationship with Ordinary Annuity

Annuity due future value can also be found by multiplying the ordinary annuity future value by $(1+r)$ :
$FVA(Due)n = FVAn imes (1+r)$
Using the previous example: $1,261.00 imes (1+0.05) = 1,261.00 imes 1.05 = 1,324.05$

FVA(Due) Equation Solution

The future value of an annuity due can be calculated using the formula:
$FVA(DUE)_n = PMT imes rac{(1+r)^n - 1}{r} imes (1+r)$

Applying this to the example:
$FVA(DUE)3 = 400 imes rac{(1.05)^3 - 1}{0.05} imes (1.05)$ $FVA(DUE)3 = 400 imes (3.1525) imes (1.05) = 1,324.05$

FVA(Due) Financial Calculator Solution

For a three-year annuity due of $400 at 5%:

N = $3$
I/Y = $5$
PMT = $-400$
PV = $0$
FV = $1,324.05$
Note: When working with Annuity Due, set the financial calculator to [BGN] mode by pressing [2nd] [BGN] [2nd] [SET]. The [BGN] indicator will appear on the top right corner of the screen.

Present Value of an Ordinary Annuity (PVAn)

Example: Calculating Present Value of a 3-year Ordinary Annuity

Consider a three-year ordinary annuity with $400 payments if the discount rate is 5%.

Cash Flow Breakdown and Discounting to Year 0:

Payment 1 (at end of Year 1): Discounted for 1 year. Value at Year 0 = $400 imes (1.05)^{-1} = 380.95$
Payment 2 (at end of Year 2): Discounted for 2 years. Value at Year 0 = $400 imes (1.05)^{-2} = 362.81$
Payment 3 (at end of Year 3): Discounted for 3 years. Value at Year 0 = $400 imes (1.05)^{-3} = 345.54$

Total Present Value ( $PVA_3$ ) at Year 0 = $380.95 + 362.81 + 345.54 = 1,089.30$

PVA Equation Solution

The present value of an ordinary annuity can be calculated using the formula:
$PVA_n = PMT imes rac{1 - rac{1}{(1+r)^n}}{r}$

Applying this to the example:
$PVA3 = 400 imes rac{1 - rac{1}{(1.05)^3}}{0.05}$ $PVA3 = 400 imes rac{1 - 0.86383759}{0.05}$
$PVA3 = 400 imes rac{0.13616241}{0.05}$ $PVA3 = 400 imes (2.723248) = 1,089.30$

PVAn Financial Calculator Solution

For a three-year ordinary annuity of $400 at 5%:

N = $3$
I/Y = $5$
PMT = $-400$
FV = $0$
PV = $1,089.30$ (Computed Present Value)

Present Value of an Annuity Due (PVA(Due)n)

Example: Calculating Present Value of a 3-year Annuity Due

Consider a three-year annuity due with $400 payments if the discount rate is 5%.

Cash Flow Breakdown and Discounting to Year 0:

Payment 1 (at beginning of Year 1 / Time 0): Value at Year 0 = $400 imes (1.05)^0 = 400.00$
Payment 2 (at beginning of Year 2 / Time 1): Discounted for 1 year. Value at Year 0 = $400 imes (1.05)^{-1} = 380.95$
Payment 3 (at beginning of Year 3 / Time 2): Discounted for 2 years. Value at Year 0 = $400 imes (1.05)^{-2} = 362.81$

Total Present Value ( $PVA(DUE)_3$ ) at Year 0 = $400.00 + 380.95 + 362.81 = 1,143.76$

Relationship with Ordinary Annuity

Annuity due present value can also be found by multiplying the ordinary annuity present value by $(1+r)$ :
$PVA(Due)n = PVAn imes (1+r)$
Using the previous example: $1,089.30 imes (1+0.05) = 1,089.30 imes 1.05 = 1,143.76$

