CH 3 What Do Interest Rates Mean and What Is Their Role in Valuation?
Present Value Concept: Fixed-Payment Loan Terms • Installment Loans,
such as auto loans and home mortgages are frequently of the fixed-payment type.
Formula for Duration
t = years till cash payment is made CPt = cash payment (interest + principal) at time t n = years to maturity
Simple Loan Terms: Loan Principal
the amount of funds the lender provides to the borrower.
Simple Loan Terms: Interest Payment
the cash amount that the borrower must pay the lender for the use of the loan principal.
Duration is additive
the duration of a portfolio of securities is the weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each security
Simple Loan Terms: Simple Interest Rate: -includes convention:
the interest payment divided by the loan principal; the percentage of principal that must be paid as interest to the lender. -Convention is to express on an annual basis, irrespective of the loan term. -interest rate (i) = additional interest payment/ principal
Simple Loan
the lender provides the borrower with an amount of funds that must be repaid to the lender at the maturity date, along with an additional payment for the interest -requires one payment of one amount equal to loan principal + interest
Discounting the future
the process of calculating today's value of dollars received in the future
Interest rates are among the most closely watched variables in the economy.
yield to maturity (YTM) is the most accurate measure of interest rates.
Yield to Maturity: Bonds 4. One-Year Discount Bond (P = $900, F = $1000) n=1 CPT i
$900 = $1,000 / (1 + i), or i = 11.11% PV=-900 FV=1000 n=1 CPT i= 11.11% Formula= (F-P)/P = (1000-900)/900= 11.11%
Duration and Interest-Rate Risk Duration can be used to show that the approximate change in price is related to duration, as follows:
%∆Price = (new-old)/old = % change in price of security from t to t+1 = rate of capital gain DUR= duration i= current interest rate
Additional Example holding a 10 year 20% coupon bond in fund's portfolio, and the interest rate is currently 10%. What loss would the fund be exposed to if the interest rate rises to 11% tomorrow? -table 3.4 gives duration of 10 year 20% coupon bond = 5.72 years
%∆Price = − DUR × (∆i / (1 + i) = -5.72 x (.11-.10/(1+.10)) =-5.20%
Example 3.10 holding a 10 year 10% coupon bond in fund's portfolio, and the interest rate is currently 10%. What loss would the fund be exposed to if the interest rate rises to 11% tomorrow? -table 3.3 gives duration of 10 year 10% coupon bond = 6.76
%∆Price = − DUR × (∆i / (1 + i) =-6.76 x (.11-.10/(1+.10)) =-6.15% USE the current interest rate for denominator
EX 3.9 25% portfolio in a bond with a 5 year duration and 75% in a bond with a 10 year duration. What is the duration of the portfolio?
(0.25 x 5) + (0.75 x 10)= 8.75 years
Key facts about duration (4)
1. All else equal, when the maturity of a bond lengthens, the duration rises as well 2. All else equal, when interest rates rise, the duration of a coupon bond fall 3. All else equal, the higher the coupon rate of the bond, the shorter the bond's duration. 4. Duration is additive
Maturity and the Volatility of Bond Returns Key findings from Table 3.2 (1 of 2)
1. Only bond whose return = yield is one with maturity = holding period 2. For bonds with maturity > holding period,as rates increase, price falls, implying capital loss 3. Longer is maturity, greater is price change associated with interest rate change
Maturity and the Volatility of Bond Returns Conclusion from Table 3.2 analysis
1. Prices and returns more volatile for long-term bonds than short-term bonds because they have higher interest-rate risk 2. No interest-rate risk for any bond whose maturity equals holding period -no risk for short term bonds, but substantial for long term. --the price at the end of the holding period is already fixed at face value for short term bonds
Present Value Applications There are four basic types of credit instruments which incorporate present value concepts:
1. Simple Loan -repaid at maturity 2. Fixed Payment Loan -payments periodically till maturity 3. Coupon Bond -payments periodically till maturity 4. Discount Bond -repaid at maturity
Table 3.1 Yields to Maturity on a 10% Coupon Rate Bond Maturing in 10 Years (Face Value = $1,000) Three interesting facts in Table 3.1
1. When bond is at par, yield equals coupon rate 2. Price and yield are negatively related 3. Yield greater than coupon rate when bond price is below par value
