# 8.3: Annuities - Mathematics

For most of us, we aren’t able to put a large sum of money in the bank today. This idea is called a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings annuities.

An annuity can be described recursively in a fairly simple way. Recall that basic compound interest follows from the relationship

(P_{m}=left(1+frac{r}{k} ight) P_{m-1})

For a savings annuity, we simply need to add a deposit, (d), to the account with each compounding period:

P_{m}=left(1+frac{r}{k} ight) P_{m-1}+d

Taking this equation from recursive form to explicit form is a bit trickier than with compound interest. It will be easiest to see by working with an example rather than working in general.

Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. In this example: (r = 0.06) (6%) (k = 12) (12 compounds/deposits per year) (d =$100) (our deposit per month)

Writing out the recursive equation gives

(P_{m}=left(1+frac{0.06}{12} ight) P_{m-1}+100=(1.005) P_{m-1}+100)

Assuming we start with an empty account, we can begin using this relationship:

(P_{0}=0)

(P_{1}=(1.005) P_{0}+100=100)

(P_{2}=(1.005) P_{1}+100=(1.005)(100)+100=100(1.005)+100)

(P_{3}=(1.005) P_{2}+100=(1.005)(100(1.005)+100)+100=100(1.005)^{2}+100(1.005)+100)

Continuing this pattern, after (m) deposits, we’d have saved:

(P_{m}=100(1.005)^{m-1}+100(1.005)^{m-2}+cdots+100(1.005)+100)

In other words, after (m) months, the first deposit will have earned compound interest for (m-1) months. The second deposit will have earned interest for (m­-2) months. Last months deposit would have earned only one month worth of interest. The most recent deposit will have earned no interest yet.

This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any easier! To simplify things, multiply both sides of the equation by 1.005:

(1.005 P_{m}=1.005left(100(1.005)^{m-1}+100(1.005)^{m-2}+cdots+100(1.005)+100 ight))

Distributing on the right side of the equation gives

(1.005 P_{m}=100(1.005)^{m}+100(1.005)^{m-1}+cdots+100(1.005)^{2}+100(1.005))

Now we’ll line this up with like terms from our original equation, and subtract each side

Almost all the terms cancel on the right hand side when we subtract, leaving

(1.005 P_{m}-P_{m}=100(1.005)^{m}-100)

Solving for (P_m)

(0.005 P_{m}=100left((1.005)^{m}-1 ight))

(P_{m}=frac{100left((1.005)^{m}-1 ight)}{0.005})

Replacing (m) months with (12N), where (N) is measured in years, gives

(P_{N}=frac{100left((1.005)^{12 mathrm{V}}-1 ight)}{0.005})

Recall 0.005 was (frac{r}{k}) and 100 was the deposit (d). 12 was (k), the number of deposit each year. Generalizing this result, we get the saving annuity formula.

Annuity Formula

(P_{N}=frac{dleft(left(1+frac{r}{k} ight)^{N k}-1 ight)}{left(frac{r}{k} ight)})

(P_N) is the balance in the account after N years.

(d) is the regular deposit (the amount you deposit each year, each month, etc.)

(r) is the annual interest rate in decimal form.

(k) is the number of compounding periods in one year.

If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.

For example, if the compounding frequency isn’t stated:

If you make your deposits every month, use monthly compounding, (k=12).

If you make your deposits every year, use yearly compounding, (k=1).

If you make your deposits every quarter, use quarterly compounding, (k=4).

Etc.

When do you use this

Annuities assume that you put money in the account on a regular schedule (every month, year, quarter, etc.) and let it sit there earning interest.

Compound interest assumes that you put money in the account once and let it sit there earning interest.

Compound interest: One deposit

Annuity: Many deposits.

Example 7

A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years? Solution In this example, (egin{array}{ll} d =$100 & ext{the monthly deposit} r = 0.06 & 6\% ext{ annual rate} k = 12 & ext{since we’re doing monthly deposits, we’ll compound monthly} N = 20 & ext{we want the amount after 20 years} end{array})

Putting this into the equation:

(P_{20}=frac{100left(left(1+frac{0.06}{12} ight)^{20(12)}-1 ight)}{left(frac{0.06}{12} ight)})

(P_{20}=frac{100left((1.005)^{240}-1 ight)}{(0.005)})

(P_{20}=frac{100(3.310-1)}{(0.005)})

(P_{20}=frac{100(2.310)}{(0.005)}=$46200) The account will grow to$46,200 after 20 years.

