Section 6: Savings

Concepts you’ll learn1. Compounding interest (review)2. The time value of money (review)3. Inflation – “real” vs. “core”4. Savings accounts5. Certificates of Deposit (CDs)6. Money markets

Problems you’ll solve– Calculate future value of money saved over time– Calculate future value of money saved over time, adjusted for

inflation– Compare and contrast various savings instruments

©2014 D. M. Kaufman. All rights reserved

Savings – Definitions• Savings: Portion of disposable income not spent on consumption of consumer goods or services, but accumulated or invested –

either through paying down debt, depositing money in a savings account, or purchasing securities.• Cost Basis: The original amount of savings or other investment, so called because of the “cost” associated with the initial

savings account deposit or purchase of a security. (This concept is something of a counterpart to “principal” associated with debt.)

• Securities: Financing or investment instruments (some negotiable, others not) bought and sold in financial markets. Savings accounts are not really securities, but other savings instruments, such as Certificates of Deposit (CDs) and money markets, are. Securities also include investment vehicles such as stocks, bonds, and annuities (covered in a later section).

• Interest Rate: A sum paid for the use of money by the entity receiving funds as a savings deposit. This is very similar to the interest paid on debt, the critical differences being that 1) the flow of the money is reversed, since the “owner” of the money is reversed, and 2) despite the reversal of money, the entity in control of the terms remains the same. For example, if you deposit money with a bank, the bank will determine the interest rate it’s willing to pay for the use of your money - -you don’t get to dictate terms to them.

• Savings Account: Bank or other depository institution account 1) from which withdrawals can be made, 2) interest accrues on the account balance, 3) does not have any maturity date, and 4) usually does not require a minimum balance.

• Certificate of Deposit (CD): Short or medium term, interest bearing, FDIC insured security offered by banks. CDs offer interest in exchange for tying up invested money for the duration of the CDs maturity – usually anywhere from 3 months to 6 years. Money withdrawn prior to maturity is subject to penalties and fees.

• Money Market: Market for short-term debt securities, such as banker’s acceptances, commercial paper, Treasury Bills, etc., with a maturity of 1 year or less – frequently 1 month or less. It’s not necessary for you to understand all the underlying securities, just understand that their short-term maturities ensure that money market accounts remain very liquid and very low risk.

• Liquidity: The ease with which an asset can be converted into cash quickly and with minimal impact to the price received. Cash is the definition of liquid, and checking accounts, savings accounts, and money markets are all highly liquid as well. CDs are not as liquid. Least liquid are assets like real estate, private businesses, etc.

• Yield Curve: A curve depicting prevalent relationship between interest rates (or yields) and maturity dates for a given class of securities – e.g. CDs and bonds with identical safety ratings. Typically, the further our the maturity date, the higher the yield, because the purchaser of the security is agreeing to tie up their money for a longer period of time. Occasionally, though, yield curves will become inverted, meaning that longer term maturities are yielding lower than short term maturities. I won’t bother you with the macroeconomic reasons for this, but it’s useful for you to know that inverted yield curves often precede recessions, stock market crashes, etc., so watch out for them.

• Federal Deposit Insurance Corporation (FDIC): The U.S. corporation insuring deposits in the U.S. against bank failure. The FDIC was created in 1933 to maintain public confidence and encourage stability in the financial system through the promotion of sound banking practices. The FDIC will insure deposits up to $100,000 as long as the bank is a member firm.

• Inflation: The rate at which the general level of prices for goods and services is rising, and, consequently, purchasing power is falling.

• Consumer Price Index (CPI): A measure that examines the weighted average of prices of a basket of consumer goods and services, such as transportation, food and medical care. The CPI is calculated by taking price changes for each item in the predetermined basket of goods and averaging them; the goods are weighted according to their importance. Changes in CPI are used to assess price changes associated with the cost of living. The CPI is sometimes referred to as “headline inflation.” In other words, it’s considered a good indicator of “real” inflation levels.

• Core Inflation: A measure of inflation that typically starts with CPI and then excludes certain items which face volatile price movements. Core inflation eliminates products that can have temporary price shocks because these shocks can diverge from the overall trend of inflation and give a false measure of inflation. The items included in core inflation are determined by the federal government and are subject to change. The government typically uses core inflation to determine cost-of-living adjustments (COLA) to pension and disability payments to government employees and beneficiaries.

Have You Noticed Yet...?

“Shift” Key: Press this first to activate any of the orange calculations on the function keys

“Clear” and “Clear All” – you’ll need to hit Shift then Clear All to dump your calculator’s memory in many

instances. I do it all the time just out of habit.

Shift Functions

xP/YR: Number of periods per year

NOM%: Nominal Interest Rate

EFF%: Effective Interest Rate

P/YR: Payments per year

AMORT: Amortize

Regular Functions

...Some of the definitions on the previous page sound an awful lot like the Debt section of the class. In fact, savings is basically the reverse

of debt, and the TVM functions are exactly the same (except you’ll seldom need to analyze any amortized payments).