PVA(Due) Equation Solution

The present value of an annuity due can be calculated using the formula:
$PVA(DUE)_n = PMT imes rac{1 - rac{1}{(1+r)^n}}{r} imes (1+r)$

Applying this to the example:
$PVA(DUE)3 = 400 imes rac{1 - rac{1}{(1.05)^3}}{0.05} imes (1.05)$ $PVA(DUE)3 = 400 imes (2.723248) imes (1.05) = 1,143.76$

PVA(Due) Financial Calculator Solution

For a three-year annuity due of $400 at 5%:

N = $3$
I/Y = $5$
PMT = $-400$
FV = $0$
PV = $1,143.76$
Note: Ensure the financial calculator is in [BGN] mode. If it’s not, set it by pressing [2nd] [BGN] [2nd] [SET]. This will show [BGN] on the screen.

Annuity Examples Using Financial Calculator

Example 1: Finding Future Value of an Annuity Due (FVA(Due))

Problem: If you deposit $500 at the beginning of each year for the next 5 years in a savings account paying 6% interest, how much will you have at the end of year 5?

Solution Steps:

Since deposits are made at the beginning of each year, set the financial calculator to [BGN] Mode.
Input:
- N = $5$
- I/Y = $6$
- PV = $0$
- PMT = $-500$
Compute FV $ ightarrow$ FV = $2,987.66$

Example 2: Finding Number of Periods (N)

Problem: Suppose you decide to make end-of-year deposits of $1,200 per year. Assuming you earn 6% annually, how long would it take to reach your $10,000 goal?

Solution Steps:

Since deposits are end-of-year, the calculator should be in [END] mode (default).
Input:
- I/Y = $6$
- PMT = $-1,200$
- PV = $0$
- FV = $10,000$
Compute N $ ightarrow$ N = $6.96$ years

Example 3: Finding Payment Amount (PMT) - Two-Step Problem

Problem: You will start making 35 deposits of $3,000 per year in your Individual Retirement Account (from $t=1$ to $t=35$ ). With the money accumulated at $t=35$ , you will then buy a retirement annuity of 20 years with equal yearly payments from a life insurance company (payments from $t=36$ to $t=55$ ). If the annual rate of return over the entire period is 8%, what will be the annual payment of the retirement annuity?

Solution Steps:
This problem involves two annuities.

Step 1: Find the Future Value (FV) of the 35-payment accumulation annuity (from $t=1$ to $t=35$ ) at $t=35$ .

Input:
- N = $35$
- I/Y = $8$
- PV = $0$
- PMT = $-3000$
**Output: FV = $516,950.41$
*This FV at $t=35$ becomes the PV for the next annuity stream.*

Step 2: Find the PMT of the 20-payment retirement annuity (from $t=36$ to $t=55$ ).

Input:
- N = $20$
- I/Y = $8$
- PV = $-516,950.41$ (This is the amount available at $t=35$ that needs to be paid out; entered as negative because it's conceptually an initial outlay to fund the annuity)
- FV = $0$ (The annuity is exhausted after 20 years)
**Output: PMT = $52,652.54$

Example 4: Finding Interest Rate (I)

Problem: You lend your friend $100,000. He will pay you $13,000 per year for ten years. What is the interest rate you are charging your friend?

Solution Steps:

Input:
- N = $10$
- PV = $-100,000$ (The initial loan outflow)
- PMT = $13,000$ (The annual payment received)
- FV = $0$ (The loan balance is zero at the end)
**Output: I/Y = $5.0787$

Perpetuities

Definition

A perpetuity is a special type of annuity that continues forever, meaning the payments are expected to last indefinitely.

Examples of Perpetuities

British Consol: A historical type of perpetual bond issued by the British government.
Preferred Stock: Often pays a fixed dividend indefinitely, making it a form of perpetuity.