Notice that you deposited into the account a total of $24,000 ($100 a month for 240 months). The difference between what you end up with and how much you put in is the interest earned. In this case it is ($46,200 -$24,000 = $22,200). Example 8 You want to have$200,000 in your account when you retire in 30 years. Your retirement account earns 8% interest. How much do you need to deposit each month to meet your retirement goal?

Solution

In this example,

We’re looking for (d).

(egin{array}{ll} r = 0.08 & 8\% ext{ annual rate} k = 12 & ext{since we’re doing monthly deposits, we’ll compound monthly} N = 30 & ext{30 years} P_{30}=$200,000 & ext{The amount we want to have in 30 years} end{array}) In this case, we’re going to have to set up the equation, and solve for (d). 200,000=frac{dleft(left(1+frac{0.08}{12} ight)^{30(12)}-1 ight)}{left(frac{0.08}{12} ight)} 200,000=frac{dleft((1.00667)^{360}-1 ight)}{(0.00667)} 200,000=d(1491.57) d=frac{200,000}{1491.57}=$ 134.09

So you would need to deposit ($134.09) each month to have ($200,000) in 30 years if your account earns 8% interest

Try it Now 2

A more conservative investment account pays 3% interest. If you deposit $5 a day into this account, how much will you have after 10 years? How much is from interest? Answer (egin{array}{ll} d =$5 & ext{the daily deposit} r = 0.03 & 3\% ext{ annual rate} k = 365 & ext{since we’re doing daily deposits, we’ll compound daily} N = 10 & ext{we want the amount after 10 years} end{array})

(P_{10}=frac{5left(left(1+frac{0.03}{365} ight)^{365 imes 10}-1 ight)}{frac{0.03}{365}}=$21,282.07) We would have deposited a total of ($ 5 cdot 365 cdot 10=$18,250), so$3,032.07 is from interest

## Why Sky-High Annuity Rates Can Be Misleading

Insurance companies issue annuities, and you have to be careful about the claims they make. Image source: Getty Images.

Interest rates have been extremely low for a long time, and investors who need income are taking desperate measures to find it. Knowing that, some insurance companies have started to market annuities by offering annuity rates that are far higher than bank CDs and bond funds offer. Yet some of these high annuity rates come with a huge catch, and if you don't understand it, you could get caught unaware of the price you're paying to earn those rates. Below, we'll take a look at the tricky market strategies that some insurance companies use and why they're exceptionally misleading.

## Present value annuities

For present value annuities, regular equal payments/installments are made to pay back a loan or bond over a given time period. The reducing balance of the loan is usually charged compound interest at a certain rate. In this section we learn how to determine the present value of a series of payments.

Consider the following example:

Kate needs to withdraw ( ext, ext<1 000>) from her bank account every year for the next three years. How much must she deposit into her account, which earns ( ext<10>\%) per annum, to be able to make these withdrawals in the future? We will assume that these are the only withdrawals and that there are no bank charges on her account.

To calculate Kate's deposit, we make (P) the subject of the compound interest formula:

We determine how much Kate must deposit for the first withdrawal:

We repeat this calculation to determine how much must be deposited for the second and third withdrawals:

Notice that for each year's withdrawal, the deposit required gets smaller and smaller because it will be in the bank account for longer and therefore earn more interest. Therefore, the total amount is:

We can check these calculations by determining the accumulated amount in Kate's bank account after each withdrawal:

## Mathematical Interest Theory: Second Edition

Mathematical Interest Theory gives an introduction of how investments grow over time. This is done in a mathematically precise manner. The emphasis is on practical applications that give the reader a concrete understanding of why the various relationships should be true. Among the modern financial topics introduced are: arbitrage, options, futures, and swaps. The content of the book, along with an understanding of probability, will provide a solid foundation for readers embarking on actuarial careers. On the other hand, Mathematical Interest Theory is written for anyone who has a strong high-school algebra background and is interested in being an informed borrower or investor. The content is suitable for a mid-level or upper-level undergraduate course or a beginning graduate course.

