Professor Jessie

From Single Cash Flows to Multiple Cash Flows

In Chapter 4, we mastered moving a single cash flow through time — one deposit, one withdrawal, one rate, one period. But the real world rarely works that way. Most financial scenarios involve multiple cash flows occurring at regular intervals: saving monthly for a down payment, receiving quarterly dividend checks, making yearly contributions to a retirement fund.

Why This Chapter Matters

Think about the financial decisions you'll face in life. A car loan means paying the same amount every month for 4-5 years. A mortgage means paying every month for 15-30 years. A retirement plan means contributing regularly over a 40-year career. These are all annuities — streams of level, recurring cash flows — and this chapter gives you the tools to analyze them.

The good news: everything builds on Chapter 4. The underlying math is the same compounding and discounting we already know. We're just applying it to many cash flows at once instead of one at a time.

Future Value of Multiple (Unequal) Cash Flows

When you make several deposits of different amounts at different times, you can find the total future value by computing each cash flow's future value individually, then adding them together. Each deposit compounds for the remaining time until your target future date.

Worked Example

Suppose you deposit $100 today, $125 next year, and $150 at the end of Year 2. If interest rates are 7%, what's the total future value at the end of Year 3?

• $100 deposit (Year 0): Compounds for 3 years → $100 × (1.07)³ = $122.50
• $125 deposit (Year 1): Compounds for 2 years → $125 × (1.07)² = $143.11
• $150 deposit (Year 2): Compounds for 1 year → $150 × (1.07)¹ = $160.50

Total FV = $122.50 + $143.11 + $160.50 = $426.11

Where CF_t is the cash flow at time t, r is the interest rate, and (n − t) is the number of periods each cash flow compounds. The earlier you deposit, the longer it grows — which is why starting early matters so much.

Future Value of an Annuity

An annuity is a stream of level (equal) cash flows paid at the end of each time period. When the payments are identical and regular, we don't need to compute each one separately — there's a shortcut formula that handles the entire stream at once.

The Annuity FV Formula

Where PMT is the payment amount, r is the interest rate per period, and n is the number of payments. The bracket term is called the future value interest factor for an annuity.

Worked Example

You deposit $100 at the end of each year for 5 years at 8% interest. What's the future value?

Even though you only deposited $500 total ($100 × 5), you end up with $586.66 — the extra $86.66 is compound interest earned along the way. Each $100 deposit earns interest from the moment it's made until the end of Year 5.

Multiple Annuities (Different Payment Levels)

What if your cash flows change at some point — say, $100/year for the first 3 years, then $150/year for the next 2? You can split this into two separate annuities, compute each one's future value, and add them together. The key is making sure each annuity is measured from its own starting point to the same final target date.

Present Value of Multiple (Unequal) Cash Flows

Just as we can compound multiple cash flows forward, we can discount multiple cash flows back to the present. This is arguably even more useful in practice — most investment decisions require computing the present value of expected future cash inflows.

The Approach

Discount each cash flow individually back to Year 0, then sum all the present values:

Worked Example

You expect to receive $100 today, $125 next year, and $150 at the end of Year 2, at 7% interest. What's the total present value?

• $100 (Year 0): Already at present → $100.00
• $125 (Year 1): $125 ÷ (1.07)¹ = $116.82
• $150 (Year 2): $150 ÷ (1.07)² = $131.02

Total PV = $100.00 + $116.82 + $131.02 = $347.84

Notice that $347.84 today is economically equivalent to the three cash flows spread across time. If someone offered you a single lump sum of $347.84 today versus those three payments, you should be indifferent — they're worth exactly the same at a 7% discount rate.

Present Value of an Annuity

When the multiple cash flows are equal and regular — an annuity — we again have a shortcut formula. This is one of the most used formulas in all of finance because most loans are structured as annuities: you borrow a lump sum today (the present value) and repay it through equal periodic payments.

The Annuity PV Formula

Worked Example

Someone makes $100 payments at the end of each year for 5 years at 8% interest. What's the present value?

This means receiving $100/year for 5 years is worth the same as receiving $399.27 as a lump sum today. The remaining $100.73 ($500 total payments minus $399.27 PV) represents the time value discount — the "cost" of waiting for your money.

Real-World Example: David Price's MLB Contract

When the Boston Red Sox signed pitcher David Price, his contract paid $30 million/year for 3 years, then $31 million for 3 years, then $32 million for the final year — a total of $217 million over 7 years. But the present value was much less. By splitting the contract into multiple annuities and discounting at 5%, the PV came to approximately $178.89 million. This is why "total contract value" headlines can be misleading — the actual economic value depends on the timing of payments.