(not really used for savings problems)

Savings Example #1: Simple Future Values

• So, let’s say you put $5,000 into a savings account yielding 2%, compounded monthly. What’ll you have in 5 years?– As always with TVM problems, if you know 4 variables you can always solve

for the 5th. In this case:• PV = 5,000• I/YR = 2• PMT = 0• Life = 5 shift N (or 60 periods)• Solve for the FV. You should get $5,525.39. Good times.

• Now, what if, instead of the savings account, you find a money market that pays 2.5%? What’ll you have after 5 years then? (You should get $5,665.01)

• Finally, let’s say you are willing to lock your money up for 5 years without the option to withdraw any. You find a CD with a 5 year maturity yielding 6.5%. What’ll you have after 5 years? (You should get $6,914.09)

See the trade-off? The CD requires you to sacrifice liquidity for a higher yield.

Savings Example #2: Regular Contributions

• Now, let’s say you start with $5,000 in a money market yielding 3%, but you now add $100 to the account every month. How much will you have in 5 years?– PV = 5,000– I/YR = 3– PMT = 100 (this time the payment isn’t negative – it’s still your

money, so it’s the same sign as the PV)– Life = 5 shift N (or 60 periods)– Solve for the FV. You should get -$12,272.76. Now, the number is

negative, but conceptually it’s really not. It’s the money that your money market account would “owe” you if you had made a loan to it. So, it’s yours. If it makes you feel better, you can input the PV and the PMT as negative, since it’s flowing from your pocket to your account, and then you’ll get a positive FV. But... it’s not really worth the trouble.

This is kind of like solving amortized loan problems, no?

Savings Example #3: Inflation Adjustments

• I’ll admit it right up front: you’re going to hate this.

Remember what inflation is. It’s the percentage by which the purchasing power of a dollar (or other unit of money) declines in a given year)

Savings Example #3: Inflation Adjustments

• So, simple example: You have $1,000 stuffed under your mattress. The current inflation rate is 3.5%. After one year, how much is your $1,000 worth in “today’s dollars,” or adjusted for inflation?– (“Today’s dollars” is a reference to purchasing power. That is, if you have

$1,000 stuffed in your mattress for a year, at the end of the year you’ll still have $1,000, but the question becomes: what can you afford to buy with that $1,000?

• Can you buy just as much as you could buy with $1,000 at the beginning of the year (which suggests zero inflation)? If so, it’s still $1,000 in “today’s dollars.”

• Can you buy more (i.e. deflation, like in the “1930 – 1939” decade in the chart on the previous page)? If so, the $1,000 is now worth more money in “today’s dollars.”

• Can you buy less? This is usually the case, since inflation is usually positive. So, a year from now the $1,000 will be worth less in “today’s dollars.”

• You hear this kind of stuff all the time. “Today’s dollars” and “adjusted for inflation” are necessary distinctions when discussing historical records like box office records, salary growth, etc.

– Okay, so, in the question above we’re basically saying that the $1,000 is worth 3.5% less at the end of the year. The calculation is 1,000*(1 - .035) = $965. So you can buy $35 less of goods and services than you could at the beginning of the year.

Savings Example #3: Inflation Adjustments

• Less simple example: What if you keep the $1,000 stuffed in your mattress for 5 years?– Beware! Inflation figures are typically given on an annual basis – there is

no monthly compounding.– You know what this means, right? To solve problems spanning multiple

years, you need to tell your calculator to compound the effects of inflation annually.

– Remember how to do this? On your calculator, press 1 shift P/YR (the “PMT” button). This tells your calculator that there is only 1 compounding period per year.

– Now, input the following:• FV = 1,000 (remember this is the amount of money you’ll have in the future, so

we’re talking about the FV here)• I/YR = 3.5• PMT = 0• Life = 5 shift N (or 5 periods)• Solve for the PV (remember you’re interested in the present value of the $1,000).

You should get -$841.97. Again, this is still your money (negative number notwithstanding), but it’s now worth quite a bit less in “today’s dollars.”

Notice how inflation erodes the value of your money over longer periods of time. This can kill your savings plans over long term horizons if you’re not careful.

Savings Example #4: Putting It All Together

1. You put $2,500 into a savings account at 1.5% interest. You contribute $100 each month into the same savings account. Inflation is at 3% and expected to remain there. What will your money be worth in today’s dollars at the end of the 10 years?

– Make sure your calculator is set up for 12 periods per year, then:• PV = 2,500• I/YR = 1.5• PMT = 100• Life = 10 shift N (or 120 periods)• Solve for FV. You should get -$15,842.35.

– Now, you need to adjust for inflation. Set up your calculator for 1 period per year, then:• FV = $15,842.35• I/YR = 3• PMT = 0• Life = 10 shift N (or 10 periods)• Solve for PV. You should get -$11,788.20.

– Here’s the kicker. Did your money increase or decrease in value?• Well, you put in $2,500 up front, plus another $100 each month for 120 months, or another $12,000.