Mathematical Interest Theory includes more than 240 carefully worked examples. There are over 430 problems, and numerical answers are included in an appendix. A companion student solution manual (see TEXT/15) has detailed solutions to the odd-numbered problems. Most of the examples involve computation, and detailed instruction is provided on how the Texas Instruments BA II Plus and BA II Plus Professional calculators can be used to efficiently solve the problems. This is important for readers wishing to pass the SOA/CAS joint financial mathematics exam FM/2. However, this part of the book can be skipped without disturbing the flow of the exposition.

Most questions from this textbook are available in WebAssign . WebAssign is a leading provider of online instructional tools for both faculty and students.

#### Reviews & Endorsements

This is an excellent book on interest theory one of the four books recognized by the Society of Actuaries and the Casualty Actuarial Society as a basis of study for the interest theory component of their joint Financial Mathematics (FM) exam. What I particularly like about 'Mathematical Interest Theory' is that many problems are intrinsically multi-stepped requiring use of several core functions. By providing a multitude of superior problems, the authors are able to familiarize the student, not only with core actuarial function, but also develop their skills in studying the interaction between these functions and real-world problems.

-- Russell Jay Hendel, UMAP Journal

Students pursuing an actuarial career as well as those seeking a mathematically based finances course stand to benefit from this informative, up-to-date, and above all, skillfully written treatise. Instructors and students of interest theory owe Daniel and Vaaler a debt of gratitude for their fine efforts.

-- Susan Staples, Texas Christian University

We use the Vaaler and Daniel text as our primary learning resource for actuarial science majors as they prepare for the SOA FM/CAS2 financial mathematics exam. Each concept is followed by several illustrative and detailed examples that help students master the big ideas in interest theory.

## Annuities

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I think your argument works. As I note above, the main issue I would have with the statement itself, if we do not assume $G$ is abelian, is that the function $x longmapsto iota_1pi_1(x)iota_2pi_2(x) ag<1>$ is not known to be a group homomorphism. If $A$ is abelian, and $f,gcolon G o A$ are group morphisms, then the pointwise product/sum of $f$ and $g$ is a homomorphism, since $egin (f+g)(xy) &= f(xy)+g(xy) &= f(x)+f(y)+g(x)+g(y) &= f(x)+g(x)+f(y)+g(y) &= (f+g)(x) + (f+g)(y). end$ But if the target is not abelian, and the image of $f$ and $g$ are not known to commute element-wise, then we don't know whether $f+g$ is a homomorphism. So we would have to interpret the final assertion as a statement that the map of underlying sets given in $(1)$ equals the identity map, emphasizing that we are not asserting, a priori, that the map is a homomorphism. You don't use the fact that this map is a homomorphism, so you are fine.

We can avoid this issue by replacing the assertion that the map $(1)$ is equal to the identity map with the assertion that $ker(pi_1)capker(pi_2)=$ , which makes sense whether $G$ is abelian or not. This eqality follows from the assertion that the map $(1)$ is equal to the identity, since if $xinker(pi_1)capker(pi_2)$ , then $(1)$ imlies that $x=e$ .

That said, you are correct that the conclusion also holds for nonabelian groups, even with the weaker assertion about the kernels.

From the fact that we have $pi_iiota_i$ be the identity we know that $pi_i$ is surjective and $iota_i$ is injecive. We also know that we can express $G$ as semidirect products, $G=ker(pi_1) times iota_1(H)$ , and $G= ker(pi_2) timesiota_2(K)$ . The semidirect products are direct products if and only if $iota_1(H)$ and $iota_2(K)$ are also normal.

From the fact that $pi_2iota_1$ and $pi_1iota_2$ are the trivial maps we know that $iota_1(H)subseteq ker(pi_2)$ and $iota_2(K)subseteq ker(pi_1)$ . So it suffices to show that $iota_2(K)=ker(pi_1)$ (the argument is clearly symmetric in $H$ and $K$ ).

