© 2000 John Petroff
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© 2000 John Petroff |
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There is a small number of well accepted relationships between rates of return and between rates of return and values, which can be looked upon as theorems. These relationships are indeed used throughout finance writings as postulates.
1- Total yield equals market yield of comparable assets
This relationship is a direct consequence of dominance principles mentioned in Chapter 1 Section B. Indeed, if two assets are truly similar, investors will bid the price up for the asset with the higher return, which will lower its return. Observing that bond yields and stock rates of return for companies in the same industry are very similar confirms that the relationship is universal.
2- Inverse relationship between rates of return and all financial asset prices
All rates of return R are some variation of periodic income (price change plus cash received) I divided by price P
R = I / P
The equation clearly establishes the inverse relationship. It is naturally also present in the formula for price (or value) V which is some variation of income flows I discount with (that is divided by) a market yield 1+R
V = Sum( I / (1+R)t)
This equation confirms the inverse relationship between price and rate of return.
Empirical evidence of the relationship working in financial markets is most vivid at each Federal Reserve Board rates announcements which cause instantaneous jump in bond and stock prices. Sometime, the rate change is anticipated, and prices change ahead of the actual announcement. This relationship always holds and is universal.
3- Direct relationship between price change and maturity
The previous theorem stated that all financial asset prices are changed when market rates change. The price change is greater the longer the maturity. This is a direct consequence of the formula for discounting of an annuity presented in Section C-2 of this chapter as
BP = C ((1 - (1 + i)-n) / i) + P (1 + i)-n
As maturity increases, (1+i)-n decreases, the present value of the principal P becomes negligible and the value of the bond is the present value of the coupon interest payments which tends to C/i. Because the mathematical proof is a little incricate with the discounted value of the principal and that of the coupon working in opposite direction, a simplified demonstration is used here. Two formulas for extreme values are derived and a numerical example shows the relationship with clarity.
First let us take a very short maturity. If the financial asset has just one year to maturity, when the rate changes by a factor of d, the price change will be
(P1-P0)/P0 = P1/P0 -1 = ((P/(1+di)) / (P/(1+i))) - 1
= (1+i)/(1+di) - 1
Now, let us take a long maturity. If the financial asset has an infinite life, the same rate change d will cause the price change to be
(P1-P0)/P0 = P1/P0 -1 = ((P/di) / (P/i)) - 1
= 1/d -1
For the values chosen in this example, one may note that the difference in price change is far from trivial: the price change of a bond with over 20 years to maturity is ten times larger than the price change for a bond with less than two years to maturity as a result of just 0.01 drop in interest rate. The relationship holds for all values of coupon rate, yield to maturity and yield change.
4- Direct relationship between interest rate risk and maturities
Interest rate risk is measured by the change in portfolio value as a result of a change in interest rates. This relationship is a direct consequence of the previous theorem: the longer the maturities in the portfolio the larger will be their price change resulting from a rates change, and consequently interest rate risk.
5- Size of bond premium (or discount) is function of excess (or deficiency) of coupon rate over yield to maturity
A bond premium (or discount) Pr is the difference between bond value BV and par value P
Pr = BV - P
Replacing bond value by the formula previously derived in Section C-2 of this chapter and substituting the coupon C by the product of par value P by coupon rate CR
Pr = P.CR ((1 - (1 + i)-n) / i) + P (1 + i)-n - P
The relative size of the premium/discount over par value Pr/P is
Pr/P = CR ((1 - (1 + i)-n) / i) + (1 + i)-n - 1
or after simplication
Pr/P = (CR - i) (1 - (1 + i)-n) / i
Thus, Pr/P is indeed a function of the difference between coupon rate and market yield, CR-i.
Corollary: a bonds sells at a premium if the coupon rate is higher than market yields. A bonds sells at a discount if the coupon rate is lower than market yields.
This relationship always holds.
6- Size of bond premium or discount is function of maturity
In the relative premium (or discount) formula above
Pr/P = (CR - i) (1 - (1 + i)-n) / i
the greater is n, the smaller is (1 + i)-n, the greater is (1 - (1 + i)-n) / i and also consequently, the greater is the entire relative premium (or discount). For a perpetual bond, and n equal infinity, the relative premium is
Pr/P = (CR - i)/i = CR/i -1
This relationship always holds.
7- Price approaches par value as time approaches nearer maturity
This relationship is essentially identical to the previous one. It is, nevertheless, useful to state it because it clearly shows the evolution of bond value over time.
This relationship is universal.
8- If a bond sells at a discount
CR<CY<YTM
It was established in theorem 5 above that based on the relationship of relative premium (or discount)
Pr/P = (CR - i) (1 - (1 + i)-n) / i
a bonds sells at a discount if CR-i is negative (i.e. the coupon rate is smaller than yield to maturity).
Since current yield CY is
CY = C / BV
and the discount implies that the bond price BV is smaller than par value P, then
C/P < C/BV or CR<CY
Furthermore, current yield accounts only for coupon income, but yield to maturity incorporates in addition price appreciation (and we know that price will necessarily rise as time approaches maturity according to theorem 7). Therefore, current yield is smaller than yield to maturity for a bond selling at a discount. Combining the two inequality:
CR<CY<YTM
This relationship always holds. A parallel can be established between stocks and bonds: if a stock is underpriced its dividend yield is less than its risk adjusted required rate of return.
9- If a bond sells at a premium CR>CY>YTM
The demonstration is exactly the reverse of the previous one.
Non-theorem 10- Returns on risky financial assets vary more than returns on safer assets
Since modern portfolio theory (established in Section E of this chapter) states that the greater the volatility (i.e. BETA) the higher the rate of return, this implies that higher risk and return assets have greater dispersion than lower risk and return assets. This relationship can be relied to make predictions about future performance most of the time, but not always and not for each and every financial asset.
Empirical evidence abounds in support of the relationship in patterns of small stocks, neglected stocks and less than investment grade bonds over the business cycle. Yet, the relationship should not be considered as a true theorem because it does not always hold. This is the case where a necessary condition is met (more volatile assets have higher return), but the condition is not sufficient (higher return assets do not always have greater volatility in the future).
We conclude this section, by observing that, except for the last relationship, all the other theorems that have been developed on the basis of bond formula are also applicable to all other financial assets.
See review questions Q-2G.1 through Q-2G.4.
See research assignment R-2G.1.
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