Week 14 - Loan and Interest Rates, Bond Valuation Flashcards
What is the result of quotation of interest rates?
tradition, legislation, or unfortunately quoted in deliberately deceptive ways to mislead borrowers and investors
What is interest rate per period
most straightforward way to express an interest rate
it represents the rate per compounding period (eg monthly, quarterly, semi-annually)
NOT annualised, making it hard to compare loans or investments
What is an example of interest rate per period?
A bank offers a 2% monthly interest rate on a loan.
This does not mean the annual interest rate is 2% × 12 = 24%, because compounding effects are ignored.
What is a stated annual interest rate/ quoted rate (or nominal)?
most available rate with periodic compounding, quoted by financial institutions often used in advertising
Example of stated annual rate
A credit card states an APR of 12%, meaning 1% per month.
However, if interest compounds monthly, the actual interest paid is higher
What is effective interest rate?
most informative rate because it accounts for compounding, it shows the true cost of borrowing or true return on investment
important when comparing different loans or investments with varying compounding periods
Example of EAR
If APR is 12% with monthly compounding, the EAR is:
(1+ 0.12/12)ˆ12 - 1 = 12.68%
12.68% is the actual interest paid, not the 12% quoted
Why does EAR matter?
different industries quote interest rates differently
some countries require financial institutions to disclose APR to protect consumers
lenders may advertise a low APR while hiding high compounding effects misleading borrowers
What is the stated annual interest rate known as in the US?
annual percentage rate (APR) or nominal rate
What is APR
annual percentage rate/ stated annual rate
commonly quoted but can be misleading because it ignores compounding
What is the APR formula?
APR = the interest rate per period x the number of periods per year
Example of APR
A credit card with a 1.5% monthly rate will have an APR of:
1.5%×12=18%
However, if interest compounds monthly, the actual cost (EAR) will be higher.
How is the APR used in the US and UK?
In many countries, legal rules require APRs to be quoted
In the US and UK, certain laws require the lender to disclose the APR on all consumer loans to protect consumers
Normally, the APR is the quoted interest rate in financial news or newspapers
How does the APR vary internationally
Some countries require APR to include fees & additional costs.
Others use a simple interest rate times number of periods formula
What is the APR if the monthly rate is 0.5%?
0.5 x 12 = 6%
What is the APR if the semi-annual rate is 0.5%?
0.5 x 2 = 1%
What is the monthly rate of the APR is 12% with monthly compounding?
12%/12 = 1%
What is the frequency of compounding?
some investments pay interest more than once a year (eg semi-annual interest rate that compounds twice a year)
The more frequently interest compounds, the higher the actual return (for investments) or higher cost (for loans)
Normally financial institutions will use the quoted annual interest rate (r_s) to standardise the comparison between different products
How can the same quoted APR correspond to many different compounding frequencies per year?
10% compounded semi-annually 10%/2 = 5% every 6 months
10% compounded quarterly 10%/4 = 2.5% every 3 months
10% compounded monthly 10%/12 = 0.83% every month
10% compounded daily 10%/365 = 0.02739% every day
Is more frequent compounding better?
the true interest rate (EAR) increases as compounding frequency increases
More frequent compounding → Higher effective annual rate (EAR)
For loans: More frequent compounding means you pay more interest than you might expect.
For investments: More frequent compounding means you earn more on your money
What is the future value of a lump sum formula?
FV=PV×(1+ rₛ/m)ˆmN
FV= future value
PV = present value
rₛ = quoted annual interest rate
m = number of compounding periods per year
n = number of years
This formula helps determine how much an investment will grow over time with compounding interest
What is the present value of annuity (PVA) formula?
PVA = (Cₘ/ rₛ/m )(1 - 1/(1+ rₛ/m)^mN)
PVA = Present Value of an annuity
Cₘ = Regular annuity payment per period
rₛ/m = Periodic interest rate
mN = Total number of periods
This formula is used to calculate the present value of a series of future cash flows, such as loan payments or pension payouts
What is the future value of annuity (FVA) formula?