Present Value of a Perpetuity

The present value (PV) of a perpetuity is given by the formula:
$PV_{Perpetuity} = rac{PMT}{r}$
Where:

$PMT$ = The constant dollar amount provided by the perpetuity each period.
$r$ = The annual interest (or discount) rate.
More formally, this can be represented as an infinite sum: PV{Perpetuity} = }This formula simplifies the infinite series: $

Examples: Perpetuity Calculation

Find the Annual Cash Flow (PMT)
Problem: Suppose the value of a perpetuity is $38,900 and the discount rate is 12% p.a. What must be the annual cash flow from this perpetuity?
Solution: Using the formula $PMT = PV imes r$
$PMT = 38,900 imes 0.12 = 4,668$
Find the Required Rate of Return (r)
Problem: An asset that generates $890 per year forever is priced at $6,000. What is the required rate of return?
Solution: Using the formula $r = rac{PMT}{PV}$
r = rac{890}{6,000} = 0.148333 ext{ or } 14.83 ext{%}

Uneven Cash Flows

Types of Uneven Cash Flow Problems

There are two main categories of problems involving uneven cash flows:

Annuity plus additional initial/final payment: These problems can often be solved using the standard Time Value of Money (TVM) Worksheet on a financial calculator, sometimes by separating the annuity component from the lump sum.
Irregular cash flows: These involve cash flows that do not follow a repeating pattern. For such problems, the Cash Flow (CF) Worksheet on a financial calculator is typically used.

Present Value of Uneven Cash Flows

The present value of a series of uneven cash flows is calculated by discounting each individual cash flow back to time zero and summing them up. The formula is:
$PVC FN = rac{CF1}{(1+r)^1} + rac{CF2}{(1+r)^2} + ext{…} + rac{CFN}{(1+r)^N} =$ ext{ }(1+r)^{t} $<ul><li>$ CF_t $= The cash flow at time$ t $.</li><li>$ r = The annual interest (or discount) rate.

BAII Plus Cash Flow Worksheet - Explained

For handling irregular cash flows, the BAII Plus financial calculator has a specific Cash Flow Worksheet.

Key Function Buttons:

[CF]: Used to access the Cash Flow Worksheet and to input the initial cash flow (CF0).
[C0n]: Represents the amount of the nth cash flow. You cycle through cash flows by pressing the down arrow after entering a C0n.
[F0n]: Stands for the frequency of the nth cash flow. This is used if a particular cash flow amount repeats for a number of periods consecutively.
[I]: Used within the NPV/IRR worksheet to enter the annual interest (or discount) rate.
[NPV]: Used to access the Net Present Value (NPV) Worksheet and compute NPV.
[IRR]: Used to access the Internal Rate of Return (IRR) Worksheet and compute IRR.

Clearing the Worksheet: To clear all variables in the Cash Flow Worksheet, press [CF], and then [2nd] [CLR WORK].

Example 1: Uneven Cash Flows - Calculating Net Present Value (NPV)

Problem: An asset promises the following cash flows: $5,000 at the end of each of the first three years, $7,000 at the end of each of the following four years, and $9,000 at the end of each of the following five years. If your required rate of return is 10%, how much is this asset worth to you (i.e., what is its Present Value)?

Solution Steps (using Financial Calculator):

Enter Cash Flows:
- Press [CF] to access the Cash Flow Worksheet.
- Enter CF0 = 0 (assuming no initial outlay, if it's a valuation). Then press [ENTER] and the down arrow.
- C01 = 5,000 $, F01 =$ 3 (for 3 years of $5,000). Press [ENTER] and the down arrow after each.
- C02 = 7,000 $, F02 =$ 4 (for 4 years of $7,000). Press [ENTER] and the down arrow after each.
- C03 = 9,000 $, F03 =$ 5 (for 5 years of $9,000). Press [ENTER] and the down arrow after each.
Compute NPV:
- Press [NPV].
- Enter I = 10 (for 10% interest rate). Press [ENTER].
- Press the down arrow.
- Press [CPT] (Compute) NPV
  ightarrow $**NPV =$ 46,612.68

