---
title: Interest, Saving, and Investment
url: https://doc.liz6.com/en/general-education/economics-foundations/21-interest-saving-and-investment
locale: en
area: general-education
tags:
- General Education
- Economics Foundations
- General education
- Economics foundations
date: 2026-09-12
modified: 2026-09-12
description: An investment of 100 today yields 108 after one year, which appears to be an "earnings of 8." However, if an alternative arrangement with the same risk yields 110 after one year, is this project still worth choosing? Interest rates place amounts at different points in time on a comparable scale, while risk, liquidity, and financing constraints determine whether such comparisons can translate into actual action.
---
# Interest, Saving, and Investment
Prerequisites: [Economic Choice and Analysis](/en/general-education/economics-foundations/01-models-and-evidence) · [Price Indexes and Inflation](/en/general-education/economics-foundations/18-price-indexes-and-inflation) · [Money, Banking, and Credit](/en/general-education/economics-foundations/20-money-banking-and-credit)
An investment of 100 today yields 108 after one year, which appears to be an "earnings of 8." However, if an alternative arrangement with the same risk yields 110 after one year, is this project still worth choosing? Interest rates place amounts at different points in time on a comparable scale, while risk, liquidity, and financing constraints determine whether such comparisons can translate into actual action.
This article first maps cash flows onto a timeline and then calculates present value and real returns. All examples are fictional, single-period, or short-term instructional cases and do not constitute investment advice for specific products; taxes are ignored and cash flows are assumed to be certain, except where conditions are explicitly changed.
## Distinguishing Principal, Interest, and Total Repayment
Depositing 100 at an annual interest rate of 5% results in a total of `100×1.05=105` after one year, consisting of principal 100 and interest 5. When compounded annually at 5% for two years with interest remaining in the account, the total is `100×1.05²=110.25`; calculating only `100+5+5=110` ignores the earnings on the first year's interest during the second year.
Interest rates must be paired with term, compounding method, and currency. Monthly interest rates cannot be directly compared with annual rates, and nominal quotes may differ from actual financing costs including fees. Listing the actual amounts paid and received on each date is more reliable than focusing solely on a prominent percentage.
## Why Future 108 Is Not Equivalent to Today's 108
If depositing 100 today in a comparable arrangement yields 105 after one year, then 105 received one year from now is equivalent to 100 today under this opportunity. Conversely, discounting a future amount $V$ back to today gives `V/(1+r)`, where $r$ is the discount rate matched to the term and risk.
| Time | Small Project Cash Flow |
| --- | ---: |
| Today t=0 | Invest −100 |
| One Year Later t=1 | Receive +108 |
With a discount rate of 5%, the present value of the future repayment is `108/1.05≈102.86`, resulting in a Net Present Value (NPV) of approximately +2.86 after subtracting the initial investment. This is better than the comparable opportunity; however, if the discount rate is 10%, the present value is approximately 98.18, and the NPV is approximately −1.82, reversing the conclusion.
Net present value is not extra cash appearing in an account, but rather the net value of a scheme compared on the same time scale. The break-even discount rate for this project is 8%, because `108/1.08=100`. For multi-period or alternating positive and negative cash flows, there may be multiple or no such internal rates of return, so single-period examples cannot be unconditionally generalized.
## Nominal Return vs. Purchasing Power Return
If account balances increase by 5% over a year and the prices of relevant goods increase by 2%, the change in purchasing power is `1.05/1.02−1≈2.94%`. The commonly cited "nominal interest rate minus inflation rate" is an approximation when changes are small; the precise relationship requires division.
Decisions are made today, while future inflation is unknown; therefore, ex-ante real interest rates depend on expected inflation. After one year, using actual inflation calculates ex-post real returns. The difference between the two is not necessarily an arithmetic error at the time, but rather that predictions did not materialize.
Borrowers' incomes and commodity prices do not necessarily adjust in sync. Unexpected inflation may reduce the real burden of fixed nominal debt, but this does not mean every borrowing household benefits: whether wages, assets, expenses, and debts are fixed determines the specific outcome.
## How Interest Rate Changes Affect Existing Asset Prices
Consider a security with no default and no interim payments, paying a fixed 110 after one year. At a comparable discount rate of 5%, its value today is approximately 104.76; at 10%, it is 100; and at 20%, it is approximately 91.67.
The promised 110 in the future has not changed; what has changed is the cost of forgoing other opportunities today. Therefore, rising discount rates lower the present value of this fixed future payment. This explains the often inverse relationship between the prices of assets with fixed cash flows and their related yields, but it does not mean that all asset prices move solely via this mechanism under any interest rate change.
Actual stocks, real estate, or bonds may simultaneously change future cash flows, risk, and liquidity. If a price decline is observed, one should ask separately how much the discount rate changed and how much expected income changed, rather than attributing all changes to a single policy rate.
## Adding Risk and Financing Constraints
Previously, we treated 108 as a certain repayment. Real projects may yield 108 only in some scenarios and incur losses in others. One should first list scenarios and cash flows, then compare them using methods appropriate to the risk; a high promised yield does not equal a high comparable expected real return.
Even if a project has a positive NPV after appropriate assessment, entrepreneurs may lack the 100 in cash, lack collateral, or banks may be unable to verify quality. Thus, the project cannot start. Interest rates are not the only adjustment variable in financing markets; credit limits, down payments, and contract conditions also constrain investment.
Conversely, firms with surplus cash may not invest even if interest rates fall, due to weak demand. The ability to finance and the existence of worthwhile projects are two separate conditions; a bank's ability to create deposits does not automatically guarantee an increase in real investment.
## How Savings and Investment Connect in Accounts
Household savings are the portion of income not consumed, which can be used to hold deposits, bonds, and other assets; macroeconomic investment primarily refers to the formation of new capital and changes in inventories. Buying existing stocks does not directly create a new machine.
In a simplified economy with no external sector and no government capital investment (so G consists only of current government consumption), `Y=C+I+G`. Defining national savings as `S=Y−C−G`, the ex-post accounting identity is `S=I`. This is an identity derived from the same set of definitions, meaning it is not the case that every dollar saved by a household directly and one-to-one becomes the factory they wish to fund.
If households plan to consume less, but firms do not increase planned investment, the short term may first see an accumulation of unexpected inventories, triggering production cuts and income declines; ultimately, actual savings and investment remain equal in the accounts. One cannot deduce from the ex-post identity that everyone's ex-ante plans are necessarily compatible. The next lesson will expand on this adjustment process.
## Testing with a Timeline
Two projects both require an investment of 100 today: Project A yields 108 after one year, and Project B yields 115 after two years. With a discount rate of 5% per year, which has a higher Net Present Value?
Expand Reasoning
The NPV of A is `108/1.05−100≈2.86`; for B, it is `115/1.05²−100≈4.31`. Under the premise that cash flows are certain, risks are identical, and these terms are comparable, B has a higher NPV. One cannot directly compare the repayment of 115 with 108, nor can one treat B's two-year return as a one-year return. If there are constraints on capital occupation duration or scale, feasible schemes must also be taken into account.
Expand Reasoning
The original scheme had an NPV of 0; the new scheme is `102/1.05−100≈−2.86`. The benefit of the declining discount rate was outweighed by the decline in expected cash flows. Analysis must track both returns and discount conditions simultaneously; one cannot conclude solely along the single arrow of "rate cuts favor investment."