FV_A = Cₘ [(1+ rₛ/m)^mN - 1]/ rₛ/m
FVA = Future Value of an annuity
Cₘ = Regular annuity payment
rₛ/m = Periodic interest rate
mN = Total number of periods
This formula calculates how much a series of payments will be worth in the future when interest compounds
For a £1 investment with 10% quoted rate
annually: £1×(1+0.10) = £1.10
semi-annually (rₛ∕𝑚 = 0.10∕2 =0.05) £1×(1+0.05)^2=£1.1025
quarterly (rₛ∕𝑚 = 0.10∕4 = 0.025) £1×(1+0.025)^4=£1.1038
monthly (rₛ∕𝑚 = 0.10∕12 = 0.0083)£1×(1+0.0083)^12=£1.1042
daily (rₛ∕𝑚= 0.10∕365 = 0.0002739)£1×(1+0.0002739)^365=£1.1051
continuous compounding: £1×𝑒^(0.10×1) = £1.105171
continuous compounding: 𝐹𝑉_𝑁=𝑃𝑉×𝑒^(rₛ𝑁)
What is the effective annual rate (EAR)
the true annual interest rate, the actual return (for investments) or cost (for loans), considering the effects of compounding that occurs during the year
the interest rate expressed as if it were compounded once per year
Does more compounding mean higher EAR?
If the APR (quoted annual rate) remains the same, increasing the number of compounding periods per year (m) will always increase the Effective Annual Rate (EAR). This happens because interest is applied more frequently, leading to more growth over time
Since different investments may have different compounding periods, we use EAR to make a fair comparison. The investment with the higher EAR is the better choice (assuming everything else is equal)
What is the EAR formula?
EAR = (1+ quoted rate/m)^m - 1
m is the number of times interest is compounded for year
What is the effective annual rate of 14.9% compounded monthly?
m = 12
EAR = (1+ 0.149/12)^12 - 1
EAR = 0.1596 or 15.96%
A bank is offering 12% compounded quarterly. You put £100 in an account, what is the EAR?
And how much will you have at the end of 2 years?
EAR = (1+ 0.12/4)^4 - 1 = 0.1255
Pt2 Method 1
£100 x (1 + 0.12/4)^8 = 126.68
Method 2
£100 x (1 + 0.1255)^2 = 126.68
If you have an effective rate, how can you compute the APR?
Rearrange the EAR equation
APR = m x [(1+EAR)^1/m -1]
APR = Quoted Annual Interest Rate
EAR = Effective Annual Rate
m = Number of compounding periods per year
What do you need to ensure when working with interest rates and cash flows?
Ensuring that the interest rate and the time period match
If there is a mismatch between the compounding frequency and the frequency of cash flows, adjustments are necessary to ensure consistency in calculations
How do you adjust the interest rate for monthly cash flows?
Monthly Cash Flows → Monthly Rate
If you’re working with monthly cash flows, you must use a monthly rate, not the annual rate. If you have an APR based on monthly compounding, you can convert it to the monthly rate by dividing the APR by 12 (for monthly compounding)
How do you adjust the interest rate for annual cash flows?
Annual Cash Flows → Effective Annual Rate (EAR)
If the cash flows are annual (meaning one payment per year), you should use the Effective Annual Rate (EAR), not the APR. The APR is typically for periods shorter than one year (monthly, quarterly, etc.), but the EAR takes into account all the effects of compounding over a year
Suppose you want to buy a new computer. The store is willing to allow you to make monthly payments. The entire computer system costs £3,500. The loan period is for 2 years. The interest rate is 16.9% with monthly compounding.
What is your monthly payment?
Monthly rate = 0.169/12 = 0.0141
Number of months = 2 x 12 = 24
3500 = C/0.0141 x [ 1 - 1/(1+ 0.0141)^24]
C = 172.88
What are the 3 basic types of loans?
- Pure discount loans
- Interest-only loans/ bullet bonds
- Amortised loans (fully or partially)
What is a pure discount loan/ zero coupon loan?
the borrower receives money today (the principal amount) and repays a single lump sum at the end of the loan (maturity date)
instead of paying interest over the life of the loan, the borrower repays the full principal at the end, along with the interest built into the principal amount. The interest is implicitly included in the loan’s face value, and it is not paid periodically
Example of pure discount loan
Loan Amount: £10,000
Maturity Period: 5 years
The loan will mature to £15,000 (representing the interest and principal).