Example 2: Uneven Cash Flows - Calculating Internal Rate of Return (IRR)

Problem: You lend your friend $100,000. He will pay you $12,000 per year for ten years and a balloon payment of $50,000 at t=10 $. What is the interest rate you are charging your friend?Solution Steps (using Financial Calculator):<ol><li>Enter Cash Flows:<ul><li>Press [CF] to access the Cash Flow Worksheet.</li><li>Enter CF0 =$ -100,000 $ (The initial loan given, an outflow). Press [ENTER] and the down arrow.</li><li>C01 =$ 12,000 $, F01 =$ 9 (for 9 years of $12,000 payments). Press [ENTER] and the down arrow after each.

C02 = 62,000 $, F02 =$ 1 (This is the 10th year payment: $12,000 annuity payment + $50,000 balloon payment). Press [ENTER] and the down arrow after each.

Compute IRR:

Press [IRR].
Press [CPT] (Compute) IRR
ightarrow $**IRR =$ 8.6543 $</li></ul></li></ol>Alternative Solution using TVM Worksheet (applicable because balloon payment simplifies the uneven cash flow to a known FV):<ol><li>Input:<ul><li>N =$ 10 $</li><li>PV =$ -100,000 $</li><li>PMT =$ 12,000 $</li><li>FV =$ 50,000 $</li></ul></li><li>**Output: I/Y =$ 8.6543 $</li></ol><h4 id="b1ee6e69-0e12-48f1-86ad-110d8a3793c0" data-toc-id="b1ee6e69-0e12-48f1-86ad-110d8a3793c0" collapsed="false" seolevelmigrated="true">Semi-Annual and Other Compounding Periods</h4><h5 id="6363ba02-5bd8-48ac-a4d1-92fd782010eb" data-toc-id="6363ba02-5bd8-48ac-a4d1-92fd782010eb" collapsed="false" seolevelmigrated="true">Compounding</h5>Compounding refers to the ability of an asset to generate earnings, which are then reinvested to generate their own earnings. It's the process of earning returns on previous returns.<ul><li>Annual Compounding: Interest is added to the principal once a year.</li><li>Semi-annual Compounding: Interest is added to the principal twice a year.</li><li>Continuous Compounding: Interest is added to the principal constantly and at every instant.</li></ul>Impact of Compounding Frequency: The future value (FV) of a lump sum will always be larger if interest is compounded more often, assuming the stated interest rate remains constant. This is because earlier reinvestment opportunities allow earnings to generate further earnings over a longer duration.<h5 id="1310088d-5f2f-4044-84b8-f3d13b13fa82" data-toc-id="1310088d-5f2f-4044-84b8-f3d13b13fa82" collapsed="false" seolevelmigrated="true">Compounding Frequency (M)</h5>Saying that the compounding period is less than one year is equivalent to saying that the frequency of compounding (M) is more than once per year.<table style="min-width: 50px;"><colgroup><col style="min-width: 25px;"><col style="min-width: 25px;"></colgroup><tbody><tr><th colspan="1" rowspan="1" style="text-align: left;">Compounding Period</th><th colspan="1" rowspan="1" style="text-align: left;">Compounding Frequency (M)</th></tr><tr><td colspan="1" rowspan="1" style="text-align: left;">Annual</td><td colspan="1" rowspan="1" style="text-align: left;">$ 1 $</td></tr><tr><td colspan="1" rowspan="1" style="text-align: left;">Semi-annual</td><td colspan="1" rowspan="1" style="text-align: left;">$ 2 $</td></tr><tr><td colspan="1" rowspan="1" style="text-align: left;">Quarter</td><td colspan="1" rowspan="1" style="text-align: left;">$ 4 $</td></tr><tr><td colspan="1" rowspan="1" style="text-align: left;">Month</td><td colspan="1" rowspan="1" style="text-align: left;">$ 12 $</td></tr><tr><td colspan="1" rowspan="1" style="text-align: left;">Day</td><td colspan="1" rowspan="1" style="text-align: left;">$ 365 $</td></tr></tbody></table><h4 id="1d6adb6d-5b9a-48d9-806f-b8d28829cf00" data-toc-id="1d6adb6d-5b9a-48d9-806f-b8d28829cf00" collapsed="false" seolevelmigrated="true">Nominal Interest Rate</h4><h5 id="2eadd6e4-6810-43c1-8cb4-ece1e102bfac" data-toc-id="2eadd6e4-6810-43c1-8cb4-ece1e102bfac" collapsed="false" seolevelmigrated="true">Definition of Nominal Interest Rate ($ r_{NOM} $)</h5>The nominal interest rate is also known as the simple rate ($ r_{SIMPLE} $), quoted rate, annual percentage rate (APR), or stated rate. It is an annual rate that ignores the effects of compounding within the year.