Let $xinker(pi_1)$ . Then $x(iota_2pi_2(x))^<-1>in ker(pi_1)$ , since $iota_2(K)subseteqker(pi_1)$ . Applying $pi_2$ to this product, we get $egin pi_2left( xigl(iota_2pi_2(x)igr)^<-1> ight) &= pi_2(x)left(pi_2iota_2pi_2(x) ight)^<-1> &= pi_2(x) left(pi_2(x) ight)^ <-1> &= e, end$ with the next-to-last equality because $pi_2iota_2$ is the identity map. Thus, $x(iota_2pi_2(x))^<-1>inker(pi_1)capker(pi_2)$ , so it is the identity element, and hence $x = iota_2pi_2(x)$ , proving that $ker(pi_1)subseteq iota_2(K)$ , giving equality.

Since $iota_2(K)$ and $ker(pi_1)$ are both normal, and $G= ker(pi_1) times iota_2(K)$ , it follows that $G= ker(pi_1) imesiota_2(K)$ . And from a symmetric argument we also get that $ker(pi_2)=iota_1(H)$ , yielding $G=iota_1(H) imesiota_2(K)$ . Since the $iota_j$ are embeddings, this gives $Gcong H imes K$ , as desired.

Annuities can be either fixed or variable. Each type has its pros and cons.

### Fixed Annuities

With a fixed annuity, the insurance company guarantees the buyer a specific payment at some future date—which might be decades in the future or, in the case of an immediate annuity, right away. In order to deliver that return, the insurer invests money in safe vehicles, such as U.S. Treasury securities and highly rated corporate bonds.

While safe and predictable, these investments also deliver unspectacular returns. ﻿ ﻿ What's more, the payouts on fixed annuities can lose purchasing power over the years due to inflation, unless the buyer pays extra for an annuity that takes inflation into account. Even so, fixed annuities can be a good fit for people who have a low tolerance for risk and don't want to take chances with their regular monthly payouts.

### Variable Annuities

With a variable annuity, the insurer invests in a portfolio of mutual funds chosen by the buyer. The performance of those funds will determine how the account grows and how large a payout the buyer will eventually receive. Variable annuity payouts can either be fixed or vary along with the account's performance.

People who choose variable annuities are willing to take on some degree of risk in the hope of generating bigger profits. Variable annuities are generally best for experienced investors, who are familiar with the different types of mutual funds and the risks they involve. ﻿ ﻿

If an annuity buyer is married, they can choose an annuity that will continue to pay income to their spouse should they die first.

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Part I THE DETERMINISTIC LIFE CONTINGENCIES MODEL 1