Perpetuities: Annuities That Never End

A perpetuity is a special type of annuity where the cash flows continue forever — there's no end date. While this may seem theoretical, it has very real applications. The most common example is preferred stock, which pays a fixed dividend indefinitely with no maturity date.

The Perpetuity Formula

When n approaches infinity, the annuity PV formula simplifies beautifully:

That's it — just divide the payment by the discount rate. If a preferred stock pays $5 per year forever and the required return is 8%, its value is: $5 ÷ 0.08 = $62.50.

Why It Works

As n gets larger, the term (1 + r)⁻ⁿ shrinks toward zero. In the annuity formula, this means the "(1 − 0) ÷ r" simplifies to "1 ÷ r", leaving just PMT ÷ r. Cash flows far in the future contribute almost nothing to present value — they're discounted so heavily they become negligible. Only the near-term payments really matter.

Ordinary Annuity vs. Annuity Due

So far, all our annuity formulas assume payments occur at the end of each period — this is called an ordinary annuity. But some payments happen at the beginning of each period — this is called an annuity due. The difference matters.

The Adjustment

Because annuity-due payments occur one period earlier, they have one extra period to grow (for FV) or one fewer period to shrink (for PV). The adjustment is simple — multiply the ordinary annuity result by (1 + r):

For example, the FV of a 5-year, $100 ordinary annuity at 8% is $586.66. The same annuity due would be: $586.66 × 1.08 = $633.59. That extra compounding period for each payment makes a meaningful difference.

Compounding Frequency: More Often = More Growth

In the real world, interest isn't always compounded once a year. Bonds often pay semiannually, savings accounts compound monthly, and credit cards compound daily. The frequency of compounding affects how much your money grows — and the effect might surprise you.

The Impact of Frequency

Consider a $100 deposit at 12% annual interest. How does compounding frequency change the future value after one year?

Two key takeaways from this table: (1) Higher compounding frequency produces a higher future value — you earn interest on interest more often. (2) The relative increase diminishes — going from annual to semiannual adds $0.36, but going from monthly to daily adds only $0.07. There are diminishing returns to more frequent compounding.

Keeping Variables Consistent

When working with intra-year compounding, you must keep all variables consistent with the compounding period. If compounding monthly: divide the annual rate by 12 for the periodic rate, and multiply the number of years by 12 for the total number of periods. Mixing annual rates with monthly periods (or vice versa) is a guaranteed way to get wrong answers.

APR vs. EAR: The Truth About Interest Rates

When you see an interest rate quoted, it might not tell the whole story. There are two different ways to express an annual interest rate, and understanding the difference can save you real money.

The EAR Formula

Where m is the number of compounding periods per year. For a 12% APR compounded monthly: EAR = (1 + 0.12 ÷ 12)¹² − 1 = (1.01)¹² − 1 = 12.68%. The true annual cost is 0.68% higher than the quoted rate.

Amortized Loans: Car Payments, Mortgages, and More

An amortized loan is repaid through equal periodic payments that cover both interest and principal. Each payment is split: part goes toward interest on the remaining balance, and part goes toward reducing the principal. Over time, the interest portion shrinks and the principal portion grows.

Finding the Payment

Rearrange the present value of annuity formula to solve for PMT — this is what banks do when they calculate your monthly payment:

Worked Example: Car Loan

You need a $10,000 car loan for 4 years at 9% APR. Monthly compounding means r = 0.75% and n = 48 months:

Over 4 years, you'll pay 48 × $248.85 = $11,944.80 total. That means you pay $1,944.80 in interest on a $10,000 loan — nearly 20% of the principal. This is why the interest rate and loan term matter enormously.

The Amortization Schedule

An amortization schedule breaks down each payment into its interest and principal components:

Notice the pattern: in early payments, most of the money goes to interest. By the final payment, almost everything goes to principal. This is why making extra principal payments early in a loan can save you thousands in interest — you reduce the balance before it accrues more interest.

Key Takeaways

• Multiple cash flows are valued by computing each one's FV or PV separately, then summing
• An annuity = level, recurring cash flows at period-end; use shortcut formulas instead of computing each payment individually
• FV of annuity: PMT × [((1 + r)ⁿ − 1) ÷ r]
• PV of annuity: PMT × [(1 − (1 + r)⁻ⁿ) ÷ r]
• A perpetuity pays forever: PV = PMT ÷ r (preferred stock is the classic example)
• Annuity due (payments at period start) = ordinary annuity × (1 + r)
• More frequent compounding → higher FV, but with diminishing returns
• APR = quoted rate (ignores compounding); EAR = true annual rate (includes compounding) — always compare using EAR
• Amortized loans: each payment covers interest + principal; early payments are mostly interest, later payments are mostly principal
• Making extra principal payments early in a loan saves significant interest