<h5 id="820bd255-228d-45e9-8c0f-4e873ed34da7" data-toc-id="820bd255-228d-45e9-8c0f-4e873ed34da7" collapsed="false" seolevelmigrated="true">Key Characteristics</h5><ul><li>$ r_{NOM} $is typically stated in contracts.</li><li>When$ r_{NOM} $is quoted, the compounding frequency (e.g., compounded quarterly, compounded daily) must be specified to provide clarity on how interest is actually calculated.</li><li>Examples:<ul><li>8%, compounded quarterly</li><li>8%, compounded daily (365 days)</li></ul></li></ul><h4 id="3eea0868-600a-423f-896b-3922a5f3c953" data-toc-id="3eea0868-600a-423f-896b-3922a5f3c953" collapsed="false" seolevelmigrated="true">Periodic Rate ($ r_{PER} $)</h4><h5 id="8f4baaf6-a793-4478-9bb6-53823ddc6073" data-toc-id="8f4baaf6-a793-4478-9bb6-53823ddc6073" collapsed="false" seolevelmigrated="true">Definition of Periodic Rate ($ r_{PER} $)</h5>The periodic rate is the amount of interest charged (or earned) each compounding period. This could be monthly, quarterly, semi-annually, etc., depending on the stated compounding frequency.<h5 id="9b796eca-aa53-4d67-a3b0-49557049fd95" data-toc-id="9b796eca-aa53-4d67-a3b0-49557049fd95" collapsed="false" seolevelmigrated="true">Formula</h5>The periodic rate is calculated by dividing the nominal rate by the number of compounding periods per year: $ r{PER} = rac{r{NOM}}{M} $ Where$ M $is the number of compounding periods per year (e.g.,$ M=4 $for quarterly,$ M=12 $for monthly compounding).<h5 id="6b39be56-00d9-46a5-b5a0-f73277b69433" data-toc-id="6b39be56-00d9-46a5-b5a0-f73277b69433" collapsed="false" seolevelmigrated="true">Examples</h5><ul><li>For 8%, compounded quarterly:$ r_{PER} = 8 ext{%} / 4 = 2 ext{%} $per quarter.</li><li>For 8%, compounded monthly:$ r_{PER} = 8 ext{%} / 12 = 0.666… ext{ %} ext{ or } 0.667 ext{%} $per month.</li></ul><h4 id="91e190f9-61d2-48bf-bc85-4723edd0b24e" data-toc-id="91e190f9-61d2-48bf-bc85-4723edd0b24e" collapsed="false" seolevelmigrated="true">Effective Annual Rate (rEAR)</h4><h5 id="2a041d72-ec30-4900-898f-89cc2c67f8f5" data-toc-id="2a041d72-ec30-4900-898f-89cc2c67f8f5" collapsed="false" seolevelmigrated="true">Definition of Effective Annual Rate ($ r_{EAR} $)</h5>The effective annual rate, also known as the equivalent annual rate, is the actual annual rate of return earned or paid on an investment or loan, taking into account the effect of compounding over a one-year period. It is the rate that would produce the same future value under annual compounding as would more frequent compounding at a given nominal rate.<h5 id="0a95260b-973a-4d92-8227-333b220eeb6c" data-toc-id="0a95260b-973a-4d92-8227-333b220eeb6c" collapsed="false" seolevelmigrated="true">Purpose</h5>To accurately compare investments or loans with different compounding intervals, it is essential to convert their nominal rates to their effective annual rates ($ r_{EAR} $).<h5 id="92dba79c-4cec-4f78-96b6-62becacf4b78" data-toc-id="92dba79c-4cec-4f78-96b6-62becacf4b78" collapsed="false" seolevelmigrated="true">Relationship between Nominal and Effective Rate</h5>The formula to calculate the effective annual rate from a nominal rate and compounding frequency is: $ r{EAR} = (1 + rac{r{NOM}}{M})^M - 1 $ Where$ M $is the number of compounding periods per year.