A coupon bond is identified with 3 pieces of information
1. corporation or gov agency that issues the bond 2. maturity date 3. bond's coupon rate: the dollar amount of the yearly coupon payment expressed as a percentage of the face value of the bond -100 coupon payment, 1000 face value, and 10% rate (100/1000)
Maturity and the Volatility of Bond Returns Key findings from Table 3.2 (2 of 2)
4. Longer is maturity, more return changes with change in interest rate 5. Bond with high initial interest rate can still have negative return if i ↑
Perpetuity/ Consol
A perpetual bond with no maturity date and no repayment of principal that makes periodic fixed payments forever P= price of perpetuity C= yearly payment i= YTM -as i goes up the price of the bond falls
Fixed Payment Loan
AKA fully amortized loan the lender provides the borrower with an amount of funds, which must be repaid by making the same payment every period, consisting of part of the principal and interest for a set number of years -EX: borrow 1000 you may pay 126 every year for 25 years
Discount Bond
AKA zero coupon bond -bought at a price below its face value (at a discount), and the face value is repaid at the maturity date -no interest payments, just pays off face value -EX: Face value of 1000 may be bought for 900 and in a years time be repaid the face value of 1000 -US treasury bills and US savings bonds
Current Yield (2 properties)
Current yield (CY) is just an approximation for YTM - easier to calculate. -However, we should be aware of its properties: 1. If a bond's price is near par and has a long maturity, then CY is a good approximation. 2. A change in the current yield always signals change in same direction as yield to maturity Formula: ic = C / P
Example 3.1 Simple Present Value What is the present value of $250 to be paid in two years if the interest rate is 15%?
FV= 250 n=2 i= 15 CPT PV= 189.04 or $250 / (1 + 0.15)^2 = $250 / 1.3225 = $189.04
The term present value (PV) can be extended to mean the...
PV of a single cash flow or the sum of a sequence or group of cash flows.
Simple Loan Terms: Maturity Date vs Loan Term
Maturity Date: the date the loan must be repaid - the Loan Term is from initiation to maturity date.
Reinvestment Risk
Occurs if investor holds a series of short bonds over long holding period (holding period is longer than maturity and must reinvest it) - i at which reinvest is uncertain -Gain from i ↑, lose when i ↓
Example 3.5 Perpetuity What is YTM? Price =$2000 pays $100 annually forever
P= C/i or (current yield) i= C/P i=100/2000= .05= 5%
Yield to Maturity: Bonds 3. Coupon Bond (Coupon rate = 10% = C/F) Face Value =1000 n=8 coupon payment= 100 yearly YTM= 12.25% CPT PV
PV of bond= Sum of PV of all coupon payments + PV of final payment of face value n=8 i=12.25 FV=1000 PMT=100 CPT PV= 889.20
Present Value Concept: Simple Loan Simple loan of $100 n= 1 i= 10% CPT FV
PV= -100 n= 1 year i=10% CPT FV= 110 or for two years just change n=2 for FV=121
Yield to Maturity: Loans 2. Fixed Payment Loan (i = 7%) Mortgage loan =$100,000 n=20 years What is the yearly payment
PV=-100,000 n=20 i=7 FV=0 CPT PMT=9,439.29
Distinction Between Interest Rates and Returns Rate of Return: we can decompose returns into two pieces:
Rate of return= the payments to the owner plus the change in its values, expressed as a fraction of its purchase price -the return on a bond will not necessarily equal the interest rate on that bond
Figure 3.1 Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953-2016
Real and nominal rates often do not move together -When nominal rates were very high in USA in 1970s, the real rates were actually very low, often negative. The cost of borrowing was thought to be high but it was actually very low.
Table 3.3 Calculating Duration on a $1,000 Ten-Year 10% Coupon Bond When Its Interest Rate Is 10%
Step 1: CPT PV -n=1, i=10, FV=100 then n=2, n=3 and so on Step 2: divide PV by total PV in column 3 -total must = 100% Step 3: (column 1 x column 4)/100 Duration is a weighted average of the maturities of the cash payments
Table 3.4 Calculating Duration on a $1,000 Ten-Year 10% Coupon Bond When Its Interest Rate Is 20%
When the interest rate is higher, the cash payments in the future are discounted more heavily and become less important in PV terms relative to the total PV of all payments. The relative weight (column 4) will drop making the effective maturity fall.
