Finance Homework
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Chap3.pptxChapter Three
Interest Rates and Security Valuation
Various Interest Rate Measures
Coupon rate
periodic cash flow a bond issuer contractually promises to pay a bond holder
Required rate of return (r)
rates used by individual market participants to calculate fair present values (PV)
Expected rate of return or E(r)
rates participants expect to earn by buying securities at current market prices (P)
Realized rate of return (r)
interest rate actually earned on investments
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Required Rate of Return
The fair present value (PV) of a security is determined using the required rate of return (r) as the discount rate
CF1 = cash flow in period t (t = 1, …, n)
~ = indicates the projected cash flow is uncertain
n = number of periods in the investment horizon
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3
Expected Rate of Return
The current market price (P) of a security is determined using the expected rate of return, or E(r), as the discount rate
CF1 = cash flow in period t (t = 1, …, n)
~ = indicates the projected cash flow is uncertain
n = number of periods in the investment horizon
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Realized Rate of Return
The realized rate of return, ( r ), is the discount rate that just equates the actual purchase price, ( ), to the present value of the realized cash flows, (RCFt), where t (t = 1, …, n)
RCF1 = realized cash flow in period t (t = 1, …, n)
r = realized rate of return on a security
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Bond Valuation
The present value of a bond (Vb) can be written as:
Par = the par or face value of the bond, usually $1,000
INT = the annual interest (or coupon) payment
T = the number of years until the bond matures
r = the annual interest rate (often called yield to maturity (YTM))
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Bond Valuation
A premium bond has a coupon rate (INT) greater than the required rate of return (r) and the fair present value of the bond (Vb) is greater than the face or par value (Par)
Premium bond: If INT > r; then Vb > Par
Discount bond: If INT < r, then Vb < Par
Par bond: If INT = r, then Vb = Par
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Equity Valuation
The present value of a stock (Pt) assuming zero growth in dividends can be written as:
D = constant dividend paid at end of every year
Pt = the stock’s price at the end of year t
rs = the interest rate used to discount future cash flows
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Equity Valuation Continued
The present value of a stock (Pt), assuming constant growth in dividends, can be written as:
D0 = current dividend per share
Dt = dividend per share at time t = 1, 2, …, ∞
g = the constant dividend growth rate
rs = required return on the stock
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Equity Valuation Concluded
The return on a stock with zero dividend growth, if purchased at current price P0, can be written as:
The return on a stock with constant dividend growth, if purchased at price P0, can be written as:
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Relation between Interest Rates and Bond Values
Interest
Rate
Bond Value
12%
10%
8%
874.50
1,000
1,152.47
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11
Impact of a Bond’s Maturity on its Price Sensitivity
Absolute Value of Percent Change in a
Bond’s Price for a Given Change in Interest Rates
Time to Maturity
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Impact of a Bond’s Coupon Rate on its Price Sensitivity
Bond
Value
Interest Rate
Low-Coupon Bond
High-Coupon Bond
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13
Impact of r on Price Volatility
Bond Price
Interest Rate
How does volatility change with interest rates?
Price volatility is inversely related to the level of the initial interest rate
r
14
You may wish to point out that actual price changes are curvilinear, duration based predicted price changes are linear with r.
Volatility varies along line: prices are a nonlinear function of interest rates, Blue line is actual price change, green line is predicted price change.
Concept: At higher interest rates, volatility is lower: Reason is that discounting far out cash flows more heavily to begin with at higher interest rates, this increases the near term percentage PV weights in relation to the long term weights.
The four variables that affect volatility are coupon and maturity (which are captured by duration), change in ytm or change in r and the starting ytm or the starting r. (r = ytm)
Duration
Duration is the weighted-average time to maturity (measured in years) on a financial security
Duration measures the sensitivity (or elasticity) of a fixed-income security’s price to small interest rate changes
Duration captures the coupon and maturity effects on volatility.
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So the only other two variables needed to predict volatility are r and the change in r.
Duration Continued
Duration (D) for a fixed-income security that pays interest annually can be written as:
D = Duration measured in years
t = 1 to T, the period in which a cash flow is received
N = the number of years to maturity
CFt = cash flow received at end of period t
r = yield to maturity or current required rate of return
PVt = present value of cash flow received at end of period t
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Note that if the security makes semiannual or monthly payments then the cash flow, the interest rate and the number of periods must be adjusted to reflect the payment frequency.
Duration Example – Annual Interest
Bond with 10% coupon, face value of $1,000, 4 year maturity, current yield to maturity (ytm) of 8%, and current price of $1,066.24.
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Duration – Various Number of Interest Payments per Year
Duration (D) for a fixed-income security, in general, can be written as:
m = the number of times per year interest is paid
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Closed Form Duration Equation
P0 = Price
INT= Periodic cash flow in dollars, normally the semiannual coupon on a bond or the periodic monthly payment on a loan.
r = periodic interest rate = APR / m, where m = # compounding periods per year
N = Number of compounding or payment periods (or the number of years * m)
Dur = Duration = # Compounding or payment periods; Durationperiod is what you actually get from the formula
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This version is in closed form, no summation needed. It is convenient for longer term securities.
If you divide by Price*m you get the duration in years. m= # of compounding or payment periods per year.
I believe the citation is:
Caks, J., Lane, W. R., Greenleaf, R. W., & Joules, R. G. (1985). A SIMPLE FORMULA FOR DURATION. Journal Of Financial Research, 8(3), 245.
Features of Duration
Duration and coupon interest
The higher the coupon or promised interest payment on the bond, the shorter its duration.
Due to the fact that the larger the coupon or promised interest payment, the more quickly investors receive cash flows on a bond and the higher are the present value weights of those cash flows in the duration calculation
Duration and rate of return
Duration decreases as the rate of return on the bond increases
Duration and maturity
Duration increases with maturity, but at a decreasing rate
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Duration and Modified Duration
Given an interest rate change, the estimated percentage change in a(n) (annual coupon paying) bond’s price is given by
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21
This really doesn’t demonstrate the usage of modified duration (MD). MD is used for non-annual payment securities.
Duration and Modified Duration Continued
Modified duration (DurMod) is a more direct measure of bond price elasticity
It is found as:
where rperiod = APR/m
Using modified duration to predict price changes:
22
Note that the annual change in interest rates is plugged into the prediction model. An example calculation is provided at the end of this file.
Duration Based Prediction Errors
23
Duration is an accurate predictor of price changes only for very small interest rate changes. For day to day fluctuations duration works quite well but when interest rates move significantly, such as when the Fed makes an announcement of a rate change, the predicted pricing errors can become significant. The prediction errors arise because bond prices are not linear with respect to interest rates.
Convexity
Convexity (CX) is the degree of curvature of the price-interest rate curve around some interest rate level
Convexity is desirable
The greater the convexity of a security or portfolio, the more insurance or interest rate protection an investor or FI manager has against rate increases and the greater the potential gains after interest rate falls
Convexity diminishes the error in duration as an investment criterion
All fixed-income securities are convex
As interest rates change, bond prices change at a nonconstant rate
© 2019 McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
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Practice Problem
Using Modified Duration
Predicted Price Change Using Modified Duration
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25
Dur Annual = 4.3774 years.
n
n
r
F
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semi
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semi
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1
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.
1
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4
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($958.42
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Predicted
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Dur
period
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.
1
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/
7548
.
8
(
=
=
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4
=
annual
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P
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.
0
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.
4
=
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%
1147
.
2
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