<h5 id="20c8f4fe-d2cf-4e32-bcb2-b4b8dba0da51" data-toc-id="20c8f4fe-d2cf-4e32-bcb2-b4b8dba0da51" collapsed="false" seolevelmigrated="true">Computing$ r_{EAR} $Example</h5>Problem: What is the effective annual return (EAR) for an investment that pays 12% interest, compounded monthly?Solution (using Formula): Here,$ r{NOM} = 0.12 $and$ M = 12r{EAR} = (1 + rac{0.12}{12})^{12} - 1 $ $ r{EAR} = (1 + 0.01)^{12} - 1r{EAR} = (1.01)^{12} - 1 $ Solution (using Financial Calculator ICONV Worksheet):<ol><li>Press [2nd] [ICONV] to access the worksheet.</li><li>Enter NOM =$ 12 $, then press [ENTER] and the down arrow.</li><li>Enter C/Y =$ 12 $ (for monthly compounding), then press [ENTER] and the down arrow.</li><li>Press [CPT] (Compute) EFF$
ightarrow $EFF =$ 12.68 $</li></ol><h5 id="b053cf99-870a-488e-8dbc-1038e03bbd27" data-toc-id="b053cf99-870a-488e-8dbc-1038e03bbd27" collapsed="false" seolevelmigrated="true">Variation of Effective Return with Compounding Intervals</h5>For a constant nominal rate (e.g., 10%), the effective return increases with more frequent compounding:<ul><li>EAR (Annual Compounding):$ 10.00 ext{%} $</li><li>EAR (Semi-annual Compounding):$ 10.25 ext{%} $</li><li>EAR (Quarterly Compounding):$ 10.38 ext{%} $</li><li>EAR (Monthly Compounding):$ 10.47 ext{%} $</li><li>EAR (Daily Compounding - 365 days):$ 10.52 ext{%} $</li></ul><h4 id="03ffb85d-daf5-45b9-b5a2-26a242b2042b" data-toc-id="03ffb85d-daf5-45b9-b5a2-26a242b2042b" collapsed="false" seolevelmigrated="true">When Is Each Rate Used?</h4>Understanding when to use each type of interest rate is crucial for accurate financial calculations.<ul><li>Nominal Interest Rate ($ r_{NOM} $):<ul><li>Used in contracts and quoted by banks/brokers.</li><li>Not generally used directly in calculations or shown on financial timelines (unless$ M=1 $).</li></ul></li><li>Periodic Rate ($ r_{PER} $):<ul><li>Always used in calculations when dealing with non-annual compounding.</li><li>Is reflected on specific compounding period timelines.</li><li>If$ M=1 $(annual compounding), then$ r{NOM} = r{PER} = r_{EAR} $.</li></ul></li><li>Effective Annual Rate ($ r_{EAR} $):<ul><li>Used to compare returns on investments or costs of loans that have different payment/compounding frequencies per year. It provides a standardized annual comparison.</li><li>Used in calculations when annuity payments do not match the compounding periods. In such cases, the$ r_{EAR} $is often used to ensure consistency over an annual period.</li></ul></li></ul><h4 id="e1ce129f-2875-4234-ba82-2482fc7cf096" data-toc-id="e1ce129f-2875-4234-ba82-2482fc7cf096" collapsed="false" seolevelmigrated="true">Finding PV and FV with Non-Annual Periods</h4>When compounding occurs more frequently than annually, the formulas for PV and FV must incorporate the periodic rate ($ r_{PER} $) and the total number of compounding periods ($ M imes N $).<h5 id="3b68a282-797b-433f-9566-06b3a2af760f" data-toc-id="3b68a282-797b-433f-9566-06b3a2af760f" collapsed="false" seolevelmigrated="true">Future Value (FV) Formula</h5>$ FV = PV imes ext{ }(1 + rac{r}{M})^{M imes N} = PV imes (1 + r_{PER})^{M imes N} $<h5 id="8b53761b-e907-41ba-b4df-6825e6918ccd" data-toc-id="8b53761b-e907-41ba-b4df-6825e6918ccd" collapsed="false" seolevelmigrated="true">Present Value (PV) Formula</h5>$ PV = rac{FV}{(1 + rac{r}{M})^{M imes N}} = rac{FV}{(1 + r_{PER})^{M imes N}} $Where:<ul><li>$ N $= number of years</li><li>$ M $= frequency of compounding per year</li><li>$ r $= nominal interest rate per annum ($ r_{NOM} $)</li><li>$ r{PER} $= periodic interest rate (i.e.,$ rac{r{NOM}}{M})