Yield to Maturity: 1. Simple Loan Interest Rate CPT i PV= -100 FV=110 n =1
Yield to maturity = interest rate that equates today's value with present value of all future payments Simple interest: (i = 10%) $100 = $110 / (1 + i), or I = 10% PV=-100 FV=110 n=1 CPT i= 10% For simple loans, the simple interest rate = YTM
Duration
average lifetime of a debt security's stream of payments -effective maturity: the term to maturity that accurately measures interest rate risk --shorter for a coupon bond than a zero coupon bond PG 56/57
The concept of present value (or present discounted value) is based on the..
commonsense notion that a dollar of cash flow paid to you one year from now is less valuable to you than a dollar paid to you today. -This notion is true because you could invest the dollar in a savings account that earns interest and have more than a dollar in one year.
Return on a bond formula EX: What is the rate of return of a bond bought for $1000 and sold one year later for $800? Face value is $1000 and coupon rate is 8%.
coupon PMT= 1000*.08=80 (80+ (800-1000))/1000= -12% or Return = C/Pt + (Pt+1 - Pt)/Pt =(80/1000) + (800-1000)/1000= -12%
Different debt instruments have very different streams of cash payments to the holder (known as cash flows), with very different timing. All else being equal,...
debt instruments are evaluated against one another based on the amount of each cash flow and the timing of each cash flow. -This evaluation, where the analysis of the amount and timing of a debt instrument's cash flows lead to its yield to maturity or interest rate, is called present value analysis.
If i = 5% and pie =0% then.. If i= 10% and pie = 20% then..
ir= i-pie 5% − 0% = 5% 10% − 20% = − 10% When the interest rate is low, there are greater incentives to borrow (pay less) and fewer incentives to lend (receive less).
Distinction Between Real and Nominal Interest Rates (2 of 3) • Real interest rate ir =i− pie We usually refer to this rate as the..
ex ante real rate of interest because it is adjusted for the expected level of inflation. -After the fact, we can calculate the ex post real rate based on the observed level of inflation.
Capital Gain yield
g= (new-old)/old change in bonds price relative to initial purchase price R= ic +g
EX 3.8 Calculate the gain or loss on a 10 year zero coupon bond for which the interest rate has increased from 10% to 20%. The bond has a face value of $1,000.
g= new-old / old g= ((Pt-1)- Pt)/Pt new= 1,000/(1+.20)^9= 193.81 old= 1,000/ (1+.10)^10= 385.54 g= (193.81-385.54)/385.54= -49.7%
Yield on a Discount Basis One-Year Bill (P = $900, F = $1000, days = 365) -2 characteristics.
idb = ($1,000−$900) / $1,000 × (360/365) = 9.9% Two Characteristics 1. Understates yield to maturity; longer the maturity, greater is understatement 2. Change in discount yield always signals change in same direction as yield to maturity
Until 1997, real interest rates in the US were not observable because only nominal rates were reported. This changed in 1997 when the US Treasury began to issue.....
indexed bonds, whose interest and principal payments were adjusted for changes in price level
Present Value Concept: Simple Loan The previous example reinforces the concept that $100 today is preferable to $100 a year from now since today's $100 could be....
lent (or deposited) at 10% interest to be worth $110 one year from now, or $121 in two years or $133 in three years. Formula= 100 X (1+i)^n
PV Concept: Fixed-Payment Loans
loans where the loan principal and interest are repaid in several payments, often monthly, in equal dollar amounts over the loan term.
Global: Negative T-Bill Rates? It Can Happen In November 1998, rates on Japanese 6-month government bonds were...
negative! Investors were willing to pay more than they would receive in the future. -Same thing happened in the U.S. in September of 2008, then Sweden (July 2009), Denmark (July 2012), the Eurozone (June 2014), Switzerland (December 2014), and Japan again in early 2016. -Best explanation is that investors found the convenience of the Treasury bills (in the U.S. case) worth something—more convenient than cash. But that can only go so far—the rate was only slightly negative.
Relationship Between Price and Yield to Maturity It's also straight-forward to show that the value of a bond (current price) and yield to maturity (YTM/interest rate) are..
negatively related. -If the interest rate i increases (YTM increases), the PV of any given cash flow is lower; hence, the price of the bond must be lower.
Nominal interest rate
no allowance for inflation
Coupon Bond
pays the owner of the bond a fixed interest payment (coupon payment) every year until the maturity date, when a specified final amount (face or par value) is repaid -Treasury bonds, notes, and corporate bonds -EX: $1000 Face value may pay you a coupon payment of $100 per year for 10 years and then repay you the face amount of $1000 at maturity
The greater the duration of a security, the greater the
percentage change in the market value of the security for a given change in interest rates -Therefore, the greater the duration of a security, the greater its interest-rate risk