2,500+12,000=$14,500 that you’ve put in, but your money is only worth $11,788.20 adjusted for inflation. You got killed.

2. You buy a $25,000 CD with a 5 year maturity, yielding 7.25%. Inflation is expected to average 4% over the next five years. What will your money be worth in today’s dollars at the end of the 5 years?

– @ 12 P/YR• PV = 25,000• I/YR = 7.25• PMT = 0• Life = 5 shift N• Solve for FV. You should get -$35,883.77.

– Now, @ 1 P/YR• FV = 35,883.77• I/YR = 4• PMT = 0• Life = 5 shift N• Solve for PV. You should get -$29,493.84.

– So did your purchasing power grow? Yes, but inflation still put the brakes on a bit.

Savings Example #5: Saving For a Goal

1. You’re a senior in high school, and you want to buy yourself a new car when you graduate from college (in five years). You have $10,000 today, and the car you want to buy costs $25,000 today. Inflation is running at 3.5% and is expected to remain there for the next five years. The following investment vehicles are available to you: 1) a 5 year CD yielding 6%, 2) a savings account yielding 1.5%, and 3) a money market account yielding 3%. How much money will you need to save to pay cash for this car in five years, and how will you save it?• To find out how much you need, solve for the FV of the car given the inflation rate.

• PV = $25,000• I/YR = 3.5• PMT = 0• Life = 5 shift N• Press FV. You should get -$29,692.16 (remember that inflation is an annual rate – your calculator should be set up for 1 P/YR).

This is what you’ll have to pay out to buy the car with cash in 5 years.• Now, pick your investments. It’s best to get the highest interest rate you can, so... put your initial $10,000 in the CD at

6%, and then make monthly deposits into the money market account over five years. But... how much will that monthly payment need to be in order to have $29,692.16 in five years?

• First, find the future value of the $10,000 CD in 5 five years (CD interest compounds monthly, so set your calculator up for 12 P/YR).

• PV = 10,000• I/YR = 6• PMT =0• Life = 5 shift N• Press FV. You should get -$13,488.50. This is what you’ll get from your CD in five years.

• So, in five years you’ll have $13,488.50, and you’ll need another $29,692.16 - $13,488.50 = $16,203.66. How much will you have to put in your money market account each month to have that amount in five years?

• PV = 0 (you start with nothing in the money market)• I/YR = 3• Life = 5 shift N• FV = 16,203.66• Press PMT. You should get -$250.65.

• As a side note, what if you’d had a little extra money up front to get your money market account started – say $1,000? You’d reflect it as a negative (-1,000) PV in the money market section of the problem. It needs to be negative because it’ll “flow” in the same direction as the monthly payment you’re solving for. Take another look at example #2 in this section if you’re confused on this.

Did You Notice...

• ...Something interesting about inflation? • Since inflation is usually positive, and therefore reduces

the purchasing power of a given unit of money over time, it favors borrowers and hurts lenders.– Borrowers get to repay with “cheaper” money over time– Lenders receive “cheaper” money over time– Assuming, of course that we’re talking about an amortized loan with a fixed

interest rate (i.e. the payment amounts never change). Floating interest rates can kill this proposition.

• So, why are banks in the loans business? Interest rates for their loans are typically set high enough that they’ll earn real money in “today’s dollars” over time.

• But, that’s not necessarily true of the rates you’ll get when you loan money to a bank by opening a savings account, buying a CD, etc.

• So, to keep your purchasing power growing over time, you’ll likely need to use investment vehicles other than pure savings instruments. We’ll discuss this more in later sections.

Section 6: Practice Problems• True or false?

– You can withdraw money without penalty from a CD whenever you want.– In general you can expect higher interest rates with less liquidity.– Inverted yield curves typically spell really good news for the economy.– Core inflation is the same as the CPI.– Core inflation’s definition is fixed.– The interest rate associated with savings instruments is typically not high enough to keep purchasing

power ahead of inflation.• You deposit $15,000 in a savings account yielding 1.75%, compounded

monthly. How much will you have in the account after 4 years, assuming you don’t put any more money in and don’t take any out?

• You deposit $7,500 in a money market account yielding 2.25%. You put another $200 in each month for 7 years. How much money is in the account at the end of the 7 years?

• You stuff $10,000 in a mattress for 20 years. Inflation averages 4% during that time. At the end of the 20 years, how much is the $10,000 worth in today’s dollars?

• You buy a CD for $100,000 with a 5 year maturity, yielding 7%. During the same time period, you also put $250 each month into a savings account yielding 1.25%. Inflation averages 3.5% during those 5 years. At the end of the 5 years, how much money have you, and what’s it worth in today’s dollars? Did your purchasing power grow?

• You want to buy a car in five years. It costs $30,000 today, and inflation is expected to run at 3.75% during the five years. You have $6,500 today. A $5,000, 5 year CD yielding 6.5% is available for purchase. You can also open a money market account yielding 2%. If you buy the CD, how much will you need to deposit into the money market account each month in order to pay cash for the car in five years?