Borrower receives £10,000 now, but must repay £15,000 at the end of 5 years.
What are 2 examples of pure discount loans?
- US government treasury bills
- Zero-coupon bonds
the principal amount is repaid at some future date without any periodic interest payments
Suppose your firm borrows £6,805,832 today from a bank and will pay back £10,000,000 in 5 years. What is the interest (rate) cost of the loan?
PV = £6,805,832
FV = £10,000,000
10,000,000 = 6,805,832 x (1+r)^5
r = 8%
What are interest only loans
they require payment of interest each period and the repayment of the principal at the end of the loan term
Example of interest-only loan
Suppose you take out an interest-only mortgage with the following terms:
Loan Amount: £100,000
Interest Rate: 5% per year
Loan Term: 10 years
Interest-Only Period: 5 years
Principal Repayment: At the end of 10 years
Monthly payments are based only on the interest:
MonthlyInterestPayment = (5% x 100000)/12
=£416.67
The borrower only makes these monthly payments during the first 5 years, without reducing the principal.
After the Interest-Only Period (Years 6-10):
Once the interest-only period ends, the borrower starts repaying the principal, either as a lump sum or with new monthly payments that include both interest and principal. For example, if the remaining term is 5 years, the new monthly payments would include principal amortization based on the £100,000 loan and the interest rate.
What are two types of amortised loans?
- Fixed principal
- Fixed payments (more common)
What are amortised loans?
loan where the borrower repays the principal in instalments over time, with each instalment consisting of both an interest payment and a principal repayment. As the loan progresses, the amount of principal outstanding decreases, and the interest payments also decrease accordingly
ie the principal will be progressively reduced to zero
- as part of the principal get repaid, interest is calculated only on the remainder of the principal
Periodic payments = interest + repayment of a portion of the principal
Eg housing loans (mortgages, car loans etc)
What is amortisation schedule?
a table that describes the periodic payments, interest and principal balance after each payment
It helps the borrower understand how much of their payment is going towards interest and how much is going towards reducing the loan principal
Amortisation schedule example: A company borrows £4,000 using a 4 year loan at 8% interest from the bank. What is the amortisation schedule for?
Fixed principal: principal reduction is constant each year
periodic payment = £1000 + interest
Fixed payment: periodic payment is constant each year
annuity
given PV, r and t, what is C (periodic payment)?
answer £1,207.68
Fixed principal looks like (table)
Columns:
Year, Beginning Balance, Periodic Payment, Interest Paid, Principal Paid, Ending Balance
How is the principal balance reduced?
By the same (fixed) amount each period
At the end of the last period, the principal is reduced to zero
The periodic payment is different each year
The periodic payment is reduced each year
Interest paid is decreasing over time
What does it mean when the periodic payment C is in the form of an ordinary annuity?
it means the payments are made at the end of each period.
How do corporations raise capital to use for long term investments?
3 sources:
1. Cheapest is through using retained earning
2. Borrow money
3. Issue equity (last resort)
What are bonds?
a debt instrument that requires the issuer to repay to the investors the amount borrowed plus interest over a specified period of time
What 3 parties are involved in bonds?
- Issuers
- Investors
- Other parties
What do issuers do in terms of bonds?
the parties (government or corporations) that borrow money and issue debt securities
such as debtors and borrowers
What is the issuers role?
Issuers raise capital by selling bonds to investors. In exchange, they agree to pay back the principal (face value) of the bond at maturity along with periodic interest payments (coupon payments)
What do investors do in terms of bonds?
the parties (persons, government or corporations that borrow money and issue debt securities)
such as creditors, lenders, bondholders
What is the role of investors?
Investors provide the capital to issuers by purchasing bonds. In return, investors receive interest payments over the life of the bond and the face value (principal) of the bond at maturity
What are bondholders?
Another term for investors holding bonds. They earn periodic interest (coupons) and are repaid the principal when the bond matures
What do other parties do in terms of bonds?
underwriters: investment banking firms that act as agents to distribute bonds to investors. They will charge a fee for the service.