1 Introduction and motivation 3

1.2 Deterministic versus stochastic models 4

1.3 Finance and investments 5

2 The basic deterministic model 7

2.2 An analogy with currencies 8

2.4 Calculating the discount function 11

2.5 Interest and discount rates 12

2.7 Values and actuarial equivalence 13

2.9 Regular pattern cash flows 18

2.10 Balances and reserves 20

2.11 Time shifting and the splitting identity 26

2.11 Change of discount function 27

2.12 Internal rates of return 28

2.13 Forward prices and term structure 30

2.14 Standard notation and terminology 33

3.3 Constructing the life table from the values of qx 41

3.5 Choice of life tables 44

3.6 Standard notation and terminology 44

4.3 The interest and survivorship discount function 50

4.4 Guaranteed payments 53

4.5 Deferred annuities with annual premiums 55

4.6 Some practical considerations 56

4.7 Standard notation and terminology 57

5.2 Calculating life insurance premiums 61

5.3 Types of life insurance 64

5.4 Combined insurance&ndashannuity benefits 64

5.5 Insurances viewed as annuities 69

5.6 Summary of formulas 70

5.7 A general insurance&ndashannuity identity 70

5.8 Standard notation and terminology 72

6 Insurance and annuity reserves 78

6.1 Introduction to reserves 78

6.2 The general pattern of reserves 81

6.4 Detailed analysis of an insurance or annuity contract 83

6.6 Nonforfeiture values 88

6.7 Policies involving a return of the reserve 88

6.8 Premium difference and paid-up formulas 90

6.9 Standard notation and terminology 91

7 Fractional durations 98

7.2 Cash flows discounted with interest only 99

7.4 Immediate annuities 104

7.5 Approximation and computation 105

7.6 Fractional period premiums and reserves 106

7.7 Reserves at fractional durations 107

7.8 Standard notation and terminology 109

8 Continuous payments 112

8.1 Introduction to continuous annuities 112

8.2 The force of discount 113

8.3 The constant interest case 114

8.4 Continuous life annuities 115

8.5 The force of mortality 118

8.6 Insurances payable at the moment of death 119

8.8 The general insurance&ndashannuity identity in the continuous case 123

8.9 Differential equations for reserves 124

8.10 Some examples of exact calculation 125

8.11 Further approximations from the life table 129

8.12 Standard actuarial notation and terminology 131

9 Select mortality 137

9.2 Select and ultimate tables 138

9.3 Changes in formulas 139

9.4 Projections in annuity tables 141

10 Multiple-life contracts 144

10.2 The joint-life status 144

10.3 Joint-life annuities and insurances 146

10.4 Last-survivor annuities and insurances 147

10.5 Moment of death insurances 149

10.6 The general two-life annuity contract 150

10.7 The general two-life insurance contract 152

10.8 Contingent insurances 153

10.9 Duration problems 156

10.10 Applications to annuity credit risk 159

10.11 Standard notation and terminology 160

11 Multiple-decrement theory 166

11.4 Determining the model from the forces of decrement 170

11.5 The analogy with joint-life statuses 171

11.6 A machine analogy 171

11.7 Associated single-decrement tables 175

12 Expenses and Profits 184

12.2 Effect on reserves 186

12.3 Realistic reserve and balance calculations 187

12.4 Profit measurement 189

13 Specialized topics 199

13.2 Variable annuities 203

Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL 209

14 Survival distributions and failure times 211

14.1 Introduction to survival distributions 211

14.2 The discrete case 212

14.3 The continuous case 213

14.5 Shifted distributions 216

14.6 The standard approximation 217

14.7 The stochastic life table 219

14.8 Life expectancy in the stochastic model 220

14.9 Stochastic interest rates 221

15 The stochastic approach to insurance and annuities 224

15.2 The stochastic approach to insurance benefits 225

15.3 The stochastic approach to annuity benefits 229

15.4 Deferred contracts 233

15.5 The stochastic approach to reserves 233

15.6 The stochastic approach to premiums 235

15.7 The variance of rL 241

15.8 Standard notation and terminology 243

16 Simplifications under level benefit contracts 248

16.2 Variance calculations in the continuous case 248

16.3 Variance calculations in the discrete case 250

16.4 Exact distributions 252

16.5 Some non-level benefit examples 254

17 The minimum failure time 259

17.2 Joint distributions 259

17.3 The distribution of T 261

17.4 The joint distribution of (T,J) 261

17.6 The common shock model 271

Part III ADVANCED STOCHASTIC MODELS 279

18 An introduction to stochastic processes 281

18.4 Finite-state Markov chains 287

18.5 Introduction to continuous time processes 293

18.6 Poisson processes 293

19 Multi-state models 304

19.2 The discrete-time model 305

19.3 The continuous-time model 311

19.4 Recursion and differential equations for multi-state reserves 324

19.5 Profit testing in multi-state models 327

19.6 Semi-Markov models 328

20 Introduction to the Mathematics of Financial Markets 333

20.2 Modelling prices in financial markets 333

20.5 Option prices in the one-period binomial model 339

20.6 The multi-period binomial model 342

20.8 A general financial market 348

20.9 Arbitrage-free condition 351

20.10 Existence and uniqueness of risk neutral measures 353

20.11 Completeness of markets 358

20.12 The Black&ndashScholes&ndashMerton formula 361

Part IV RISK THEORY 375

