FIN 331 – Business Finance CHAPTER 6 - Interest Rates and Bond Valuation How the time value equations can be used for the valuation of Bonds Zero coupon bonds / Pure Discount Bonds Example of Pure Discount Loan: Treasury Bill. Describe them and draw time line. Q: What equation do we use here? Present value of a future cashflow PV = FV/ (1 + r) t
Example: Consider a 10-year zero coupon bond, it is priced to yield 9%. What is the current price?
Note that I said “priced to yield” What I am implying is that the market forces have determined that 9% is required on a particular security with its particular risk and payment structure. That required return is established in the market place. YTM is the IRR on a bond. We’ll discuss IRR in detail a little later.
Example Say we have a 20-year zero-coupon bond that is selling for $275. What is its yield-to-maturity?
Pricing and YTM for Coupon-Paying Corporate Bonds Coupon paying bonds pay an interest payment every 6 months and return the principle (usually $1,000) at the end of the loan.
Example Assume we have a 20-year bond paying a 10.25% annual coupon rate (=> .1025*1000/2 =) or $51.25 every six months. If it is priced to yield 11%, what is its price?
all at 51.25 | | | | |
1,000 ||
|-------- |-------- |-------- |-------- |---.....................----- |-------- | | | 1 2 3 4 39 40 months
PV r = 11%/2
For equations:
use (1) pv of annuity and (2) pv of future cash flow.
PV = Coupon (PVIFAr,t) + 1000 (PVIFr,t) :General solution for Bond Price
For calculator:
What if the required return goes up? Say to 12%, then PV = Bond is at a discount!! What if required return goes down? Say to 10%, then PV = Bond is at a premium!!
Solving for yield: However, when we try to calculate the YTM for a coupon paying corporate bond, we do not have a closed form solution. We must solve for YTM using trial-anderror or have our financial calculator solve for us.
Let’s try another price and see what the yield is! If the PV = 1035, YTM =
Interest Rate Risk: Discussion in Class!
Interest Rate Levels
What is a rate of return (or interest rate)? Simple: I offer to give you $115 in one year if you give me $100 now.
Return on your $100 investment:
i.e. Given a $100 investment and a 15% rate of return we have:
What elements contribute to your required return? INFLATION …IS A COSTLY THING EXAMPLE: Your usual basket of groceries cost $150 last year, this year the same basket costs $160.
If that basket is used to measure inflation, then inflation for the year was:
We see that inflation reduces the purchasing power of money, i.e. the dollar. Note in the example that one dollar last year bought 1/150 of your basket, while now one dollar only buys 1/160 of your basket. NOMINAL VERSUS REAL RATES Nominal rates of return – an observed rate of return that is not adjusted for inflation, like the 15% mentioned in the preceding discussion. Although the investment yielded a 15% return, if inflation was 15%, we cannot buy any more goods than before.
Real rates of return - are inflation adjusted. In the example above, our nominal rate was 15% but, since we can only buy the same amount of goods, we received a 0% real return from that investment.
The relationship between nominal and real interest rates is given in the Fischer Equation: Nominal Interest Rate = real rate + expected inflation
Which is often used to show the real rate of return as:
The exact formulation that is much more useful in more complex analysis is:
The real rate has been about 3% for much of the 1900s. Nominal rates have fluctuated from 4% to 18% over the time, though. As a result, much of the variation in nominal interest rates is due to changes in expected inflation.
In the example where you lend me $100, your nominal rate of return was 15% but, your real return on that investment was:
rloan to SLB = rloan to SLB = Term Structure of Interest Rates Rf = (Default) risk-free rate of return = Real rate of return + Inflation Premium + Interest Rate “Risk” Premium Term structure of interest rates considers Rf for pure discount bonds. Rf is compared for different maturities. Inflation Premium – based on expected inflation, investors must be compensated for inflation Interest Rate Risk – the risk that higher interest rates will reduce the value of the security (alternatives)
Upward sloping term structure:
Downward sloping term structure:
Yield curve looks at Rf for securities of similar default risk, usually Treasury Yield Curve. Upward sloping yield curve:
Downward sloping yield curve:
Define R1 as the required return (YTM) on security 1 R1 = Specifically a risk premium is the return required on a risky investment in excess of the return required on a riskless investment. Where DRP, TaxP, and LP are a function of the individual security Specifically a risk premium is the return required on a risky investment in excess of the return required on a riskless investment. DRP – additional return demanded for default risk TaxP – additional return demanded for tax cost LP – additional return demanded for lack of liquidity (risk that you cannot sell quickly without loss in value)
For any particular investment, required rate of return can be defined as: R1 = return on U.S. Treasury Security + risk premium1 Why? 1. The treasury securities are considered default risk free 2. Investors require a higher return for default-, tax-, liquidity- risk
Assume you had accurately used the 6.667% as your estimate of inflation (your inflation premium): Rloan to SLB = r + IP + RPloan to SLB .15 = .03 + .066667 + RPloan to SLB .15 - .03 - .066667 = RPloan to SLB .0533 = RPloan to SLB Inflation impacts other investors in the same way. Thus, investors must be compensated for expected inflation in their investments.