What is the role of underwriters?
Underwriters act as intermediaries in bond issuance. They may take on some risk by guaranteeing to sell a certain amount of bonds and can charge a fee for the service. The underwriting fee is typically a percentage of the total amount raised in the bond offering
How is a bond typically structured?
as an interest-only loan, meaning the issuer (borrower) makes periodic interest payments to the bondholder (investor) but does not repay the principal (face value) until maturity
What is the face value F (par value) of the bond?
the amount the issuer agrees to repay the bondholder at maturity (end of the loan)
it is typically £1,000, $1,000, or €1,000, but it can vary
It is a fixed amount agreed upon at issuance
most the time it isnt equal to the bond price
What are coupons C?
the periodic payments that issuers promise to make and bondholder receives
Example of coupon payment
Example: If a bond has a 5% coupon rate and a £1,000 face value, it pays £50 per year in interest (or £25 semi-annually if paid twice a year)
How are coupons C usually expressed?
as a percentage of the par value, it is usually constant and determined upon issue
What is the coupon payment calculation?
Coupon payment = (Coupon rate (c) x Par value (F)) / Number of coupon payments per year
coupon rate isnt the discount rate for discounting
Is the coupon rate and discount rate the same?
No
The coupon rate is fixed when the bond is issued and determines the periodic coupon payment.
The discount rate (or yield to maturity, YTM) is the market rate used to price the bond
What is maturity?
the number of years till the date when the issuer repays the face value to the bondholder
What is the current yield?
measures the bond’s return based on its current market price (P) rather than its face value
What is the current yield or CY formula?
CY = Annual coupon payment/ market price
Example of Current yield
A bond with a 6% coupon rate, £1,000 face value, and a market price of £950 has:
CY = (0.06x1000)/950
=6.32%
Note: Current yield does not account for capital gains or losses if the bond is sold before maturity.
What does the maturity date of a bond refer to?
refers to the date when the principal (face value) is repaid, and the bond contract ends
Until maturity, the issuer makes periodic coupon payments (if applicable)
Can there be more than one coupon payment in a calendar year?
yes there may be more than one coupon payment and can correspond to many compounding periods
What count as short term bonds?
1-5 years
What count as intermediate bonds?
5-12 years
What counts as a long term bond?
12+ years
What is the yield to maturity?
the annual return an investor will earn if they hold a bond until maturity.
It is the discount rate that equates the present value (PV) of the bond’s future cash flows to its current market price (P)
What is the yield to maturity also called?
the market interest rate, implicit rate of interest, required rate of return
Why is the yield to maturity important?
Reflects the bondholder’s return based on current market conditions
Changes over time as investor risk preferences and market conditions shift
Used to price bonds and compare investment opportunities
What does the yield depend on to change over time?
depending on the investors’ risk attitudes and investment opportunities
The yield to maturity is the rate of return on the bond to an investor given 3 critical assumptions. What are they?
- The investor holds the bond to maturity
- The issuer makes all of the coupon and principal payments in full on the scheduled dates (the issuer doesnt default on any of the payments)
- The investor is able to reinvest coupon payments at the same yield
Explain the assumption that the investor holds the bond to maturity
YTM assumes that the investor does not sell the bond before maturity.
If the investor sells the bond early, the actual return may be higher or lower than the YTM due to price fluctuations in the bond marke
Explain the assumption that the issuer makes all coupon and principal payments in full and on time
The bond issuer must pay all interest (coupons) and principal as scheduled.
If the issuer defaults, investors do not receive full payments, leading to lower actual returns than YTM suggests.
Credit risk is a factor here—higher-rated bonds (AAA) have a lower risk of default than lower-rated bonds (junk bonds)
Explain the assumption that the investor can reinvest coupon payments at the same yield (reinvestment risk)
YTM assumes that every coupon payment is reinvested at the same YTM rate.
In reality, interest rates fluctuate over time, and the investor may not find reinvestment opportunities at the same YTM rate.
This is known as reinvestment risk:
If interest rates fall, future coupon payments earn less than expected.
