Interest Rates and Time Value of Money
14M Estimated
The time value of money quantifies the relationship between cash flows that occur on
different dates. Given the choice between receiving either $100 or $105 today, the larger
amount is clearly preferable. However, to decide between receiving $100 today or $105 in
one year, it is necessary to convert the amount of the future cash flow into its present
equivalent. And to do this conversion, we need to use an interest rate, also known as a
yield.
An interest rate can be thought of in three ways:
1. Required rates of return: An investor who is willing to lend $100 today in exchange
for receiving $105 in one year requires a 5% rate of return.
2. Discount rate: If a 5% discount rate is used, $105 in one year is equivalent to $100
today.
3. Opportunity cost: The decision to spend $100 today means giving up the
opportunity to invest the same amount and have it grow to $105 over the next
year.
Determinants of Interest Rates
Interest rates are determined by the interaction between the forces of supply and demand.
In financial markets, funds are supplied by savers who would prefer a higher rate of return
and demanded by borrowers who want to pay a lower interest rate. The equilibrium
interest rate that emerges from the collective interactions between borrowers and savers
will be higher or lower depending on the relative strength of each side.
However, there is not a single interest rate that is used for all loans. For a given investment,
an investor will require an interest rate, , that can be calculated as the sum of the real riskfree rate and any applicable premiums to compensate for various risks:
where
is the real risk-free rate that investors require as compensation for a risk-free loan
with no expected inflation. It can be thought of as the pure opportunity cost (in real
terms) of using funds for current consumption rather than investing.
is the inflation premium that compensates lenders for expected inflation. Investors
must earn a nominal rate of return that includes both the real-risk free rate and this
premium in order to preserve the purchasing power of their funds.
is the default risk premium that lenders charge for the possibility that borrowers will
not meet their obligations on time and in full. Although government securities are
exposed to certain risks, such as unexpected inflation, they are referred to as “riskfree” because they carry no default risk.
is a liquidity premium that compensates the investor for the extra cost of converting
an investment to cash relatively quickly. Government securities typically carry little-tono liquidity premium because they are traded in active markets with many willing
buyers and sellers. Yields on corporate debt, particularly securities issued by smaller
firms, reflect a liquidity premium because investors are more likely to be forced to
accept a discount to market value when selling their positions.
is the maturity premium earned by investors who are willing to lend over longer
periods, which requires greater exposure to interest rate risk. Even longer-dated
government bonds that are free of default risk will have yields that reflect a maturity
premium relative to the yields on short-term government securities.
The nominal risk-free rate reflects the real risk-free rate and the premium for expected
inflation. If calculated over more than one period, a multiplicative formula should be used
to capture the impact of compounding:
However, in practice, this rate is often approximated using an additive formula:
Example: Interest Rate Components
An investor gathers the following data on three newly issued bonds:
1-year government bond, 3.0% yield
1-year ABC corporate bond, 4.2% yield
10-year government bond, 3.8% yield
If investors require a 0.5% liquidity premium for corporate bonds, what are the
components of the required return on a 10-year ABC bond?
Solution
The 3.0% yield on a 1-year government bond reflects the nominal risk-free rate, which
is the sum of the real risk-free rate and the premium for expected inflation.
The yield on the 1-year ABC bond reflects a total premium of 1.2% over the nominal
risk-free rate. If investors require a 0.5% liquidity premium, the additional 0.7% is
compensation for default risk.
Because the 10-year government bond has no exposure to either default risk or
liquidity risk, the 0.8% difference in yield compared to the 1-year government bond
represents a maturity premium.
With a 3.0% nominal risk-free rate, a 0.5% liquidity premium, a 0.7% default risk
premium, and a 0.8% maturity premium, the total required return on a 10-year ABC
corporate bond is 5.0%.