21 Compound distributions 377

21.2 The mean and variance of S 379

21.3 Generating functions 380

21.4 Exact distribution of S 381

21.5 Choosing a frequency distribution 381

21.6 Choosing a severity distribution 383

21.7 Handling the point mass at 0 384

21.8 Counting claims of a particular type 385

21.9 The sum of two compound Poisson distributions 387

21.10 Deductibles and other modifications 388

21.11 A recursion formula for S 393

22 Risk assessment 403

22.3 Convex and concave functions: Jensen&rsquos inequality 406

22.4 A general comparison method 408

22.5 Risk measures for capital adequacy 412

23.2 A functional equation approach 422

23.3 The martingale approach to ruin theory 424

23.4 Distribution of the deficit at ruin 433

23.5 Recursion formulas 434

23.6 The compound Poisson surplus process 438

23.7 The maximal aggregate loss 441

24 Credibility theory 449

24.1 Introductory material 449

24.2 Conditional expectation and variance with respect to another random variable 453

## What Are the Cons of Annuities?

Nothing in the financial sphere is immune to disadvantages, and annuities are no exception. For example, the fees charged in conjunction with some annuities can be rather overbearing. In addition, the safety of an annuity is enticing, but their returns can sometimes be weaker than what you might earn through traditional investing.

### Variable Annuities Can Be Pricey

Variable annuities can get very expensive. Any time you consider one, you need to understand all the fees that come with it to be sure that you pick the best option for your goals and situation.

Variable annuities have administrative fees, as well as mortality and expense risk fees. Insurance companies charge these, which often run about 1-1.25% of your account’s value, to cover the costs and risks of insuring your money. Investment fees and expense ratios vary depending on how you invest with a variable annuity. These fees are similar to what you would pay if you invested independently in any mutual fund.

Fixed and indexed annuities, on the other hand, are actually fairly cheap. Many of these contracts don’t come with any annual fees and have limited other expenses. But in an effort to let you customize your contract, companies will often offer additional benefit riders for these. Riders come with an additional fee, but they are completely optional. Rider fees typically vary up to 1% of your contract value annually, and variable annuities may offer them too.

Surrender charges are common for both variable and fixed annuities. A surrender charge applies when you make more in withdrawals than you’re allowed to. Insurance companies usually limit withdrawal fees during the early years of your contract. Surrender fees are often high and can also apply for an extended period of time, so beware of these.

### Returns of an Annuity Might Not Match Investment Returns

The stock market will make gains in a good year. That could mean more money for your investments. At the same time, your investments will not grow by the same amount that the stock market grew. One reason for that difference in growth is annuity fees.

Let’s say you invest in an indexed annuity. With an indexed annuity, the insurance company will invest your money to mirror a specific index fund. But your insurer will likely cap your gains through something called a “participation rate.” If you have a participation rate of 80%, then your investments will only grow by 80% of the amount that the index fund grew. You could still make great gains if the index fund performs well, but you could also be missing out on returns.

If your goal is to invest in the stock market, then you should consider investing in an index fund on your own. That might seem daunting if you don’t have investing experience, so consider using a robo-advisor. A robo-advisor will manage your investments with much lower fees than an annuity.

Another thing to keep in mind is that you will likely pay lower taxes if you invest on your own. Contributions to a variable annuity are tax-deferred, but any withdrawals you make will be taxed at your regular income tax rate, not the long-term capital gains tax rate. The capital gains tax rates are lower than the income tax rates in many places. So you’re more likely to save on taxes if you invest your after-tax dollars instead of investing in an annuity.

### Getting Out of an Annuity May Be Difficult or Impossible

This is a major concern relating to immediate annuities. Once you contribute the money to fund an immediate annuity, you cannot get it back or even pass it on to a beneficiary. It may be possible for you to move your money into another annuity plan, but doing so could also leave you subject to fees.

On top of the fact that you can’t get your money back, your benefits will disappear when you die. You cannot pass that money to a beneficiary, even if you have a lot of funds left when you die.

The reason life insurance agents pump annuities up to their clients is simple. The commissions from an annuity sale are massive. Furthermore, the money salespeople receive for selling this product does not appear out of thin air. Hidden costs to you, the investor, ultimately provide for the agent’s commission check.

In fact, one of the reasons annuities have such exorbitant early withdrawal charges is that the money helps enable commissions on this product to remain so high.