If interest rates rise, future coupons could be reinvested at a higher return than YTM
How to calculate bond price as the present value of cash flows?
Bond price = TΣt=1 C/(1+r)^t + F/(1+r)^T =
PV(Coupons) + PV(Face value) = C x [1- 1/(1+r)^t] /r + F x 1/(1+r)^T
where C = c x F
(c stands for coupon rate)
P = Market price of the bond
C = Coupon payment per period (C=c×F)
c = Coupon rate
F = Face value of the bond
r = Yield to Maturity (YTM)
T = Number of periods to maturity
usually bond price = 1000 same as Face value
What is the implicit rate of interest (YTM) investors earn? formula
(c x F)/r x [1- 1/(1+r)^t] + F x 1/(1+r)^T
What is the relationship between price and yield to maturity?
x axis yield to maturity %
y axis Price
downward sloping, inverse relationship
What is the relationship between par value and bond price if the yield-to-maturity = coupon rate?
if YTM = coupon rate
then par value = bond price
The bond is selling at par (i.e., at its face value).
Investors are willing to pay exactly the face value because the coupon payments provide a return equal to the market rate
What is the relationship between par value and bond price if the yield-to-maturity > coupon rate?
if YTM > coupon rate
then par value > bond price
selling at a discount, called a discount bond
Investors require a higher return than what the bond’s coupon offers, so they pay less than the face value.
Example when YTM> Coupon rate and bond price < par value
If a bond pays a 5% coupon but the market interest rate is 7%, investors will only buy it if it is priced below face value to compensate for the lower coupon rate
What is the relationship between par value and bond price if the yield-to-maturity < coupon rate?
if YTM < coupon rate
then par value < bond price
selling at a premium, called a premium bond
Investors are willing to pay more than the face value because the coupon payments are higher than the market-required return
Example when YTM< Coupon rate and par value<bond price
If a bond pays a 7% coupon but the market rate is only 5%, investors will bid up the bond’s price above face value to earn the higher fixed payments
What is a consol bond (or perpetual)?
a fixed-income security with no maturity date, meaning it pays coupons forever without repaying the principal
What is the formula of a consol bond?
P = C/r
P = Price of the bond
C = Annual coupon payment
r = Yield to Maturity (YTM) (required return by investors)
Example of a consol bond calculation
Face Value (F) = 1,000
Coupon Rate = 8%
Annual Coupon Payment (C) = 8% × 1,000 = 80
Yield to Maturity (r) = 10% = 0.10
Time to Maturity = ∞ (Perpetual Bond)
P = 80/0.10 = 800
the price of the consol bond is $800
What are 2 causes of bond price movements?
- the passage of time
- changes in interest rates (YTM)
How does the passage of time cause bond price movements?
As a bond approaches maturity, its price gradually moves toward its face value (par).
If a bond is trading at a discount, its price will rise toward par.
If a bond is trading at a premium, its price will decline toward par.
This happens because, at maturity, the investor receives only the face value, regardless of whether they bought the bond at a premium or discount
How does the change in interest rates (YTM) cause bond price movements?
Bond prices are inversely related to market interest rates (YTM).
When interest rates (YTM) rise → bond prices fall
When interest rates (YTM) fall → bond prices rise
This happens because investors always compare a bond’s fixed coupon rate to newly available market rates
Who will care about the price change of the bond?
investors (bondholders) and issuers (companies and government)
Why will investors (bondholder) care about bond price changes?
If interest rates drop, existing bonds with higher coupon rates become more valuable (price increases).
If interest rates rise, existing bonds with lower coupon rates become less valuable (price decreases).
Investors who plan to sell bonds before maturity care about these price movements.
Why will issuers (companies and governments) care about bond price changes?
If bond prices fall (because interest rates increase), issuers may need to offer higher yields to attract investors for new bond issues.
If bond prices rise (because interest rates drop), issuers might refinance debt at lower costs by issuing new bonds at lower interest rates
A bonds value can differ substantially from its par value prior to maturity. Where will the price converge towards?
Before maturity, a bond’s market price can be above or below its par value. However, as the bond approaches its maturity date, its price gradually converges to its par value (face value)
