【耶魯大學】《金融市場》（全集）頂尖投資大師課！ P45 - 耶魯大學《金融市場_Financial Markets》中英字幕 - 45.44_永續年金和普通年金.zh_en(Av1450436343,P45)

我之前提到過一臺電腦的價格是C,或者是X, 而電腦的現金價值是Coupon over R, 這很明顯,為什麼我們稱之為電腦呢? 因為在1700年,英國政府在沒有成長日期的情況下, 英國政府仍然在1880年代做了一些調整, 他們仍然在付Coupon, 英國政府並沒有在那段時間中失敗, 雖然不是永久,但是接近永久, 我指的是幾百年, 是一段很長的時間在付Coupon。 現在,很簡單的理解, 電腦的現金價值是C over R, 我們可以回過頭來看, 電腦的成長率是C over P, 這很明顯,對吧? 如果電腦在付Coupon, 它在賣200元, 我指的是200英鎊, 那成長率是多少呢? 它是3英鎊分別200英鎊, 那如果成長率是3%的話, 對不起,如果成長率是3%的Coupon, 現在它是1.5%的成長率, 因為價格升了。 這就是Bonds的重要點, Bonds的Coupon是在付Coupon的時間中修復的, 但是Bonds的市場價格是隨著時間改變的。 所以,如果英國政府付出了 100英鎊的3英鎊的Console, 而那個Console現在在賣200英鎊, 成長率會降到1.5% 而不是3%, 但是這不是英國政府的錯, 這是他們的口號, 他們正在付Console, 這是市場做的, 所以Bonds是有風險的, 它們有市場風險, 即使沒有標準風險。 如果你買Console, 你會知道你不會永遠生活, 英國政府可能會永遠生活, 但是你不會, 所以你會想要賣Coupon, 英國政府不會保證你會賣Console的價格。 所以有市場風險的Console, 對於任何貨幣器材來說, 長期貨幣器材來說。 順帶一提, 如果銀行或公司或政府 發行Consoles, 通常公司不會發行Consoles, 因為沒有人相信它們會永遠存在。 英國的國民黨可能會相信 英國政府會永遠存在, 雖然我可以告訴你, 它們不會, 它們不會永遠存在, 但是它們想像它們會永遠存在, 所以它們願意購買英國Consoles。 但是如果英國政府做了更加嚴重的事情? 它們會說, 我們將在Consoles上的Coupon 持續增長。 所以它會開始在3英鎊, 然後它會持續增長到G每年, G是每年增長的百分比, 永遠。 那現在的房價是多少? 如果房價是R, 房價會持續增長至G。 結果房價的持續增長代表 R 和 G的現值。 So by the way, what happens if g is greater than r, or g equals r? Any idea on that? Well, I think you probably can figure this out. If g equals r, it's x divided by 0, it's infinite. That's because the series that you're summing is not convergent. If you're getting an amount that's growing at the rate of interest, the present value is infinite. And if it's growing at greater than the rate of interest, well, the formula breaks down. It's still infinite. So that's why we get puzzled about the present situation. If the interest rate is 0, then consoles don't even converge. How can that be? Doesn't the interest rate always have to be above the growth rate? How can anything... And nothing can have an infinite price. But I guess the answer is, short-term interest rates are 0, but longer-term interest rates are still positive. So, and we don't have many... The U.S. doesn't have consoles. But we have something analogous to consoles, like land, for example. That lasts forever, as far as we know. It pays, you know, you can rent it out, or you can plant crops on it, as far as we know, forever. Maybe those crop... What about land? Aren't crop values rising through time? So that G is a positive number. And if interest rates are... They can't be 0. They have to be... Long rates have to be above 0. Otherwise, assets that have growing payments would be worth an infinite amount. Now, this is an annuity present discounted value. Now, an annuity is like a console, except that it stops after a certain number of years. Now, in a typical annuity... A typical annuity is a home mortgage. When you buy a house, the typical financing you'll get is that you will pay... Now, the compounding interval is monthly. They think it's not realistic to ask ordinary people to pay their mortgage every 6 months. Because it's too hard for them. They'd have to... They wouldn't remember to save, and they wouldn't have enough money to pay it after 6 months. So, we got to move to a monthly schedule for individuals. So, a typical... This is not compounding monthly. This is the present value of X dollars every year, starting in 1 year, and then again in 2 years, then again in 3 years, and the last payment is T years. This is different from the present value for a corporate bond, or for a coupon carrying bond, because there's no principal repayment at the end. I'm talking about... I think I have another slide on mortgages. I'll come back to that. But I tell you, realistically, these financial instruments are designed around human imperfections. People find it difficult to pay back a mortgage. And what they'll find especially difficult is to pay a balloon payment at the end. They call it a balloon payment. There used to be mortgages like this. You would borrow to buy your house. I'm going back to the 1920s. The house cost $10,000. Typical house in the 1920s. You borrow $9,000. You've got your own money to put up. The $9,000, then you pay back the money at a rate interest. But at the end, you don't owe anything, except the last monthly payment. So, that's what an annuity is. This is a little history of thought. Actually, the growing console formula has been called the Gordon Rule, according to Myron Gordon, who was a professor of economics about a half century ago. But actually, it goes back to Jacob Bernoulli. I learned that from Will Getzman and his co-author Gerrit Roanhorse. So, it's an old formula.