Example 1: Compounding - Future Value

Problem: What is the FV of $100 after 3 years under 10% semi-annual compounding? What about quarterly compounding?

Solution (Semi-annual compounding):

r_{NOM} = 10 ext{%} $,$ M = 2 $,$ N = 3 $years.</li><li>$ r_{PER} = 10 ext{%} / 2 = 5 ext{%} $</li><li>Total periods =$ M imes N = 2 imes 3 = 6 $</li><li>$ FV = 100 imes (1 + 0.05)^6 = 100 imes (1.3400956) = 134.01 $</li></ul>Solution (Quarterly compounding):<ul><li>$ r_{NOM} = 10 ext{%} $,$ M = 4 $,$ N = 3 $years.</li><li>$ r_{PER} = 10 ext{%} / 4 = 2.5 ext{%} $</li><li>Total periods =$ M imes N = 4 imes 3 = 12 $</li><li>$ FV = 100 imes (1 + 0.025)^{12} = 100 imes (1.3448888) = 134.49 $</li></ul>Alternative Solution (using TVM Worksheet):<ol><li>Semi-annual compounding:<ul><li>N =$ 3 imes 2 = 6 $</li><li>I/Y =$ 10 / 2 = 5 $</li><li>PV =$ -100 $</li><li>PMT =$ 0 $</li><li>Compute FV$
ightarrow $FV =$ 134.01 $</li></ul></li><li>Quarterly compounding:<ul><li>N =$ 3 imes 4 = 12 $</li><li>I/Y =$ 10 / 4 = 2.5 $</li><li>PV =$ -100 $</li><li>PMT =$ 0 $</li><li>Compute FV$
ightarrow $FV =$ 134.49

Example 2: Compounding - Present Value of an Annuity

Problem: What’s the PV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semi-annually?

