Topic 2: Interest Rates and Asset Values


You have just learned how the interest rate is a payment for postponing the use of funds. Since funds are used to buy goods, it is also a payment for postponing consumption---for shifting consumption from now to the future. You have also learned that a dollar (and goods) to be received tomorrow is worth less now than a dollar (goods) to be received now. This is the principle of discounting. Finally you learned that an asset is the right to receive funds (goods) in the future. The value of an asset is the present value of the funds (goods) that ownership of the asset entitles one to receive. We now expand on this latter point.

Consider an asset that entitles one to receive a sequence of payments R1, R2, R3, R4, ..... Rn at the ends of years 1, 2, 3, ... n. The value of this asset is simply the present value of the sequence of payments. The present values of the individual payments in the sequence are

R1 ⁄ (1 + r),   R2 ⁄ (1 + r)2,   R3 ⁄ (1 + r)3,   R4 ⁄ (1 + r)4,   .....   Rn ⁄ (1 + r)n,

and the value of the asset is the sum of these present values

1.       PV = R1 ⁄ (1 + r) + R2 ⁄ (1 + r)2 + R3 ⁄ (1 + r)3 + R4 ⁄ (1 + r)4 + ..... + Rn ⁄ (1 + r)n

Suppose now that the payments received in all years are identical so that

                     R1 = R2 = R3 = R4 = ..... = Rn = R.

The present value then becomes

2.       PV = R ⁄ (1 + r) + R ⁄ (1 + r)2 + R ⁄ (1 + r)3 + R ⁄ (1 + r)4 + ..... + R ⁄ (1 + r)n

When we factor out all the R's in this equation it reduces to

3.       PV = Ψ R

where

4.       Ψ = 1 ⁄ (1 + r) + 1 ⁄ (1 + r)2 + 1 ⁄ (1 + r)3 + 1 ⁄ (1 + r)4 + ..... + 1 ⁄ (1 + r)n

Letting   a = 1 ⁄ (1 + r) ,   the above expression becomes simply   Ψ = a + a2 + a3 + a4 + .... + an.

A fundamental mathematical result derived in every textbook dealing with infinite series (see the general references) is that as n increases the infinite sum (1 + Ψ) approaches

                     1 + Ψ = 1 + a + a2 + a3 + a4 + .... + a = 1 ⁄ (1 - a).

It follows that as the period over which the fixed payments R are received gets infinitely long sum Ψ becomes equal to

5.       Ψ = [(1 ⁄ (1 - a)] - 1 = 1 ⁄ r.

The value of a constant infinite stream of payments of $R per year in perpetuity is thus

6.       PV = R ⁄ r.

This can be seen by substituting Equation 5 above into Equation 3.


The relationship given by Equation 6 makes perfect sense. Consider an asset that represents the right to receive payments of $10 per year in perpetuity. If the interest rate is 10 percent, one could afford to pay $10/0.10 = $100 for this asset because $10 is 10 percent interest on an investment of $100. This explains why the prices of government (and corporate) bonds tend to vary inversely with the rate of interest. Consider, for example a consol, which is a bond paying fixed nominal interest every year for ever with no repayment of the principal. The present value of such a bond is determined by Equation 6 where R is the fixed annual interest or coupon yield and r is the rate of interest on other assets of similar risk. When the interest rate rises, the price of the bond, which is the present value of the coupon yield, goes down. Plug any number for R you choose into Equation 6 and you will see that a rise in the interest rate from 5 percent to 10 percent, for example, will reduce the value of the bond by half.

Another type of bond is one that pays coupon interest per year of some amount R for a fixed number of years, after which the principal is also paid back. The value of a bond of this type is given by

7.       PV = R ⁄ (1 + r) + R ⁄ (1 + r)2 + R ⁄ (1 + r)3 + R ⁄ (1 + r)4 + ..... + (R + A) ⁄ (1 + r)n

which is equation (2) above modified to include repayment of the principal A in the final year. It is easy to see that the market value of this bond will also fall when the rate of interest rises because the denominator of every term to the right of the equal sign increases.

Finally, there are bonds like treasury bills (issued by the Treasury Branch of the Government) that pay no coupon interest at all---the purchaser of treasury bills buys only the right to receive a fixed amount at some future date, usually 1 or 3 months hence. If we let this fixed amount be T, then the market value of the bill will be simply

8.       PV = T ⁄ (1 + r).

where r must now be interpreted as a 1 month or 3 month interest rate, depending upon the circumstances. The buyer pays sufficiently less than a dollar for each dollar to be received at maturity to yield an interest rate equal to the rate prevailing for other securities of similar risk. As the market interest rate rises, the amount investors will pay to purchase the fixed future amount---that is, the value of the bill---will fall.

When the interest rate rises the penalty for consuming now rather than in the future increases, so the fixed future receipts of principal and coupon interest are worth less today. If one wants to consume the future earnings from the security now, one will have to pay more for that privilege.

You should now understand a number of things about the rate of interest. First, you should understand that it is a payment for postponing consumption. It is not the price of money, but rather, the price of postponing the use of money. Second, you should understand that the interest rate will be positive in the economy because capital, broadly defined to include knowledge, technology and human capital as well as physical capital is productive---that is, produces income. Third, you should understand that it makes no sense to add together or compare dollar amounts to be paid or received at different points of time without discounting them. All sums must be put in a common time dimension by converting them to their present or discounted values. Finally, you should understand why interest rates and bond prices (and asset prices in general) move in opposite directions.

You should now be ready for another test. Be sure to figure out your own answers before clicking on the ones provided.

Question 1
Question 2
Question 3

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