Solution (by discounting individual cash flows):

The periodic rate r_{PER} = 10 ext{%} / 2 = 5 ext{%} $</li><li>The annuity payments are annual, but compounding is semi-annual.</li><li>This means each annual payment must be discounted using the periodic rate and respective total compounding periods.</li><li>Note: When the annuity payments are annual but compounding is more frequent, one common approach is to find the effective annual rate (EAR) first for the interest rate, then use that EAR with annual payments. Another approach, which is more complex and less intuitive for annual payments with sub-annual compounding is as shown in the raw calculation, explicitly using compounding periods for each annual payment: $ PV = rac{100}{(1.05)^2} + rac{100}{(1.05)^4} + rac{100}{(1.05)^6} $ $ PV = 100(1.05)^{-2} + 100(1.05)^{-4} + 100(1.05)^{-6} $ $ PV = 100(0.907029) + 100(0.822702) + 100(0.746215) $ $ PV = 90.7029 + 82.2702 + 74.6215 = 247.59 $</li></ul>Alternative Solution (using TVM Worksheet with EAR): This is generally the preferred approach when payment frequency differs from compounding frequency.<ol><li>Step 1: Find the annual effective rate (EAR) for the 10% semi-annual compounding. $ EAR = (1 + rac{0.10}{2})^2 - 1 = (1.05)^2 - 1 = 1.1025 - 1 = 0.1025 ext{ or } 10.25 ext{%} $</li><li>Step 2: Use the EAR and treat the cash flows as an ordinary annuity to solve for PV.<ul><li>N =$ 3 $</li><li>I/Y =$ 10.25 $</li><li>PMT =$ -100 $</li><li>FV =$ 0 $</li><li>Compute PV$
ightarrow $PV =$ 247.59

Example 3: Compounding - Car Loan (Find Stated and Effective Rates)

Problem: You have decided to buy a car whose price is $45,000. The dealer offers to finance the entire amount and requires 60 monthly payments of $950 per month. What are the yearly stated and effective interest rates for this financing?

Solution Steps:
This is a monthly compounding problem.

Find the periodic (monthly) rate using the TVM Worksheet.
- N = 60 $(60 monthly payments)</li><li>PV =$ 45,000 $(The loan amount received by you, so entered as positive initial inflow)</li><li>PMT =$ -950 $(The monthly payment outflow)</li><li>FV =$ 0 $(Loan is fully paid off)</li><li>Compute I/Y$
 ightarrow $I/Y =$ 0.8103 $ (This is the periodic monthly rate, in percent, i.e.,$ 0.8103 ext{%} $per month)</li></ul></li><li>Calculate the annual nominal rate ($ r{NOM} $). $ r{NOM} = ext{Periodic Rate} imes M = 0.8103 ext{%} imes 12 = 9.7236 ext{%} $</li><li>Calculate the effective annual rate ($ r{EAR} $). $ r{EAR} = (1 + ext{Periodic Rate})^{M} - 1 = (1 + 0.008103)^{12} - 1 $ $ r_{EAR} = (1.008103)^{12} - 1 = 1.10168 - 1 = 0.10168 ext{ or } 10.168 ext{%} $</li></ol><h5 id="1b922ac1-b126-4b0e-bc6f-b9cf82de8567" data-toc-id="1b922ac1-b126-4b0e-bc6f-b9cf82de8567" collapsed="false" seolevelmigrated="true">Example 4: Compounding - Doubling Account Balance (Find N in Years)</h5>Problem: The stated interest rate for a bank account is 7% and interest is paid semi-annually. How many years will it take to double the account balance?Solution Steps: Assume an initial deposit of $1 dollar today. To double the balance, the future value must be $2.<ol><li>Determine TVM inputs for semi-annual compounding.<ul><li>I/Y =$ 7 / 2 = 3.5 $(Periodic semi-annual rate)</li><li>PV =$ -1 $(Initial deposit, outflow)</li><li>PMT =$ 0 $</li><li>FV =$ 2 $(Target future value)</li></ul></li><li>Compute N$
 ightarrow $N =$ 20.15 $ (This is the number of semi-annual compounding periods.)</li><li>Convert compounding periods to years. Since$ N $represents semi-annual periods, divide by 2 to get years: $ Years = 20.15 / 2 = 10.08$$ years.
 Therefore, it takes approximately 10.08 years to double the account balance.